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-% ============================================================
-% ANNULAR VARIATION OF THE TRIANGULAR HILBERT TRANSFORM
-% AT THE SYMMETRIC POINT
-%
-% An expository rewrite. Written so that a determined
-% high-school student can read it from page one, and so that a
-% graduate student can find every technical step.
-%
-% HOW TO READ THIS FILE
-%
-% Every line that starts with % is a remark for you, the reader.
-% The mathematics comes afterward. The order is always:
-% story -> statement -> proof.
-% Whenever a formula appears, the next sentence says what it
-% means in English. Whenever a symbol is new, its first use is
-% defined right there. Skim the technical sections the first
-% time through; come back to them when a proof points you there.
-%
-% A formula is bookkeeping. The sentence next to it is the idea.
-% ============================================================
-
-\documentclass[11pt]{article}
-\usepackage[margin=1in]{geometry}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage{microtype}
-\usepackage[colorlinks=true,linkcolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
-\newcommand{\R}{\mathbb R}
-\newcommand{\C}{\mathbb C}
-\newcommand{\dd}{\,\mathrm d}
-\DeclareMathOperator{\tr}{tr}
-\DeclareMathOperator{\Dil}{Dil}
-\newcommand{\HS}{\mathrm{HS}}
-\newcommand{\op}{\mathrm{op}}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-\numberwithin{equation}{section}
-
-\title{Annular variation of the triangular Hilbert transform\\at the symmetric point\\[4pt]
- \large a rewrite for students}
-\author{after the research paper}
-\date{}
-
-\begin{document}
-\maketitle
-\tableofcontents
-
-% ============================================================
-\section{The big picture}\label{sec:picture}
-% ============================================================
-
-% This section assumes only that you know what an integral and a
-% limit are. Everything else is built here. Read it twice.
-%
-% The whole paper is about ONE operator. It takes two functions
-% F and G of two variables (x,y) and returns a third function of
-% (x,y). The recipe: step from (x,y) in the horizontal direction
-% through F, step from (x,y) in the vertical direction through G,
-% multiply, and weight the step length t by 1/t. Then integrate
-% over t. The question: is this operator stable under EVERY way
-% of truncating the integration in t, including a way that changes
-% from point (x,y) to point (x,y)? The answer is yes, at a cost
-% that grows like the square root of the number of truncations.
-
-\subsection{The Hilbert transform in one dimension}\label{subsec:hilbert-1d}
-
-% Start in one variable. For a smooth function f that vanishes
-% far away, the Hilbert transform of f is
-\begin{equation}\label{eq:HT-1d}
- Hf(x)=\frac1\pi\,\operatorname{p.v.}\!\int_{-\infty}^{\infty}
- \frac{f(x+t)}{t}\,\dd t .
-\end{equation}
-% Read it out loud. At the point x, look at f at the point x+t,
-% stepping t to the right. Weight the value by 1/t. Positive t
-% contributes f(x+t)/t; negative t contributes the SAME sign in
-% front, and 1/t already has the opposite sign for negative t.
-% So the transform measures how lopsided f is around x: it is a
-% machine for detecting antisymmetry. A constant has zero Hilbert
-% transform. A symmetric bump has zero Hilbert transform.
-
-% Two warnings about (1.1).
-%
-% (i) The integral is singular. Near t=0, f(x+t)/t looks like
-% f(x)/t, which has infinite integral. The letters "p.v."
-% stand for principal value: integrate the truncated version
-% over eps<|t|<R and only THEN let eps go to zero and R go
-% to infinity. For smooth compactly supported f the limit
-% exists at every x.
-%
-% (ii) The transform is not bounded on L^1. It IS bounded on
-% every L^p with 1<p<infinity: a constant C_p satisfies
-% ||Hf||_p <= C_p ||f||_p.
-% "L^p" means the function has finite p-th power integral,
-% ||f||_p = (integral of |f|^p)^(1/p),
-% which you may think of as a measure of size with a knob p.
-% Large p emphasises peaks. Small p emphasises mass.
-
-% The TRUNCATIONS are the objects we are going to "vary":
-\begin{equation}\label{eq:HT-trunc}
- H_{\varepsilon,R}f(x)
- =\frac1\pi\int_{\varepsilon<|t|<R}\frac{f(x+t)}{t}\,\dd t ,
- \qquad 0<\varepsilon<R<\infty .
-\end{equation}
-% Even though Hf exists as a principal value, the truncated
-% operators H_{eps,R} are far from innocent. As eps decreases to
-% zero and R grows to infinity, the functions H_{eps,R}f do
-% converge pointwise almost everywhere -- but proving this
-% UNIFORMLY in x is a genuine theorem (a one-sided version of
-% the classical Carleson theorem for the Hilbert transform).
-% "Almost everywhere" means: for every x outside a set of
-% measure zero.
-%
-% The present paper proves the analogue of that statement in TWO
-% dimensions, for a bilinear operator, in the strongest possible
-% form: not just pointwise convergence, but control of the total
-% amount of oscillation across all truncations at once. That is
-% the meaning of "variation".
-
-
-\subsection{The bilinear Hilbert transform}\label{subsec:BHT}
-
-% Before the two-dimensional version, there is a one-dimensional
-% bilayer. Take two functions f and g of ONE variable. Send one
-% input to the right and the other to the LEFT by the same amount
-% t. Then weight by 1/t and integrate.
-\begin{equation}\label{eq:BHT}
- \mathrm{BH}(f,g)(x)
- =\operatorname{p.v.}\!\int_{-\infty}^{\infty}
- f(x+t)\,g(x-t)\,\frac{\dd t}{t}.
-\end{equation}
-% This is the bilinear Hilbert transform. "Bilinear" means linear
-% in f alone and linear in g alone, but not in f+g together. It
-% was open for decades whether it is bounded on any nontrivial
-% range of L^p spaces; Lacey and Thiele settled this in 1997.
-% The relevant point for us: the two inputs move in OPPOSITE
-% directions along the SAME axis. That opposition is what makes
-% the analysis tractable.
-
-\subsection{The triangular Hilbert transform}
-\label{subsec:triangular}
-
-% Now come to the operator of the paper. It is
-% two-dimensional: F and G are functions of (x,y). Instead of
-% moving both inputs along the same x-axis, we move them along
-% ORTHOGONAL directions.
-\begin{equation}\label{eq:triangular-def}
- B_{\varepsilon,R}(F,G)(x,y)
- =\int_{\varepsilon<|t|<R}
- F(x+t,y)\,G(x,y+t)\,\frac{\dd t}{t},
- \qquad 0<\varepsilon<R<\infty .
-\end{equation}
-% Read it out loud. Fix an output point (x,y). Step from (x,y)
-% to (x+t,y): that is a HORIZONTAL move, and we read F there.
-% Step from (x,y) to (x,y+t): that is a VERTICAL move, and we
-% read G there. Multiply the two values, weight by 1/t, sum
-% over t.
-%
-% Three points participate:
-% (x,y) the corner,
-% (x+t,y) the horizontal vertex,
-% (x,y+t) the vertical vertex.
-% They form a right triangle. The two inputs sit on the two legs
-% that leave the corner. This is why the operator is called
-% TRIANGULAR.
-%
-% Compare (1.3) to the bilinear Hilbert transform (1.2). In (1.2)
-% both inputs lie on the SAME line. In (1.3) they lie on
-% different lines. The one-dimensional opposition in (1.2) was a
-% gift; the two-dimensional orthogonality in (1.3) does not give
-% the same gift, and the operator is harder. This paper is the
-% first proof that it behaves well at the symmetric endpoint
-% (defined in the next subsection).
-
-% There is a subtlety in the word "coherent". F(x+t,y) and
-% G(x,y+t) can be highly aligned with the kernel 1/t for large
-% stretches of t; only cancellations across changes of scale keep
-% the operator bounded. In the classical Hilbert transform, those
-% cancellations are packaged by orthogonality of frequencies.
-% Here the two different directions break that orthogonality and
-% a new mechanism is required. That mechanism is what the paper
-% discovers.
-
-
-\subsection{The scalar simplex form and the symmetric point}
-\label{subsec:simplex}
-
-% Dualise the bilinear operator into a TRILINEAR form: multiply
-% the output of B by a third function G_0 and integrate.
-% Holder's inequality, the standard pairing tool, says
-% ||B(F,G)||_p <= C ||F||_3 ||G||_3,
-% || G_0 B(F,G)||_1 <= ||G_0||_q ||B(F,G)||_p
-% provided 1/q + 1/p = 1. If p=3/2 then q=3. So for a L^3
-% basis, the natural companion G_0 is also in L^3. The scalar
-% form we want is exactly
-\begin{equation}\label{eq:scalar-annular}
- \Lambda_{\varepsilon,R}(G_0,G_1,G_2)
- =\iiint G_0(a,b)\,G_1(b,c)\,G_2(c,a)\,
- \mathbf 1_{\varepsilon<a+b+c<R}\,
- \frac{\dd a\,\dd b\,\dd c}{a+b+c}.
-\end{equation}
-% Where does (1.4) come from, and why does the kernel become
-% 1/(a+b+c) instead of 1/t? Use the determinant-one substitution
-% a=-x, b=-y, t=a+b+c,
-% whose inverse is x=-a, y=-b, t=a+b+c.
-% The three variables (a,b,c) satisfy c=t+x+y. Then
-% F(x+t,y) becomes G_1(b,c)=F(b+c,-b),
-% G(x,y+t) becomes G_2(c,a)=G(-a,c+a),
-% G_0(x,y) becomes G_0(a,b).
-% Each of these maps (F,G) -> (G_1,G_2) and back has matrix
-% determinant one, and preserves the L^3 norm. Computing from
-% (1.3) yields (1.4): the compactly supported part of
-% the kernel is 1/t=1/(a+b+c).
-%
-% The three inputs in (1.4) each see a DIFFERENT PAIR of the
-% three scalar variables (a,b,c):
-% G_0 sees (a,b),
-% G_1 sees (b,c),
-% G_2 sees (c,a).
-% The pairs form a CYCLE b->c->a->b. This is the "simplex":
-% the graph with vertices a,b,c and edges the three pairs. No
-% one input sees all three variables. That is what makes the
-% operator "entangled" from the point of view of real-interpolation.
-
-% Now for the exponents. Holder in three variables says:
-% the bound
-% |\Lambda| <= C ||G_0||_{p_0} ||G_1||_{p_1} ||G_2||_{p_2}
-% applies exactly when
-% 1/p_0 + 1/p_1 + 1/p_2 = 1.
-% That is a whole SURFACE of choices, not a single point.
-%
-% The SYMMETRIC POINT of that surface is the choice
-% p_0 = p_1 = p_2 = 3.
-% It is the choice in which all three inputs are treated equally,
-% with exponent 3 for each. Symmetric-point variation and
-% symmetric-point principal values are the endpoint of the whole
-% theory: they are what we would want if we did not already
-% know a comfortable exponent tuple.
-
-% Which of these was open in 2025?
-% - The lower exponents, p_0,p_1,p_2 close to 2, are easier
-% because there is more room in the Holder relation.
-% - The symmetric point p=3,3,3 is the HARDEST nontrivial
-% choice on the boundary: each input is exactly at the
-% point where the coarea/decomposition tools start to lose.
-% - Even harder is the VARIATION version of the estimate:
-% not just a single annulus eps<|t|<R, but a weighted sum
-% of many disjoint annuli (equation (1.7) below).
-% This paper proves the hardest version.
-
-
-\subsection{Variation}\label{subsec:variation-intro}
-
-% The naive object is the single truncated operator B_{eps,R}.
-% But we care about how far the family B_{eps,R} can wander as
-% the truncation moves. For a real number r>2 (the "variation
-% exponent"), define
-\begin{equation}\label{eq:variation-def}
- V_r(F,G)(x,y)
- =\sup_{\substack{J\ge1\\0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
- \Big(\sum_{j=1}^{J}
- |B_{t_{j-1},t_j}(F,G)(x,y)|^r\Big)^{1/r}.
-\end{equation}
-% Read the outer object first. For EACH point (x,y), pick any
-% partition 0<t_0<t_1<...<t_J of the positive real line, form
-% the truncated operators on the slices (t_{j-1},t_j), take the
-% absolute value of the scalar at (x,y), raise it to the r-th
-% power, sum over the slices, take the r-th root, and THEN take
-% the supremum over all partitions. The answer is a function of
-% (x,y). The map F,G |-> V_r(F,G) is the r-th variation of the
-% family B_{eps,R}.
-%
-% Two features matter.
-%
-% (i) The partition is allowed to depend on (x,y). Different
-% points (x,y),(x',y') may use different partitions; the
-% supremum is taken pointwise FIRST and only afterwards
-% are the L^{3/2} norms taken.
-%
-% (ii) There is no limit on the number of intervals J. The
-% supremum is over every finite partition. We do not
-% prefer dyadic endpoints.
-%
-% These features are what make variation strictly stronger than
-% pointwise convergence and stronger than the maximal estimate.
-
-\subsection{The count estimate}\label{subsec:count-intro}
-
-% Theorem A below follows from a quantitative estimate for
-% finitely many annuli. For an integer n>=1, define
-\begin{equation}\label{eq:count-def}
- \mathcal S_n(F,G)(x,y)
- =\sup\Big\{\sum_{j=1}^{m}|B_{\varepsilon_j,R_j}(F,G)(x,y)|
- : 0\le m\le n,\ 0<\varepsilon_j<R_j\in\mathbb Q,\
- (\varepsilon_j,R_j)\text{ pairwise disjoint}\Big\}.
-\end{equation}
-% Same story. For each point (x,y): pick at most n annuli
-% eps_j<|t|<R_j that are pairwise disjoint (endpoints may touch;
-% "disjoint" is about the interiors), form the truncated operator
-% on each, take absolute values, sum. Take the supremum over all
-% choices. The result is \mathcal S_n(F,G), a function of (x,y).
-%
-% The main theorem is the size bound on \mathcal S_n. It is a
-% count estimate: it costs the number of annuli n but nothing
-% else. From it, variation follows by a short combinatorial
-% step (Section 1.7 below).
-
-% Here is why the count estimate is the right reduction.
-% From n annuli we lose sqrt(n). From the count estimate to
-% VARIATION at exponent r>2, we sort the increments by size
-% and bound the resulting series:
-%
-% V_r(F,G) <= sum_j 2^{j(1/r-1/2)} \mathcal S_{2^j}(F,G),
-%
-% and the series on the right converges whenever r>2. In
-% particular, the count estimate on the right directly implies
-% the full-variation estimate. The series converges because
-% (1/r - 1/2) < 0.
-
-
-\subsection{From count to variation}\label{subsec:count-to-variation}
-
-% Why does the count estimate imply the variation estimate?
-% Take any partition t_0<t_1<...<t_J at a point (x,y). Let
-% d_j = |B_{t_{j-1},t_j}(F,G)(x,y)|
-% be the increment sizes. Sort them in decreasing order:
-% d_{(1)} >= d_{(2)} >= ... >= d_{(J)}.
-% (The (j) is rank of the increment, not index of the interval;
-% the procedure is a pointwise rearrangement.)
-%
-% For every rank 2^k at which the list still has an entry,
-% the first 2^k increments together are a valid choice of 2^k
-% pairwise disjoint intervals in the count supremum. Hence
-% d_{(2^k)} <= 2^{-k} * \mathcal S_{2^k}(F,G).
-%
-% The increments of ranks 2^k <= l < 2^{k+1} form a block of
-% size 2^k, and each is at most d_{(2^k)}, so this block's
-% l^r sum is at most 2^k * d_{(2^k)}^r, i.e. its l^r norm is at
-% most 2^{k/r} d_{(2^k)}. Summing over k gives
-\begin{equation}\label{eq:count-to-variation}
- V_r(F,G)
- \le \sum_{k\ge0} 2^{k(1/r-1/2)}\, \mathcal S_{2^k}(F,G)
- \quad\text{pointwise},
-\end{equation}
-% The coefficient 2^{k(1/r-1/2)} is positive and decreases
-% geometrically when r>2, which is exactly the range we need
-% for the series to converge. Applying the L^{3/2} norm,
-% then Minkowski, and then the count estimate,
-% ||V_r(F,G)||_{3/2}
-% <= C ||F||_3 ||G||_3 \sum_k (1+k)2^{k(1/r-1/2)},
-% and the geometric series is finite because 1/r-1/2<0. That is
-% the whole content of Theorem B below, aside from measurability.
-
-% ============================================================
-\section{Statements of the main theorems}\label{sec:statements}
-% ============================================================
-
-% Here is the exact content of the paper. Everything else is
-% preparation or consequences. We use \R for the real numbers,
-% \C for the complex numbers, L^p(\R^2) as usual, and
-% ||f||_p for the L^p norm. All functions are complex-valued.
-
-\subsection{The count estimate}
-
-\begin{theorem}[Disjoint-annulus count estimate]\label{thm:count}
-There is an absolute constant $C$ such that, for every integer
-$n\ge1$ and every complex $F,G\in L^3(\R^2)$,
-\begin{equation}\label{eq:count-bound}
- \|\mathcal S_n(F,G)\|_{3/2}
- \le C\sqrt n\,\log(2+n)\,\|F\|_3\,\|G\|_3 .
-\end{equation}
-\end{theorem}
-
-% Read the right side. There are three factors.
-%
-% ||F||_3 ||G||_3 is the size of the inputs. If F=0 or
-% G=0 then the left side is zero.
-%
-% sqrt(n) is the price for the number of annuli. Doubling
-% the number of annuli costs at most sqrt(2) more.
-%
-% log(2+n) is a negligible extra cost; it appears because we
-% allow arbitrary disjoint annuli, not just dyadic ones.
-%
-% The constant C is absolute: it does not depend on n, on F, G,
-% on the annuli, on the point (x,y), or on anything else.
-%
-% The answer to "why sqrt(n)" is a rescaling trick. We
-% normalize the third input G_0 to unit size, and we can split
-% the whole energy of the operator across n annuli. Each annulus
-% contributes a bounded amount per switch, but we can choose the
-% switch thresholds adaptively. A duality rescaling by a factor
-% n^{1/2} absorbs the linear count of switches into the trace
-% of the energy (see the smooth-annuli section later).
-
-\subsection{The variation estimate}
-
-\begin{theorem}[Full annular variation]\label{thm:variation}
-For every real $r>2$ there is a finite constant $C_r$ such that
-\[
- \|V_r(F,G)\|_{3/2}
- \le C_r\,\|F\|_3\,\|G\|_3
-\]
-for all complex $F,G\in L^3(\R^2)$.
-\end{theorem}
-
-% Theorem~\ref{thm:variation} is the headline. It says the
-% family B_{eps,R} has bounded r-variation for every r>2,
-% uniformly. In particular (take r->infinity formally) it
-% implies a maximal estimate, and it implies joint convergence.
-% Both consequences are corollaries below.
-
-
-\subsection{The maximal and principal-value corollaries}
-\label{subsec:corollaries}
-
-% Take a single increment in the partition: two endpoints. Then
-% variation dominates the maximal operator over all endpoints:
-\begin{corollary}[Maximal operator]\label{cor:maximal}
-For complex $F,G\in L^3(\R^2)$, define
-\[
- B_*(F,G)(x,y)
- =\sup_{0<\varepsilon<R<\infty}|B_{\varepsilon,R}(F,G)(x,y)|
-\]
-on a common full-measure set (on which every truncated operator
-is defined), and set it to zero elsewhere. Then $B_*$ is
-measurable and
-\begin{equation}\label{eq:hard-maximal-bound}
- \|B_*(F,G)\|_{3/2}\le C_M\|F\|_3\|G\|_3
-\end{equation}
-with $C_M=C_2$ from Theorem~\ref{thm:variation}.
-\end{corollary}
-
-% The supremum is over BOTH hard truncation endpoints, not just
-% the upper one. The constant C_M is absolute. From the maximal
-% estimate the principal-value theorem follows in one more step.
-\begin{corollary}[Joint principal value]\label{cor:bilinear-pv}
-For every complex $F,G\in L^3(\R^2)$, the joint limit
-\[
- B(F,G)=\lim_{\substack{\varepsilon\downarrow0\\R\uparrow\infty}}
- B_{\varepsilon,R}(F,G)
-\]
-exists almost everywhere and in $L^{3/2}$, and
-$\|B(F,G)\|_{3/2}\le C_M\|F\|_3\|G\|_3$.
-Moreover, the entire tail converges in the maximal sense:
-\begin{equation}\label{eq:maximal-tail-convergence}
- \Big\|\sup_{\substack{0<\varepsilon<1/n\\R>n}}
- |B_{\varepsilon,R}(F,G)-B(F,G)|\Big\|_{3/2}
- \longrightarrow 0 .
-\end{equation}
-\end{corollary}
-
-% The maximal operator controls the two-parameter (eps,R)
-% truncations simultaneously: the SUPREMUM inside the norm on
-% the left of (2.4) shrinks to zero as n grows. So the
-% principle-value limit is not merely pointwise; it is
-% dominated in L^{3/2} by an integrable envelope.
-%
-% Section 6 proves all of the above from the count estimate.
-% Here we only stated them for context.
-
-% ============================================================
-\section{Strategy: how a single-annulus bound becomes a count}
-\label{sec:strategy}
-% ============================================================
-
-% The proof of Theorem~\ref{thm:count} has three layers, each
-% built to solve a problem the one beneath it could not.
-%
-% 1. HEAT FLOW. Replace the hard annulus indicator by a smooth,
-% Gaussian-derived window. The hard truncation is recovered as
-% the endpoint of a Gaussian scale integral. Smooth windows
-% have the miraculous property that their second derivatives
-% are TRACES OF MATRICES with positive spectrum.
-%
-% 2. MATRIX ENERGY. Sample the trilinear form on a fine grid.
-% It becomes a trace of a product of three matrices times
-% diagonal insertions. The second derivative of the trace
-% trace(R^*R)^{3/2} in the insertion directions is a sum of
-% squares of entries (a Hessian form). A "mixed trace
-% inequality" bounds the cyclic product by those costs.
-%
-% 3. ROUGH KERNELS. Coming back to the actual annulus
-% indicator, the endpoint errors are bounded using the
-% frequency-block estimate, which exploits disjointness of
-% the annuli. Counting disjointness is exactly what the
-% sqrt(n) loss comes from.
-%
-% Here is the logic in one sentence: SMOOTHING COSTS A TRACE,
-% DISJOINTNESS BOUNDS THE TRACE, AND RANK SORTING
-% CONVERTS A COUNT INTO A VARIATION.
-
-
-\subsection{The two difficulties, and where sqrt(n) comes from}
-\label{subsec:difficulties}
-
-% The proof has to handle TWO kinds of error, and each looks at
-% first like it costs a factor of n.
-%
-% difficulty (S) -- SMOOTH SWITCHES. The smooth annuli
-% \int_{eps^2}^{R^2} (k_s * k_s * k_s)(t) ds/s have a
-% beginning and an end at each truncation endpoint. Summing
-% over every interval, the number of switches is 2n at each
-% output point. Naively the bound is n times something, not
-% sqrt(n). Section~\ref{sec:smooth} shows how to rescale the
-% third input to absorb this linear cost into a square root.
-%
-% difficulty (H) -- HARD ENDPOINT ERRORS. The gap between the
-% smooth annulus and the hard Hilbert annulus is a sum of
-% "endpoint error" kernels E_eps, one per endpoint. Naively
-% there are 2n of them, which again looks linear. Section
-% \ref{sec:rough} groups endpoints at comparable scales. Within
-% a dyadic group, disjointness gives a uniform pointwise bound
-% on the total, so each group behaves like ONE kernel. The
-% frequency-block estimate (Section~\ref{sec:frequency}) divides
-% the groups into two classes: those with many jumps (there are
-% at most 2 sqrt(n) of them) and those with few (which are
-% handled by the frequency estimate). The sqrt(n) is the price
-% of the changeover.
-
-% The point is that both difficulties are paid for ONCE by
-% sqrt(n), and not by n. This is the content of the paper.
-
-% ============================================================
-\section{Preliminaries}\label{sec:preliminaries}
-% ============================================================
-
-% This section fixes three things: the representatives of
-% truncated operators, the change of variables to simplex
-% coordinates, and the step-function structure that makes the
-% analysis measurable. Then it records the one-dimensional
-% maximal estimates that will pay for the switch costs.
-
-\subsection{Finite truncations}\label{subsec:truncations}
-
-% The truncated operator B_{eps,R}(F,G) is a two-dimensional
-% integral in t of a product, and it is not obvious whether it
-% even makes sense at a given (x,y). The following lemma
-% handles this. We call a "null set" a set of measure zero on
-% R^2.
-\begin{lemma}[Finite truncations]\label{lem:finite-truncations}
-For $F,G\in L^3(\R^2)$, all finite annular integrals defining
-$B_{\varepsilon,R}(F,G)$ are absolutely convergent outside one
-common null set. On its complement they depend continuously on
-$(\varepsilon,R)$ in $0<\varepsilon<R<\infty$. Changing the
-representatives of $F$ or $G$ changes this family only on a
-common null set. Moreover,
-\begin{equation}\label{eq:finite-annulus-bound}
- \Big\|\int_{\varepsilon<|t|<R}
- |F(x+t,y)G(x,y+t)|\,\frac{\dd t}{|t|}\Big\|_{3/2}
- \le 2\log(R/\varepsilon)\|F\|_3\|G\|_3 .
-\end{equation}
-\end{lemma}
-\begin{proof}
-Fix $t$. Holder's inequality in $(x,y)$ and translation
-invariance give
-$\|F(\cdot+t,\cdot)G(\cdot,\cdot+t)\|_{3/2}\le\|F\|_3\|G\|_3$.
-Minkowski's integral inequality yields \eqref{eq:finite-annulus-bound}.
-Apply it to the countably many annuli $1/m<|t|<m$, $m\ge2$;
-their union of exceptional null sets works for every finite
-annulus. Absolute continuity of the $t$-integral gives endpoint
-continuity. A zero-$L^3$-difference estimate on each of those
-annuli proves the assertion about representatives.
-\end{proof}
-
-% Read (3.1) as a bookkeeping bound: on any finite annulus the
-% truncated operator is dominated by $\log(R/\eps)$ times the
-% product of input norms. The $\log$ is essential: without it the
-% principal value would not be an L^{3/2}-bounded operator on the
-% hard annulus. The lemma lets us use any representative we like;
-% the truncation acts on L^3 as a well-defined continuous family
-% of operators.
-%
-% In (1.7) the count supremum uses rational endpoints. By
-% continuity in $(\varepsilon,R)$, rational endpoints exhaust all
-% endpoints; we may (and will) replace rational by arbitrary
-% throughout. Similarly, measurability of $B_*$ and $V_r$ comes
-% from this exhaustion.
-
-
-\subsection{Simplex coordinates}\label{subsec:coordinates}
-
-% Most of the analysis is done in simplex coordinates. The
-% change of variables below has absolute determinant one, so it
-% preserves every L^p norm and commutes with the operator up to
-% name changes.
-\begin{equation}\label{eq:coords-change}
- x=-a,\quad y=-b,\quad t=a+b+c,
- \quad\text{with inverse}\quad
- a=-x,\quad b=-y,\quad c=t+x+y .
-\end{equation}
-% Pulled back along this map, the two inputs F and G become
-\begin{equation}\label{eq:pair-maps}
- G_1(b,c)=F(b+c,-b),\qquad G_2(c,a)=G(-a,c+a).
-\end{equation}
-% Each of these two maps is a norm-preserving bijection of
-% L^3(\R^2). Its companion change of the dual input is the
-% identity: G_0(a,b) is just G_0(a,b). Combining (3.2) with
-% (1.3) yields the simplex form (1.4). Thus the bilinear
-% theorem on B_{eps,R} and the trilinear theorem on
-% \Lambda_{eps,R} are equivalent, at matching constants.
-%
-% In particular, (1.4) is not merely a formal change of
-% variables. It has the SAME optimal constant as the scalar
-% flat form (Section 6). We mark this now because it justifies
-% working only with the simplex coordinates from here on.
-
-\subsection{Step choices}\label{subsec:step-choices}
-
-% The count and variation suprema are taken pointwise. But the
-% operators appearing under the supremum are not arbitrary
-% functions of (a,b); they are built from a FINITE list of
-% choices. To keep the analysis measurable and Riemann-summable
-% we restrict to choices that are locally constant on a
-% partition of \R^2 by finitely many horizontal and vertical
-% lines.
-
-\begin{definition}[Step choice]\label{def:step-choice}
-A \emph{step choice} for the annulus is a choice of data
-\[
- \text{ menu }=\big\{\,(\varepsilon_j^{(m)},R_j^{(m)},
- \beta_j^{(m)})_{j=1,\ldots,n}
- \big\}_{m=1,\ldots,M}
-\]
-together with a partition of $\R^2$ into finitely many open
-rectangles by horizontal and vertical lines, and a map from
-rectangles to $m$. The bounded rectangle is where the operator
-is genuinely varying; the values on the measure-zero boundaries
-of the rectangles are immaterial.
-\end{definition}
-
-% In plain words: we pre-commit to a finite list of annuli
-% choices, one per row of a big table. The row is chosen
-% independently in each rectangle. In the interior of a rectangle
-% the annuli and coefficients do not change as (a,b) varies.
-%
-% Why step choices? Three reasons.
-%
-% (1) Measurability. A measurable assignment of annuli to (a,b)
-% can be approximated in measure by a step choice. So proving
-% the bound for step choices with uniform constants gives the
-% bound for all measurable assignments by exhaustion.
-%
-% (2) Riemann sums. Fixing a step choice means we are reduced to
-% finitely many kernels. Riemann-summing against a smooth
-% companion amounts to a finite calculation.
-%
-% (3) Endpoints at cell boundaries are measure zero, and they
-% contribute nothing to the trilinear form.
-
-% We will be careful about two points.
-% - The cell choice can depend on (a,b) and on the specific
-% rectangular partition (of the OUTPUT point), but the finitely
-% many kernels under the sum are fixed once the partition is
-% fixed.
-% - Nothing about the step choice needs to be the same at
-% different points (a,b), nor consistent across dyadic scales.
-
-
-\subsection{Line maximal estimates}\label{subsec:line-maximal}
-
-% Every jump in a matrix energy is controlled by a maximal
-% average along a line. We record the two forms below. The
-% constants are absolute.
-
-% Let M_R and M_Z denote the CENTERED Hardy--Littlewood
-% maximal operators on the real line and on the integer lattice
-% respectively: for functions on R,
-% M_R f(x) = sup_{r>0} (2r)^{-1}\int_{x-r}^{x+r}|f(y)|dy,
-% and for sequences on Z,
-% M_Z f(j) = sup_{m>=0} (2m+1)^{-1}\sum_{|l-j|<=m}|f(l)|.
-
-\begin{lemma}[Line maximal estimates]\label{lem:line-maximal}
-On either $\R$ or $\mathbb Z$, for $1<p<\infty$,
-\begin{equation}\label{eq:line-maximal}
- \mu\{M f>\rho\}\le\frac3\rho\|f\|_1,
- \qquad
- \|Mf\|_p^p\le\frac{6p\,2^{p-1}}{p-1}\|f\|_p^p .
-\end{equation}
-Let $g_\sigma(t)=(2\pi\sigma)^{-1/2}e^{-t^2/(2\sigma)}$. There is
-an absolute constant $C$ such that, uniformly in $\sigma$,
-\begin{align}
- \sum_m h g_\sigma((m-j)h)|f(m)|
- &\le C M_{\mathbb Z}f(j), &&\sigma\ge h^2,
- \label{eq:discrete-gaussian-maximal}\\
- \int g_\sigma(t)|f(x-t)|\,\dd t
- &\le C M_{\R}f(x), &&\sigma>0 .
- \label{eq:continuous-gaussian-maximal}
-\end{align}
-\end{lemma}
-\begin{proof}
-The weak-type estimate: from any finite family of intervals,
-greedily select the widest interval not already overlapping
-another, and discard any covered by their triple enlargements;
-this yields a disjoint sub-covered set of at most 3x the original
-measure (also for integer intervals). Combining with the
-truncation argument for strong $L^p$ gives
-\eqref{eq:line-maximal}.
-
-The Gaussian bounds follow from splitting the sum or integral
-into shells $2^{k-1}a<|m-j|\le 2^ka$ with $a=\sqrt\sigma/h$.
-Each shell contributes at most $C2^k a M f$, and the resulting
-geometric series converges.
-\end{proof}
-
-% ============================================================
-\section{Dimension-free matrix inequalities}\label{sec:matrix}
-% ============================================================
-
-% This is the linear-algebra engine of the proof. Everything
-% below is finite-dimensional. The point of this section is to
-% introduce a QUADRATIC COST attached to each possible direction
-% of variation of a matrix trace, and to prove that this cost
-% controls the trilinear form we want to estimate.
-
-% Setup. Spaces $H_0,H_1,H_2$ are finite-dimensional complex
-% Hilbert spaces. Maps $W_v:H_{v+1}\to H_v$ are linear (indices
-% mod 3). The ROW STAR at vertex $v$ is
-\begin{equation}\label{eq:row-star}
- R_v=[\,W_v\quad W_{v-1}^*\,]:H_{v+1}\oplus H_{v-1}\to H_v,
-\end{equation}
-% with two Gram matrices
-\begin{equation}\label{eq:star-grams}
- T_v=R_v^*R_v,\qquad S_v=R_vR_v^* .
-\end{equation}
-% The two Grams have equal nonzero spectra, so their energies
-% coincide:
-\[
- \Phi(T_v)=\Phi(S_v),\qquad \Phi(P):=\tr(P^{3/2}) .
-\]
-
-% The energy objective is a smooth function of a positive
-% semidefinite matrix $P\ge0$, with a natural Hessian cost.
-% We first record the Hessian formula.
-\begin{lemma}[Supported Hessian]\label{lem:hessian}
-On the cone of matrices positive definite on a fixed support,
-$\Phi(P)=\tr(P^{3/2})$ is smooth, and its first and second
-derivatives at a supported direction $K$ (and at supported
-directions $K,L$) are
-\begin{equation}\label{eq:hessian}
- d\Phi_P(K)=\frac32\tr(\sqrt P\,K),\qquad
- d^2\Phi_P(K,L)=\mathcal B_P(K,L),
-\end{equation}
-where the Hessian form is
-\begin{equation}\label{eq:hessian-form}
- \mathcal B_P(K,L)
- =\frac32\operatorname{Re}\big\langle K,
- (\mathsf L_{\sqrt P}+\mathsf R_{\sqrt P})^{-1}L\big\rangle_{\HS} .
-\end{equation}
-% Here \mathsf L_A(X)=AX and \mathsf R_A(X)=XA$ are the left and
-% right multiplication operators. In a diagonal basis for $P$
-% with positive eigenvalues $p_i$, we have
-\begin{equation}\label{eq:hessian-metric}
- b_P(K)=\mathcal B_P(K,K)
- =\frac32\sum_{i,j}\frac{|K_{ij}|^2}{\sqrt{p_i}+\sqrt{p_j}} .
-\end{equation}
-% The form $b_P$ is symmetric positive-definite on supported
-% Hermitian directions, and it satisfies
-% $|\mathcal B_P(K,L)|^2\le b_P(K)b_P(L)$ (after Cauchy--Schwarz).
-\end{lemma}
-\begin{proof}
-Restrict to the support $\supp P$ (where $P$ is positive
-definite) and set $A=\sqrt P$. Differentiating the squaring map
-$A\mapsto A^2$ at $A$ multiplies the $(i,j)$ entry by
-$\sqrt{p_i}+\sqrt{p_j}>0$, so the inverse function theorem
-gives a smooth positive square root. Then
-$d\Phi_P(K)=\frac32\tr(AK)$ by the chain rule, and one more
-differentiation gives (4.4)-(4.5). Diagonalising $A$ yields
-(4.6). Cauchy--Schwarz on the Hilbert-Schmidt inner product
-along eigenvectors of $\mathsf L_A+\mathsf R_A$ gives the
-bound $|\mathcal B_P(K,L)|^2\le b_P(K)b_P(L)$ after noting that
-$(\mathsf L_A+\mathsf R_A)^{-1}$ is positive definite on
-supported Hermitian matrices.
-\end{proof}
-
-% The key use is a COMPARISON: if we enlarge the Gram, the
-% cost shrinks. This is a monotonicity of the Hessian form,
-% analogous to Loewner--Heinz monotonicity of the square root.
-
-
-% The following lemma is the monotonicity of the Hessian cost
-% under a Loewner-type order on the Gram.
-\begin{lemma}[Order comparison]\label{lem:order}
-If $P,Q\ge0$ on the same space, $a\ge1$, and $P\le aQ$, then
-every Hermitian $K$ supported on $P$ is supported on $Q$, and
-\begin{equation}\label{eq:matrix-order}
- b_P(K)\ge a^{-1/2}\,b_Q(K).
-\end{equation}
-\end{lemma}
-\begin{proof}
-$\ker Q\subseteq\ker P$ follows directly from $0\le P\le aQ$,
-which gives the support containment. The order facts: spectrum
-calculus and the integral representation
-$\sqrt X=\pi^{-1}\int_0^\infty X(X+tI)^{-1}t^{-1/2}\dd t$
-combined with inversion order
-($0<X\le Y\Rightarrow Y^{-1}\le X^{-1}$) and monotonicity of
-the multiplication operator $\mathsf L_A+\mathsf R_A$ in $A$
-give $\sqrt{P+\eta I}\le \sqrt a\sqrt{Q+\eta I}$ for all $\eta>0$.
-Then
-$(\mathsf L_{\sqrt{P+\eta I}}+\mathsf R_{\sqrt{P+\eta I}})^{-1}$,
-being the inverse of a positive operator on Hilbert--Schmidt
-space, decreases as $\eta\downarrow0$ by monotonicity of
-inversion. Taking $\eta\downarrow0$ proves (4.7).
-\end{proof}
-
-% The following lemma bounds a matrix cost of the form
-% $R^*LR$ (Gram of $R$, insertion $L$) by the size of $L$
-% times $\Phi(R^*R)$. It is the single most-used inequality in
-% the paper.
-\begin{lemma}[Inserted Gram bound]\label{lem:insertion}
-For a linear map $R:E\to F$ and a Hermitian map $L:F\to F$,
-the matrix $R^*LR$ is supported on $R^*R$, and
-\begin{equation}\label{eq:insertion-bound}
- b_{R^*R}(R^*LR)
- \le \frac34\|L\|_{\op}^2\,\Phi(R^*R) .
-\end{equation}
-\end{lemma}
-\begin{proof}
-Let $\rho_i>0$ be the positive singular values of $R$ and
-$C$ the compression of $L$ to the left singular subspace.
+The triangular Hilbert transform sits between two classical objects
+and one open problem.
+
+For a smooth compactly supported function f on the real line, the
+Hilbert transform is the principal value
+
+ Hf(x) = p.v. (1/pi) int_{-infinity}^{infinity} f(x + t) dt / t .
+
+At each x the function f is paired against the antisymmetric kernel
+1/t. Symmetric pieces vanish under the pairing, lopsided pieces
+survive. Boundedness of H on L^p for 1 < p < infinity is a
+Calderon-Zygmund theorem, and the same truncated integrals
+
+ H_{eps,R}f(x) = (1/pi) int_{eps < |t| < R} f(x + t) dt / t
+
+converge pointwise almost everywhere as eps -> 0 and R -> infinity.
+Uniformity of that convergence, and the stronger statement that the
+family { H_{eps,R}f } has bounded r-variation for r > 2, is a
+two-parameter Carleson problem.
+
+When the same kernel is applied to a bilinear product
+
+ BH(f,g)(x) = p.v. int_{-infinity}^{infinity} f(x + t) g(x - t) dt / t,
+
+the two inputs sit on opposite sides of one corner. Boundedness of
+BH on a wide range of L^p products is Lacey and Thiele (1997). The proof decomposes the frequency plane into tilted wave packets
+and selects coherent trees. Demeter and Thiele
+later isolate a two-dimensional analogue in which the two inputs
+move along perpendicular axes:
+
+ ./0803.1268.pdf
+
+For complex functions F, G of two real variables and 0 < eps <
+R < infinity the annular triangular Hilbert transform is
+
+ B_{eps,R}(F,G)(x,y)
+ = int_{eps < |t| < R} F(x + t, y) G(x, y + t) dt / t .
+
+At the corner (x,y), F is read on the horizontal leg and G on the
+vertical leg. The two inputs occupy distinct lines, so the
+one-dimensional opposition that organises the bilinear Hilbert
+transform is absent. The exponent triple 3, 3, 3/2 sits at the
+symmetric point 1/p_0 + 1/p_1 + 1/p_2 = 1 of the Holder surface,
+where every technique of multilinear analysis starts to lose room.
+
+Pairing B_{eps,R}(F,G) with a third function G_0 in L^3, and
+substituting x = -a, y = -b, t = a + b + c (a change of variables
+of absolute determinant one), puts the operator in simplex form:
+
+ Lambda_{eps,R}(G_0, G_1, G_2)
+ = iiint_{eps < |a+b+c| < R} G_0(a,b) G_1(b,c) G_2(c,a)
+ da db dc / (a + b + c) .
+
+Each factor reads one edge of the triangle with vertices a, b, c.
+The edges form the cycle b -> c -> a -> b, and no input depends on
+all three variables simultaneously. The same substitution preserves
+L^p norms, so an estimate for Lambda at (3,3,3) is equivalent to an
+estimate for B at (3,3,3/2).
+
+The two-variable dyadic model with two general inputs and a third
+of specified structure is constructed by Kovac, Thiele, and
+Zorin-Kranich. That paper also states the cyclic trace formulation
+and the equivalence of the flat and simplex forms.
+
+ ./1506.00861.pdf
+
+Tao proves sublogarithmic cancellation for the one-dimensional
+multilinear Hilbert transform, and Zorin-Kranich extends the
+technique to the simplex Hilbert transform over finitely many
+smooth scales.
+
+ ./1505.06479.pdf
+ ./1507.02436.pdf
+
+Durcik, Kovac, Skreb, and Thiele obtain square-function and
+norm-variation estimates for the triangular form over finitely many
+smooth scales. Durcik, Kovac, and Thiele obtain power-type
+cancellation for hard truncations. Permuting and interpolating the
+(4,4,2) bounds of the latter paper gives a bound of order
+sqrt(log(R/eps)) at the symmetric point. Both estimates retain a
+dependence on the range of scales, and neither places the supremum
+over output-dependent partitions inside the output norm.
+
+ ./1603.00631.pdf
+ ./1608.00156.pdf
+
+Durcik's continuous Gaussian telescoping for entangled forms is
+the analytic template for the finite-dimensional heat identities
+used below.
+
+ ./1412.2384.pdf
+ ./1506.08150.pdf
+
+The Bellman function and twisted paraproduct work of Kovac isolate
+the obstruction that a triangular cycle presents and motivate the
+energy method.
+
+ ./1108.0917.pdf
+ ./1011.6140.pdf
+
+Grafakos, Oliveira e Silva, Pramanik, Seeger, and Stovall collect
+open problems at the symmetric point. Problem 13 in Section 9 of
+that survey asks precisely for the scalar estimate proved below.
+
+ ./1701.06637.pdf
+
+Durcik-Roos, Christ-Durcik-Roos, Hsu-Lin, and Lin-Slavikova
+obtain related bounds by averaging a direction parameter
+or by introducing curvature into the kernel. Those mechanisms are
+distinct from the flat problem considered below.
+
+ ./1812.11701.pdf
+ ./2008.10140.pdf
+ ./2410.15791.pdf
+ ./2607.20206.pdf
+
+
+The r-th annular variation of the family { B_{eps,R} }, for real
+r > 2, is the function
+
+ V_r(F,G)(x,y)
+ = sup over J >= 1 and rational 0 < t_0 < ... < t_J of
+ [ sum_{j=1}^{J} |B_{t_{j-1}, t_j}(F,G)(x,y)|^r ]^{1/r}.
+
+The supremum at (x,y) is independent of the value at any other
+point. The number of increments J is unbounded, and the endpoints
+need not lie on a dyadic grid. Rational endpoints make the
+supremum measurable; endpoint continuity of each finite truncation,
+established in the preliminary lemmas, shows that the restriction
+does not change the value.
+
+Variation dominates both a maximal function and any partially
+truncated principal value. A single increment recovers the value of
+|B_{eps,R}(F,G)(x,y)| for every pair (eps, R), so V_r bounds the
+maximal operator over both endpoints. Dyadic partitions recover
+every cofinal family of truncations. An L^{3/2} bound on V_r for a
+single r > 2 therefore implies a maximal bound on the supremum of
+B_{eps,R} and joint almost-everywhere convergence of the truncations
+in the maximal sense.
+
+The main theorem is a uniform variation bound at the symmetric
+point. For every real r > 2 there is a finite constant C_r,
+independent of F and G, such that
+
+ || V_r(F,G) ||_{L^{3/2}(R^2)}
+ <= C_r ||F||_{L^3(R^2)} ||G||_{L^3(R^2)}
+
+for every pair of complex F, G in L^3(R^2). The output exponent 3/2
+is forced by Holder duality: pairing B_{eps,R} with a third L^3
+input and applying the flat-simplex equivalence leaves the output
+in L^{3/2}. Below the symmetric point the same methods give
+simpler estimates; at 3, 3, 3 every multilinear technique loses
+room.
+
+The proof passes through a discrete count estimate. Fix an integer
+n >= 1, and consider lists of at most n pairwise disjoint annuli
+(eps_1, R_1), ..., (eps_m, R_m) with rational endpoints and complex
+coefficients beta_j of modulus at most one. Define
+
+ S_n(F,G)(x,y)
+ = sup sum_{j=1}^{m}
+ |B_{eps_j, R_j}(F,G)(x,y)|
+
+where the supremum runs over every such list and every m <= n.
+Every list is selected separately at each point (x,y), and the
+empty sum is zero.
+
+The count theorem states that there is an absolute
+constant C, independent of n, with
+
+ || S_n(F,G) ||_{L^{3/2}(R^2)}
+ <= C sqrt(n) log(2 + n) ||F||_3 ||G||_3
+
+for every integer n >= 1 and every pair F, G in L^3(R^2). The
+factor sqrt(n) is the price paid for the number of annuli; the
+factor log(2+n) is a mild extra cost from allowing arbitrary
+disjoint annuli rather than dyadic ones. Neither factor depends on
+the endpoints, the coefficients, or the location.
+
+From the count theorem, the main theorem follows by a combinatorial
+exhaustion. Fix F, G, and any point (x,y), and any partition
+0 < t_0 < ... < t_J. Denote the increments
+
+ d_j = |B_{t_{j-1}, t_j}(F,G)(x,y)|
+
+and rearrange them in decreasing order d_(1) >= ... >= d_(J).
+Whenever the rank 2^k does not exceed J, the first 2^k increments
+form a permissible list inside S_{2^k}, so
+
+ d_(2^k) <= 2^{-k} S_{2^k}(F,G)(x,y).
+
+The ranks 2^k <= l < 2^{k+1} form a block of size at most 2^k, and
+each member is at most d_(2^k). The l^r norm of that block is at
+most 2^{k/r} d_(2^k). Summing over k gives the pointwise inequality
+
+ V_r(F,G)(x,y)
+ <= sum_{k >= 0} 2^{k(1/r - 1/2)}
+ S_{2^k}(F,G)(x,y).
+
+The exponent 1/r - 1/2 is strictly negative for r > 2, so the
+geometric factor decays. Taking L^{3/2} norms, applying Minkowski's
+inequality, and invoking the count theorem,
+
+ || V_r(F,G) ||_{3/2}
+ <= C ||F||_3 ||G||_3
+ sum_{k >= 0} (1 + k) 2^{k(1/r - 1/2)},
+
+which is finite. Neither a bound on the number of increments nor a
+restriction to dyadic endpoints survives in the conclusion.
+
+The substance of the argument is therefore the count theorem. Its
+statement concerns finitely many annuli at every point, but
+disjointness is a strong pointwise hypothesis: at any fixed (x,y),
+an annulus cannot sit inside another annulus at comparable radius.
+The argument exploits this by grouping endpoints at comparable
+scales and by using frequency information to avoid paying the number
+of endpoints linearly.
+
+
+The proof of the count estimate has three layered ingredients.
+
+First, replace each hard annulus kernel by its smooth Gaussian
+version, obtained as the endpoint of a scale integral of
+convolutions of three one-variable Gaussian derivative windows.
+Sampling the trilinear form on a fine grid converts it to a
+cyclic trace of three matrices, with diagonal insertions coming
+from the window factors. The trace of the positive cube root of
+the Gram that summarises the three matrices has the same
+degree-three scaling as the trilinear form.
+
+Second, add a Gaussian heat flow to the Gram. Differentiating the
+trace in the scale parameter produces positive Hessian costs in
+the insertion directions, and a mixed trace inequality bounds the
+cyclic product by three quadratic costs, one at each vertex. A
+heat identity turns those costs into derivatives of a positive
+energy, and integrating in scale gives an inequality between the
+accumulated costs and the initial and terminal energies. The
+energies are bounded by cubed L^3 norms. The dissipation is
+strict when the three variance rates are comparable.
+
+Third, the comparison between hard and smooth annuli leaves an
+odd endpoint error at every annular boundary. Estimating each
+error separately would cost a factor of n in the number of
+annuli, which is forbidden at the required count estimate.
+Instead, group endpoints at dyadic radius scales. Inside a
+dyadic group disjointness of the annuli gives a single pointwise
+Gaussian bound on the sum of all endpoint errors in that group,
+independent of the number of its discontinuities. A
+frequency-block estimate handles groups with few discontinuities;
+groups with many are few enough to handle by the pointwise bound.
+A sqrt(n) changeover absorbs the linear cost.
+
+Both layers pay their square-root losses exactly once. In the
+smooth part the switch count per output point is bounded by 2n;
+a rescaling of the third input converts the n-fold loss to
+sqrt(n). In the hard-endpoint part, dyadic grouping reduces the
+sum of error kernels to at most a constant times n, but the
+grouping plus the frequency-block disjunction produces the same
+sqrt(n) price.
+
+The technical prerequisites fix the framework.
+
+For complex F and G in L^3(R^2) and any pair of real endpoints
+with 0 < eps < R < infinity, the finite annulus operator
+B_{eps,R}(F, G)(x,y) is the integral over eps < |t| < R of the
+product against 1/t. When the truncated integral is evaluated at
+any fixed (x, y), the resulting absolutely convergent
+two-variable sum exists outside a single null set of the plane.
+
+To see this, fix F, G in L^3(R^2) and fix t. Holder's inequality
+in (x, y) and translation invariance give
+
+ || F(. + t, .) G(., . + t) ||_{3/2}
+ <= ||F||_3 ||G||_3.
+
+Integrating in t over eps < |t| < R and using Minkowski's integral
+inequality,
+
+ || int_{eps < |t| < R} |F(x + t, y) G(x, y + t)| dt / |t| ||_{3/2}
+ <= 2 log(R / eps) ||F||_3 ||G||_3.
+
+Apply the same estimate to the countably many annuli
+1/m < |t| < m for m = 2, 3, ... . The union of the exceptional null
+sets is again null. Outside that null set every finite annulus
+integral converges absolutely. The same estimate applied to a
+zero-L^3 difference shows that changing the representative of F
+or G changes the family only on the same null set. Absolute
+continuity of the t-integral gives continuity of B_{eps,R} in
+(eps, R). The rational-endpoint restriction in the definitions of
+V_r and S_n therefore does not change the suprema.
+
+The three-variable substitution has absolute determinant
+
+ | det d(x,y,t) / d(a,b,c) | = 1
+
+under x = -a, y = -b, t = a + b + c. The two-variable pair maps
+
+ G_1(b,c) = F(b + c, -b),
+ G_2(c,a) = G(-a, c + a)
+
+each preserve L^p norms of their arguments. Pairing B_{eps,R}(F,G)
+against a third L^3 function G_0, rewriting the 1/t kernel as
+1/(a+b+c) on the annulus eps < |a+b+c| < R, and substituting,
+gives the flat form
+
+ iiint_{eps < |a+b+c| < R} G_0(a,b) G_1(b,c) G_2(c,a) da db dc / (a+b+c) .
+
+This trilinear form includes the two-parameter supremum over
+(eps, R) as a single supremum over |a+b+c|. Both the count theorem
+and the variation theorem reduce to estimates on the trilinear
+form.
+
+The count and variation suprema are pointwise in (x, y), so the
+numerical choices of annulus lists can differ from point to point.
+The argument first proves the required bound for a finite menu of
+step choices, constant on finitely many axis-parallel rectangles,
+then recovers the general case by measurable exhaustion.
+
+A step choice consists of a finite set of menus, one menu per row
+of a table; each menu is a list of at most n annuli together with
+complex coefficients of modulus at most one. A partition of the
+plane by finitely many horizontal and vertical lines induces
+rectangles, and each rectangle selects one menu row. Inside a
+rectangle the annuli and their coefficients are frozen. Values on
+the rectangle boundaries are immaterial to any integral.
+
+For the switch analysis, the centered Hardy-Littlewood maximal
+operator on the line is
+
+ M_R f(x) = sup_{r > 0} (2r)^{-1} int_{x-r}^{x+r} |f(t)| dt,
+
+and the discrete analogue is
+
+ M_Z f(j) = sup_{m >= 0} (2m + 1)^{-1} sum_{|l - j| <= m} |f(l)| .
+
+Both operators satisfy mu{ M f > rho } <= 3 rho^{-1} ||f||_1 and
+
+ || M f ||_p^p <= (6 p 2^{p-1} / (p - 1)) ||f||_p^p
+
+for 1 < p < infinity. The covering argument is the classical one
+of Hardy and Littlewood (Acta Mathematica 54, 1930). A
+Gaussian-weighted average of |f| on either space is bounded by a
+constant multiple of M f, with the constant independent of the
+Gaussian variance and of the mesh. On the grid m = j + (m - j),
+
+ sum_m h g_sigma((m - j) h) |f(m)|
+ <= C M_Z f(j) for sigma >= h^2,
+
+and on the line,
+
+ int g_sigma(t) |f(x - t)| dt
+ <= C M_R f(x) for sigma > 0,
+
+where g_sigma(t) = (2 pi sigma)^{-1/2} exp(-t^2 / (2 sigma)). The
+constant C is absolute. The shell decomposition
+2^{k-1} a < |x - x'| <= 2^k a with a = sqrt(sigma) / h produces a
+geometric series with sum bounded by a constant.
+
+
+The finite-dimensional engine has four facts: a Hessian cost on the
+energy tr(P^{3/2}), an order comparison, an insertion bound, and a
+cyclic mixed-trace inequality. Every statement below is exact, with
+explicit constants.
+
+Let H_0, H_1, H_2 be finite-dimensional complex Hilbert spaces,
+indexed by Z/3, and let W_v : H_{v+1} -> H_v be linear maps. The
+row star at vertex v is
+
+ R_v = [ W_v W_{v-1}^* ] : H_{v+1} + H_{v-1} -> H_v,
+
+with Gram matrices
+
+ T_v = R_v^* R_v, S_v = R_v R_v^*.
+
+The two Grams have equal nonzero eigenvalue multisets, including
+multiplicities, so
+
+ Phi(T_v) = Phi(S_v), Phi(P) = tr(P^{3/2}) for P >= 0.
+
+A matrix K is supported on a positive P if K annihilates ker P on
+both sides. On the cone of matrices positive definite on a fixed
+support, the energy Phi is smooth, with
+
+ dPhi_P(K) = (3/2) tr( sqrt(P) K ),
+ d^2 Phi_P(K, L) = B_P(K, L),
+
+where the Hessian bilinear form is
+
+ B_P(K, L) = (3/2) Re < K, (L_{sqrt P} + R_{sqrt P})^{-1} L >_{HS}
+
+with L_A(X) = AX and R_A(X) = XA the left and right multiplication
+operators. In an orthonormal eigenbasis of P with positive
+eigenvalues p_i,
+
+ b_P(K) = B_P(K, K)
+ = (3/2) sum_{i,j} |K_{ij}|^2 / (sqrt(p_i) + sqrt(p_j)).
+
+The form B_P is symmetric positive definite on supported Hermitian
+directions, and its entries obey |B_P(K, L)|^2 <= b_P(K) b_P(L).
+
+The proof restricts to supp P and puts A = sqrt P. The squaring map
+A |-> A^2 differentiates to X |-> A X + X A, which in an eigenbasis
+multiplies entry (i,j) by sqrt(p_i) + sqrt(p_j) > 0. Inverse
+function theorem gives a smooth positive square root on the support.
+The chain rule applied to Phi(P) = tr(P sqrt P) gives
+dPhi_P(K) = (3/2) tr(A K); one more differentiation gives B_P.
+Diagonalising A yields the entry formula. Cauchy-Schwarz on the
+Hilbert-Schmidt inner product, along eigenvectors of
+L_A + R_A, gives the bilinear bound. The calculus is the square-root
+case of the divided-difference calculus for Hermitian functions,
+stated originally in Daletskii (Uspekhi Mat. Nauk 12, 1957).
+
+Both Phi and B_P are invariant under simultaneous unitary conjugation
+of all arguments, and under extension of every argument by zero.
+The positive eigenvalues and the supported entries do not change,
+so Grams on different subspaces can be compared.
+
+Order comparison. If P and Q are positive semidefinite on the same
+space, a >= 1, and P <= a Q, then every Hermitian K supported on P
+is supported on Q, and
+
+ b_P(K) >= a^{-1/2} b_Q(K).
+
+Proof. For x in ker Q, 0 <= <P x, x> <= a <Q x, x> = 0, so
+ker Q is contained in ker P and K is supported on Q. Two order
+facts are used. First, inversion reverses order on positive definite
+matrices: 0 < X <= Y implies 0 < Y^{-1} <= X^{-1}, by conjugating
+with X^{-1/2}. Second, the matrix square root preserves order; the
+classical Loewner-Heinz monotonicity is stated and proved in Kwong
+(Proc. Amer. Math. Soc. 51, 1975). For eta > 0,
+
+ P + eta I <= a (Q + eta I),
+ sqrt(P + eta I) <= sqrt(a) sqrt(Q + eta I).
+
+If A <= B, then L_A + R_A <= L_B + R_B on Hilbert-Schmidt space,
+because the difference has quadratic form
+tr(X^* (B - A) X) + tr(X^* X (B - A)) >= 0. Taking inverses
+reverses the order, so
+
+ b_{P+eta I}(K) >= a^{-1/2} b_{Q+eta I}(K).
+
+Letting eta -> 0 in an eigenbasis, every entry of K that meets the
+kernel of the limiting matrix is zero, and the entry formula gives
+the supported cost on both sides.
+
+Insertion bound. For a linear map R: E -> F and a Hermitian L on F,
+the matrix R^* L R is supported on R^* R, and
+
+ b_{R^* R}(R^* L R) <= (3/4) ||L||_{op}^2 Phi(R^* R).
+
+Proof. The insertion is Hermitian and annihilates ker R. Let
+rho_i > 0 be the positive singular values of R, and let C be the
+compression of L to the corresponding left singular subspace. Then
+
+ b_{R^* R}(R^* L R)
+ = (3/2) sum_{i,j} rho_i^2 rho_j^2 |C_{ij}|^2 / (rho_i + rho_j).
+
+For x, y > 0, x^2 y^2 / (x + y) <= (x y)^{3/2} / 2
+<= (x^3 + y^3) / 4. Each row and each column of C has squared norm
+at most ||L||_{op}^2. Summing,
+
+ b_{R^* R}(R^* L R)
+ <= (3/8) sum_{i,j} (rho_i^3 + rho_j^3) |C_{ij}|^2
+ <= (3/4) ||L||_{op}^2 sum_i rho_i^3.
+
+Joint convexity. Let (Omega, nu) be a probability space and E a
+fixed subspace. Suppose T_w >= 0 and K_w = K_w^* are measurable,
+supp T_w = E, K_w is supported on T_w almost everywhere, and
+int (||T_w||_{op} + ||K_w||_{op}) d nu(w) is finite. Then
+T = int T_w d nu(w) has support E, K = int K_w d nu(w) is supported
+on T, and
+
+ b_T(K) <= int b_{T_w}(K_w) d nu(w).
+
+Proof. Restrict every matrix to E. For every nonzero x in E, the
+positive quantity <T_w x, x> has positive integral, so T is positive
+definite on E, and the averages vanish on E^perp. Put A_w = sqrt T_w
+and A = int A_w d nu. Vector-valued Cauchy-Schwarz gives
+||A x||^2 <= <T x, x>, so A^2 <= T, and square-root order gives
+sqrt(T) >= int sqrt(T_w) d nu. The variational formula for a
+positive definite selfadjoint operator M on a real Hilbert space is
+
+ <K, M^{-1} K> = sup_X { 2 <K, X> - <X, M X> }.
+
+Applying with M = L_{sqrt T} + R_{sqrt T} on the real Hilbert space
+of Hermitian matrices, using the averaging inequality just proved
+inside the supremum, and bounding the supremum of an integral by the
+integral of pointwise suprema, gives the claim. Uniform
+invertibility in w is not assumed; every inverse is taken on E,
+where T is positive definite.
+
+Rank changes and entrywise derivatives. For rectangular R_-, R_+ of
+the same size, set R(z) = (1 - z) R_- + z R_+ and D = R_+ - R_-.
Then
-$b_{R^*R}(R^*LR)
-=\frac32\sum_{i,j}\frac{\rho_i^2\rho_j^2}{\rho_i+\rho_j}
- |C_{ij}|^2$.
-The pointwise inequality
-$x^2y^2/(x+y)\le (x^3+y^3)/4$ for $x,y>0$ and the fact that
-each row (and column) of $C$ has squared $\ell^2$ norm at most
-$\|L\|_{\op}^2$ yield
-$\le\frac34\|L\|_{\op}^2\sum_i\rho_i^3$, which equals
-the right side of (4.8).
-\end{proof}
-
-% Finally, if we form a PROBABILITY AVERAGE of Grams and their
-% insertions, the cost can only decrease. This is what lets
-% us average over Gaussian center shifts.
-\begin{lemma}[Joint convexity]\label{lem:joint-convexity}
-Let $E$ be a fixed subspace and $T_\omega\ge0$, $K_\omega$
-measurable, $\supp T_\omega=E$, $K_\omega$ supported on
-$T_\omega$, and $\int(\|T_\omega\|_{\op}+\|K_\omega\|_{\op})\dd\nu<\infty$.
-Then $\overline T=\int T_\omega\dd\nu$ has support $E$,
-$\overline K=\int K_\omega\dd\nu$ is supported on $\overline T$,
-and
-\begin{equation}\label{eq:joint-convexity}
- b_{\overline T}(\overline K)
- \le\int b_{T_\omega}(K_\omega)\,\dd\nu(\omega) .
-\end{equation}
-\end{lemma}
-\begin{proof}
-For any $x\in E$, $\langle\overline T x,x\rangle>0$, so
-$\overline T$ is positive definite on $E$, and $K$ is supported
-on it. Put $A_\omega=\sqrt{T_\omega}$ and $A=\int A_\omega\dd\nu$.
-Cauchy--Schwarz in $\omega$ gives $A^2\le\overline T$. By
-$\sqrt X$ monotonicity and
-$\langle K,\mathsf M^{-1}K\rangle
-=\sup_X\{2\langle K,X\rangle-\langle X,\mathsf M X\rangle\}$
-(varational formula for a positive selfadjoint operator), the
-cost at $(\overline T,\overline K)$ is controlled by the average
-of the costs at $(T_\omega,K_\omega)$.
-\end{proof}
-
-
-% The Hessian cost is defined on a fixed support. But our
-% matrices CHANGE rank: when entries are switched on or off
-% they go from zero to nonzero. The following lemma covers
-% this case: it is a first-derivative formula valid across
-% rank changes.
-
-\begin{lemma}[Rank changes and entrywise derivatives]\label{lem:rank-change}
-For rectangular $R_-,R_+$ of the same size, set
-$R(z)=(1-z)R_-+zR_+$ and $D=R_+-R_-$. Then
-\begin{equation}\label{eq:rank-change-derivative}
- \Phi(R_+^*R_+)-\Phi(R_-^*R_-)
- =3\int_0^1\operatorname{Re}\langle R(z)\sqrt{R(z)^*R(z)},D\rangle_{\HS}\,\dd z .
-\end{equation}
-For every $(i,j)$,
-\begin{equation}\label{eq:entry-gradient}
- |(R\sqrt{R^*R})_{ij}|
- \le\|\operatorname{row}_iR\|_2\|\operatorname{col}_jR\|_2 .
-\end{equation}
-\end{lemma}
-\begin{proof}
-Approximate by $\sqrt{R(z)^*R(z)+\eta I}$ and let $\eta\downarrow0$.
-The main claim is the spectral-calculus continuity
-$\|\sqrt{R^*R+\eta I}-\sqrt{R^*R}\|_{\op}\le\sqrt\eta$.
-Then (4.9) follows from the fundamental theorem of calculus.
-(4.10) is Cauchy--Schwarz in the matrix product.
-\end{proof}
-
-% The centrepiece of the section: a cyclic product of three
-% matrices, each with a diagonal insertion, is dominated by
-% the three Hessian costs.
-
-\begin{lemma}[Mixed trace inequality]\label{lem:mixed-trace}
-For Hermitian maps $D_v:H_v\to H_v$ put
-$D_*=\max_v\|D_v\|_{\op}$. Then
-\begin{equation}\label{eq:mixed-trace}
- |\tr(D_0W_0D_1W_1D_2W_2)|
- \le\frac53 D_*\sum_{v=0}^2 b_{T_v}(R_v^*D_vR_v) .
-\end{equation}
-The constant is independent of dimensions, and no Gram is
-assumed invertible.
-\end{lemma}
-\begin{proof}
-The proof proceeds in three stages.
-%
-First, package the three costs into one quadratic form.
-Let $H=H_0\oplus H_1\oplus H_2$ and set
-\[
- M=\begin{pmatrix}0&W_0&W_2^*\\W_0^*&0&W_1\\W_2&W_1^*&0\end{pmatrix},
- \quad D=\operatorname{diag}(D_0,D_1,D_2).
-\]
-If $P_v$ projects onto $H_v$ and $A_v=MP_vM$, $K_v=MP_vDM$,
-then after permuting blocks these equal $T_v$ and $R_v^*D_vR_v$,
-and $K_v$ is supported on $A_v\le M^2$. If $b$ denotes $b_{M^2}$
-on $H$, then
-\begin{equation}\label{eq:trace-common-metric}
- \mathcal E:=\sum_v b_{T_v}(R_v^*D_vR_v)
- \ge\sum_v b_{M^2}(K_v)\ge\tfrac13 b_{M^2}(MDM).
-\end{equation}
-%
-Second, bound a cubic trace using the eigenvalue structure
-of $M$: in a diagonal eigenbasis $m_i$ of $M$ with $q_i=|m_i|$
-nonincreasing, formula (4.6) yields
-$\mathcal E\ge\frac12 Q$ with
-\[
- Q:=\sum_{i,j:q_i+q_j>0}\frac{q_i^2q_j^2}{q_i+q_j}|D_{ij}|^2 .
-\]
-Set
-$A_i=q_i^3|D_{ii}|^2$, $E_i=q_i\sum_{j>i}q_j^2|D_{ij}|^2$,
-$A=\sum A_i$, $E=\sum E_i$. Then $Q\ge A/2+E$ (Hermitian
-symmetry with $q_j\le q_i$ for $j>i$).
-Now group the cubic trace $\tr((DM)^3)$ by the smallest index
-of each triple:
-%
-(case 1) smallest index once: the contribution is at most
-$3D_*E_i$ by Cauchy--Schwarz in the off-diagonal direction.
-% (case 2) smallest index twice: the contribution is at most
-$3D_*\sqrt{A_iE_i}$ (Cauchy--Schwarz with
-$\sum_k|D_{ik}|^2\le D_*^2$).
-% (case 3) all three indices equal: modulus at most $D_*A_i$.
-%
-Summing and using Cauchy--Schwarz,
-\[
- |\tr((DM)^3)|
- \le D_*(A+3E+3\sqrt{AE})
- \le D_*(\tfrac52A+\tfrac92E)
- \le 5D_*(\tfrac A2+E)
- \le 5D_*Q .
-\]
-%
-Third, recover the complex phase. Set
-$\tau=\tr(D_0W_0D_1W_1D_2W_2)$. The matrix $M$ has zero
-diagonal blocks, so each closed three-step block product visits
-all three vertices. The three starting points in one
-orientation give $\tau$, in the other give $\overline\tau$, so
-$\tr((DM)^3)=6\operatorname{Re}\tau$. If $\tau\ne0$ choose
-$|z|=1$ with $z\tau=|\tau|$ and replace $W_0\to zW_0$; this
-multiplies $R_0,R_1$ on the right by unitary phase diagonals,
-preserving every cost. Applying the previous bound gives
-$6|\tau|\le5D_*Q\le10D_*\mathcal E$, which is (4.11).
-\end{proof}
-
-% For complex $D_v$, split $D_v=D_v^{(0)}+iD_v^{(1)}$ into
-% real and imaginary parts, both Hermitian, of operator norm at
-% most $\|D_v\|_{\op}$. Expanding gives eight products with
-% Hermitian diagonals. Each is bounded by (4.11) with $D_*$
-% unchanged; each cost occurs four times across the eight.
-% The result is
-\begin{equation}\label{eq:complex-mixed-trace}
- |\tr(D_0W_0D_1W_1D_2W_2)|
- \le\frac{20}{3}D_*\sum_{v=0}^2\sum_{\epsilon=0}^1
- b_{T_v}(R_v^*D_v^{(\epsilon)}R_v) .
-\end{equation}
-% All directions on the right remain Hermitian.
-
-% That concludes the dimension-free matrix engine. What matters
-% for the sequel: the cyclic product is controlled by a sum of
-% three positive quadratic costs, one for each vertex.
-
-
-% ============================================================
-\section{Gaussian heat flow}\label{sec:heat}
-% ============================================================
-
-% The matrix costs from Section~\ref{sec:matrix} are second
-% derivatives. The heat flow turns those second derivatives
-% into derivatives in a scale parameter s, so that integrating
-% over s controls the accumulated costs by initial and final
-% energies. This is the "energy method": we pay the quadratic
-% costs by heat dissipation.
-
-% Setup. Fix a finite collection of vertices u_i=ih with
-% -N<=i<=N and h>0. Vertex indices v range over Z/3. Edge
-% arrays A_v(i,j) are finite, and we set n_v = (sum h^2
-% |A_v(i,j)|^3)^{1/3}. Centers p_v are real; variances sigma_v
-% positive. Weights:
-\begin{equation}\label{eq:weights}
- g_\sigma(t)=(2\pi\sigma)^{-1/2}e^{-t^2/(2\sigma)},\qquad
- w_v(i)=h g_{\sigma_v}(u_i-p_v),\qquad
- W_v(i,j)=A_v(i,j)\sqrt{w_v(i)w_{v+1}(j)} .
-\end{equation}
-% The star at vertex v has two blocks
-\begin{equation}\label{eq:heat-blocks}
- C_{v,v+1}=W_v,\quad C_{v,v-1}=W_{v-1}^*,\quad
- R_v=[C_{v,v+1}\ C_{v,v-1}] ,
-\end{equation}
-% and the weights enter two matrix cubes through
-$e_v=\Phi(T_v)=\Phi(S_v)$ on the corresponding Gram.
-% The Gaussian scores and their insertions are
-\begin{equation}\label{eq:heat-scores}
- N_v=\operatorname{diag}_i\frac{u_i-p_v}{\sigma_v},\quad
- U_v=R_v^*N_vR_v,\quad
- P_{v,w}=C_{v,w}N_wC_{v,w}^* .
-\end{equation}
-% The QUADRATIC COSTS we want to dissipate are
-\begin{equation}\label{eq:heat-costs}
- V_v=b_{T_v}(U_v),\quad
- Y_{v,w}=b_{S_v}(P_{v,w}),\quad
- Z_v=\mathcal B_{S_v}(P_{v,v+1},P_{v,v-1}) .
-\end{equation}
-% All insertions are supported on their own Gram, since a
-% sandwiched matrix $B^*DB$ annihilates $\ker B$ both ways.
-
-% The plane over which we integrate is $\Pi_d=\{p_0+p_1+p_2=d\}$,
-% with measure inherited from any pair of centers (both changed
-% to be the free coordinates; the Jacobian is always one).
-
-% We first record the heat identities. These relate the
-% derivative in s to center derivatives and the costs.
-\begin{lemma}[Heat identities]\label{lem:heat-identities}
-For fixed finite arrays and any positive rates $\alpha_v>0$,
-evaluated at $\sigma_i=\alpha_is$, $e_v$ is smooth in the
-centers and in $s$. On $\Pi_d$,
-\begin{equation}\label{eq:heat-identity}
- 2\partial_se_v=\sum_i\alpha_i\partial_{p_i}^2e_v
- -\alpha_vV_v-\sum_{w\ne v}\alpha_wY_{v,w} ,
-\end{equation}
-and
-\begin{equation}\label{eq:heat-mixed}
- \partial_{p_{v+1}}\partial_{p_{v-1}}e_v=Z_v .
-\end{equation}
-The plane integrals $J_v(d,s)=\int_{\Pi_d}e_v\,\dd\pi_d$ are
-finite, smooth, and satisfy
-\begin{equation}\label{eq:heat-plane-identities}
- \begin{aligned}
- 2\partial_sJ_v
- &=\Sigma_\alpha\partial_d^2J_v
- -\int_{\Pi_d}\Big(\alpha_vV_v+\sum_{w\ne v}\alpha_wY_{v,w}\Big)\dd\pi_d,\\
- \partial_d^2J_v&=\int_{\Pi_d}Z_v\,\dd\pi_d ,
- \end{aligned}
-\end{equation}
-with $\Sigma_\alpha=\sum\alpha_i$. On any compact
-$s$-interval $[s_0,s_1]\subset(0,\infty)$, every such plane
-integral grows at most polynomially in $|d|$.
-\end{lemma}
-\begin{proof}
-The support issue: each star factors as
-$R_v=D_{\rm row}R_v^0D_{\rm col}$ with positive invertible
-diagonal multipliers, so $\ker R_v$, $\supp T_v$,
-$\ran R_v$, $\supp S_v$ are locally fixed. Smoothness reduces
-to smoothness of the positive eigenvalues of a product of two
-positive matrices, which is standard. The rank-zero case is
-trivial.
-
-The Gaussian identities
-$\partial_pw=Nw$, $\partial_p^2w=(N^2-\sigma^{-1})w$,
-$2\partial_\sigma w=\partial_p^2 w$ combined with the chain
-rule on $e_v$, restricted to its support, give (5.6) and (5.7).
-Dominating by polynomial times $e_v$ and using
-$\int_{\Pi_d}e_v\,\dd\pi_d\le$ polynomial in $|d|$, we can
-differentiate under the integral. Choosing $p_i$ as dependent
-coordinate yields $\partial_d^2J_v=\int\partial_{p_i}^2e_v\dd\pi_d$;
-choosing $p_{v+1}$ and then $p_{v-1}$ yields
-$\partial_d^2J_v=\int\partial_{p_{v+1}}\partial_{p_{v-1}}e_v\dd\pi_d$.
-Then (5.6) and (5.7) give (5.8).
-\end{proof}
-
-
-\subsection{Strict dissipation at comparable variances}
-\label{subsec:dissipation}
-
-% The mixed term $Z_v$ does not have a sign. We trade it for
-% column costs and then compare them to the row costs at the
-% neighboring vertex. The loss is $\sqrt2$, small enough that
-% strict dissipation survives when the three variance rates
-% are comparable.
-
-\begin{lemma}[Plane dissipation]\label{lem:plane-dissipation}
-For finite arrays and any positive rates,
-\begin{equation}\label{eq:heat-plane-dissipation}
- 2\partial_sJ(d,s)\le\int_{\Pi_d}\sum_w
- \Big[-\alpha_w+\sqrt2\max\{0,\Sigma_\alpha/2-\alpha_w\}\Big]
- V_w\,\dd\pi_d .
-\end{equation}
-In particular, put $\lambda=11/10$. For each permutation of
-$(1,\lambda,\lambda)$,
-\begin{equation}\label{eq:heat-strict}
- -\partial_sJ(d,s)\ge c_*\int_{\Pi_d}\sum_wV_w\,\dd\pi_d,
- \qquad c_*=\tfrac12\min\{1-\tfrac35\sqrt2,\tfrac{11}{10}-\tfrac12\sqrt2\}>0 .
-\end{equation}
-\end{lemma}
-\begin{proof}
-The core inequality is the neighbor comparison
-\begin{equation}\label{eq:heat-neighbors}
- Y_{w+1,w}+Y_{w-1,w}\le\sqrt2\,V_w .
-\end{equation}
-Let $A=W_w$, $B=W_{w-1}^*$, $N=N_w$. The elementary inequality
-$\|Az+Bz'\|^2\le 2\|Az\|^2+2\|Bz'\|^2$ implies
-$T_w\le2Q$ with $Q=\operatorname{diag}(A^*A,B^*B)$. By
-Lemma~\ref{lem:order},
-$V_w\ge2^{-1/2}b_Q([A\ B]^*N[A\ B])$. In a block-preserving
-eigenbasis of $Q$, discarding off-diagonal blocks gives
-$b_Q([A\ B]^*N[A\ B])\ge b_{A^*A}(A^*NA)+b_{B^*B}(B^*NB)$.
-Then $A^*A\le S_{w+1}$ with insertion $P_{w+1,w}$, and
-$B^*B\le S_{w-1}$ with insertion $P_{w-1,w}$. A second use of
-Lemma~\ref{lem:order} with comparison factor one proves
-(5.10). Summing (5.8) over $v$ and using
-$2Z_v\le Y_{v,v+1}+Y_{v,v-1}$ gives (5.9); the two column
-costs carrying the score at vertex $w$ have common coefficient
-$\Sigma_\alpha/2-\alpha_w$. For the stated rates $\Sigma=16/5$;
-the two coefficients are $-1+(3/5)\sqrt2$ and $-11/10+(1/2)\sqrt2$,
-both negative. This gives (5.10).
-\end{proof}
-
-\subsection{Energy bounds and single-edge dissipation}
-\label{subsec:energy-bounds}
-
-% The initial energy is bounded by the discrete $L^3$ norms
-% once the Gaussian variance is at least the mesh square.
-\begin{lemma}[Initial energy]\label{lem:initial-energy}
-For all $d\in\R$ and all variances $\sigma_v\ge h^2$,
-\begin{equation}\label{eq:initial-energy}
- \begin{gathered}
- \int_{\Pi_d}\|W_v\|_{\HS}^3\,\dd\pi_d\le M n_v^3 ,
- \qquad M=1+\sqrt{2/\pi},\\
- \int_{\Pi_d}\sum_ve_v\,\dd\pi_d\le2\sqrt2\,M\sum_v n_v^3 .
- \end{gathered}
-\end{equation}
-The bounds also hold after replacing entries by smaller ones
-in absolute value.
-\end{lemma}
-\begin{proof}
-The Gaussian grid mass obeys
-\begin{equation}\label{eq:heat-mesh-mass}
- \sum_i h g_\sigma(u_i-p)\le 1+\frac{2h}{\sqrt{2\pi\sigma}} .
-\end{equation}
-For $\sigma\ge h^2$, the right side is at most $M$. Weighted
-Holder gives
-$\|W_v\|_{\HS}^3\le M\sum|A_v|^3w_vw_{v+1}$.
-Using $p_v,p_{v+1}$ as free coordinates on $\Pi_d$ and
-integrating gives the first estimate; summing over $v$ and
-double-counting each edge once in each star gives the second.
-\end{proof}
-
-% When variance rates become very unequal, summed star
-% dissipation no longer holds. But a SINGLE EDGE still
-% dissipates exactly, because its star has no mixed term.
-\begin{lemma}[Single-edge energy]\label{lem:single-edge-energy}
-Fix $v\ne w$ and retain only the weighted block $C=C_{v,w}$
-at vertex $v$. For arbitrary positive rates, set
-\[
- e^e_{v,w}=\Phi(CC^*), \quad
- V^e_{v,w}=b_{C^*C}(C^*N_vC), \quad
- Y^e_{v,w}=b_{CC^*}(CN_wC^*),
-\]
-\[
- J^e_{v,w}(s)=\iint_{\R^2}e^e_{v,w}\,\dd p_v\dd p_w .
-\]
-The plane integrals of $e^e,V^e,Y^e$ are independent of $d$,
-and
-\begin{equation}\label{eq:heat-single-edge}
- -2\partial_sJ^e_{v,w}=\iint_{\R^2}
- (\alpha_vV^e_{v,w}+\alpha_wY^e_{v,w})\,\dd p_v\dd p_w .
-\end{equation}
-With $n_e$ the edge array's $L^3$ norm, whenever
-$\alpha_vs_0,\alpha_ws_0\ge h^2$ and $s_1>s_0$,
-\begin{equation}\label{eq:heat-single-edge-integral}
- \int_{s_0}^{s_1}\iint_{\R^2}
- (\alpha_vV^e_{v,w}+\alpha_wY^e_{v,w})\,\dd p_v\dd p_w\dd s
- \le2M n_e^3 .
-\end{equation}
-For the full two-block star, $Y_{v,w}\le Y^e_{v,w}$.
-\end{lemma}
-\begin{proof}
-The integrands depend only on the two endpoint centers; use
-those as free coordinates. In the one-edge star the other
-column block is zero, so $Z_v=0$. After permuting blocks the
-matrices are
-$R=[C\ 0]$, $T=\operatorname{diag}(C^*C,0)$,
-$U=\operatorname{diag}(C^*N_vC,0)$, $S=CC^*$. The Hessian
-costs are exactly $V^e_{v,w}$ and $Y^e_{v,w}$; (5.8) gives
-(5.12). Since $e^e_{v,w}\le\|C\|_{\HS}^3$, the initial-energy
-lemma bounds $J^e_{v,w}(s_0)\le Mn_e^3$; integrate (5.12) and
-use that the terminal energy is nonnegative. Finally $CC^*\le S_v$
-and $CN_wC^*$ is supported on $CC^*$, so Lemma~\ref{lem:order}
-with factor one bounds $Y_{v,w}\le Y^e_{v,w}$.
-\end{proof}
-
-% With the two dissipation lemmas in hand, the heat machinery
-% is complete. We can now turn to the main smooth-annulus
-% argument.
-
-
-% ============================================================
-\section{Smooth annuli and repeated masks}\label{sec:smooth}
-% ============================================================
-
-% We prove the smooth-annulus version of the count estimate.
-% Same structure: at most n disjoint scale intervals at each
-% output point, complex coefficients $|\beta|\le1$, smooth
-% derivative windows. The result costs $\sqrt n$ but no more.
-
-% Define the smooth kernels and their scale-integrals by
-\begin{equation}\label{eq:smooth-kernels}
- k_s(t)=\frac{t}{\sqrt s}g_s(t),\qquad
- I_{\varepsilon,R}(t)
- =\int_{\varepsilon^2}^{R^2}(k_s*k_s*k_s)(t)\,\frac{\dd s}{s} .
-\end{equation}
-
-\begin{proposition}[Smooth annuli with repeated choices]
-\label{prop:smooth-switches}
-Let $G_v\in C_c^\infty(\R^2)$ be complex. At each output point
-$(a,b)$, make a step choice of at most $n\ge1$ disjoint radius
-intervals $(\varepsilon_j,R_j)$ and coefficients $|\beta_j|\le1$.
-Then for $\mathcal K_{a,b}(t)=\sum_j\beta_jI_{\varepsilon_j,R_j}(t)$,
-\[
- |\Lambda_{\mathcal K}(G_0,G_1,G_2)|
- \le C\sqrt n\prod_{v=0}^2\|G_v\|_3 .
-\]
-The constant is absolute and independent of the step partition,
-the menu, and all endpoints.
-\end{proposition}
-
-\subsection{The cost of the switches}
-\label{subsec:switch-cost}
-
-% Work on the finite grid of the heat section. The masked dual
-% array (choosing which annuli are active at each (i,j))
-% is
-\begin{equation}\label{eq:repeated-mask}
- A_0^s(i,j)=A_0(i,j)\sum_\ell\beta_\ell
- \mathbf 1_{\{\varepsilon_\ell^2<s<R_\ell^2\}} .
-\end{equation}
-% The sum runs through the list chosen at THIS index pair.
-% Keep $A_1,A_2$ fixed.
-
-% The key inputs are that the mask stays bounded by $|A_0|$ and
-% that at every switch, the total variation is linear in n.
-\begin{equation}\label{eq:mask-switch-count}
- |A_0^s(i,j)|\le|A_0(i,j)|,\qquad
- \sum_\tau|A_0^{\tau+}(i,j)-A_0^{\tau-}(i,j)|\le2n|A_0(i,j)|.
-\end{equation}
-
-\begin{lemma}[Repeated-mask dissipation]\label{lem:repeated-dissipation}
-With masked arrays above and variances $\sigma_v=\alpha_vs$ where
-$\boldsymbol\alpha$ is a permutation of $(1,11/10,11/10)$,
-\begin{equation}\label{eq:repeated-dissipation}
- \int_{s_-}^{s_+}\int_{\Pi_0}\sum_vV_v\dd\pi_0\dd s
- \le C\Big(\sum_vn_v^3+n(n_0^3+n_0^2(n_1+n_2))\Big).
-\end{equation}
-\end{lemma}
-\begin{proof}
-Between consecutive switch scales the arrays are fixed, so
-\eqref{eq:heat-strict} applies. The only new work is bounding
-the total absolute energy jump.
-
-At a switch $\tau$, freeze the weights and set
-$\delta A(i,j)=A_0^{\tau+}(i,j)-A_0^{\tau-}(i,j)$.
-Interpolate all changed entries linearly; convexity keeps
-moduli bounded by $|A_0|$. For fixed centers and index, define
-the abbreviations
-$X^2=\sum_m|A_0(i,m)|^2w_1(m)$,
-$Y^2=\sum_m|A_0(m,j)|^2w_0(m)$,
-$H^2=\sum_m|A_2(m,i)|^2w_2(m)$,
-$Q^2=\sum_m|A_1(j,m)|^2w_2(m)$.
-In $R_0=[W_0\ W_2^*]$, the changing entry has row norm at most
-$\sqrt{w_0(i)}(X+H)$, column norm $\sqrt{w_1(j)}Y$,
-derivative $|\delta A(i,j)|\sqrt{w_0(i)w_1(j)}$; the adjoint
-in $R_1=[W_1\ W_0^*]$ has row and column bounds
-$\sqrt{w_1(j)}(Y+Q)$ and $\sqrt{w_0(i)}X$. The star at vertex 2
-is untouched. Lemma~\ref{lem:rank-change} gives
-\[
- |\Delta e_0|+|\Delta e_1|\le C\sum_{i,j}w_0(i)w_1(j)|\delta A(i,j)|
- (2XY+HY+QX).
-\]
-The measure $h^{-2}w_0(i)w_1(j)\dd\pi_0$ is a probability measure,
-under which $p_0,p_1$ are independent normals with means
-$u_i,u_j$ and variances $\sigma_0,\sigma_1$, and $p_2=-p_0-p_1$.
-Adding Gaussian variances gives
-$\mathbb E w_0(m)=hg_{2\sigma_0}(u_m-u_i)$,
-$\mathbb E w_1(m)=hg_{2\sigma_1}(u_m-u_j)$,
-$\mathbb E w_2(m)=hg_{\sigma_0+\sigma_1+\sigma_2}(u_m+u_i+u_j)$, and
-Lemma~\ref{lem:line-maximal} bounds the $\mathbb E$ of
-$X^2,Y^2,H^2,Q^2$ by constant multiples of
-$X_*^2=M_{\mathbb Z}(|A_0(i,\cdot)|^2)(j)$,
-$Y_*^2=M_{\mathbb Z}(|A_0(\cdot,j)|^2)(i)$,
-$H_*^2=M_{\mathbb Z}(|A_2(\cdot,i)|^2)(-i-j)$,
-$Q_*^2=M_{\mathbb Z}(|A_1(j,\cdot)|^2)(-i-j)$,
-all independent of the switch scale. Cauchy--Schwarz under the
-probability measure controls the three products. Applying the
-$L^{3/2}$ line maximal inequality on slices gives (via
-bijection $j\mapsto-i-j$ on relevant slices) that the three
-$L^3$ norms of the majorants are at most $Cn_0,Cn_2,Cn_1$.
-Integrating, summing the switches using
-\eqref{eq:mask-switch-count}, and H\"older on the grid gives
-the total jump cost
-$\sum_\tau|\Delta\sum_vJ_v(0,\tau)|\le Cn(n_0^3+n_0^2(n_1+n_2))$.
-Discarding the terminal energy and adding the initial-energy
-bound from Lemma~\ref{lem:initial-energy} finishes the proof.
-\end{proof}
-
-
-\subsection{From the energy to the smooth kernel}
-\label{subsec:energy-to-kernel}
-
-\begin{proof}[Proof of Proposition~\ref{prop:smooth-switches}]
-Put $\lambda=11/10$ and use base variances $\lambda s$ at every
-vertex. In Lemma~\ref{lem:mixed-trace}, insert the diagonal matrix
-with entries $k_s(u_i-p_v)/g_{\lambda s}(u_i-p_v)$ at vertex $v$.
-It is Hermitian and uniformly bounded: the polynomial factor in
-$k_s$ is absorbed by the extra Gaussian decay of $g_{\lambda s}$
-relative to $k_s$.
-
-To handle the cost at vertex $v$, narrow just its row variance to
-$s$. Write $T_v^{\rm b}$ for the base Gram and $T_v^{\rm n}$ for
-the narrowed-row Gram. Then
-$(R_v^{\rm b})^*D_vR_v^{\rm b}=\sqrt s\,U_v^{\rm n}$ and
-$T_v^{\rm n}\le\sqrt\lambda\,T_v^{\rm b}$. Row scaling preserves
-support, so Lemma~\ref{lem:order} bounds the base cost by
-$CsV_v^{\rm n}$. The narrowed variances are one of the
-permutations allowed in Lemma~\ref{lem:repeated-dissipation}.
-
-Integrate the mixed trace inequality over base centers in $\Pi_0$
-and over $s$ with measure $\dd s/s$. Its right side is controlled
-by \eqref{eq:repeated-dissipation}. On the left the density at
-each vertex is $hk_s(u_i-p_v)$; integration over the center plane
-convolves the three windows, giving
-\[
- h^3\sum_{i,j,l}A_0(i,j)A_1(j,l)A_2(l,i)
- \sum_{\text{chosen at }(i,j)}\beta\,
- I_{\varepsilon,R}(u_i+u_j+u_l).
-\]
-If the original norms are nonzero, apply this to arrays normalized
-to have norms $n^{-1/2},1,1$. The right side is bounded by an
-absolute constant because $n(n^{-3/2}+2n^{-1})$ is bounded.
-Rescaling gives $C\sqrt n\prod_vn_v$. The Riemann passage of the
-preliminaries gives the proposition for smooth compactly supported
-inputs.
-\end{proof}
-
-\subsection{The hard endpoint identity}
-\label{subsec:hard-endpoint}
-
-% Smooth annuli differ from hard Hilbert annuli by a
-% difference of two odd endpoints. We record the exact
-% identity now so that the next two sections can estimate
-% the endpoint errors together.
-%
-% Put
-% psi = k_1*k_1*k_1 = -g_3''',
-% P(v) = 2int_0^v psi(z)dz (v>=0),
-% c_0 = P(inf) = 2g_3''(0) != 0.
-% Then P has a smooth even extension vanishing to second order
-% at zero. For rho>0 define
-\begin{equation}\label{eq:endpoint-kernel}
- E_\rho(t)=\frac{P(|t|/\rho)-c_0\mathbf 1_{\{|t|>\rho\}}}{t},
-\end{equation}
-% with the continuous value zero at $t=0$. Gaussian scaling and
-% the substitution $z=|t|/\sqrt s$ in (6.1) give, away from
-% the endpoint values,
-\begin{equation}\label{eq:hard-smooth}
- I_{\varepsilon,R}(t)
- =\frac{c_0}{t}\mathbf 1_{\{\varepsilon<|t|<R\}}
- +E_\varepsilon(t)-E_R(t) .
-\end{equation}
-% This is the BRIDGE between smooth and hard annuli. The hard
-% kernel is $c_0/t$ on the annulus $(\varepsilon,R)$; the
-% two endpoint errors $E_\varepsilon$ and $E_R$ account for the
-% Gaussian tail. Each $E_\rho$ is an odd dilate of one fixed
-% kernel with two jumps at $\pm\rho$.
-
-% So the hard count estimate reduces to two problems.
-% (i) The smooth-annuli estimate of Proposition~\ref{prop:smooth-switches}
-% bounds the correct $c_0/t$ piece after adding the
-% endpoint errors.
-% (ii) A frequency-block estimate, developed next, controls
-% the endpoint errors with square-root loss.
-
-
-% ============================================================
-\section{Uniform frequency-block estimate}\label{sec:frequency}
-% ============================================================
-
-% The second main new ingredient: a Fourier estimate for
-% oscillatory Gaussian windows, with coefficients that can
-% depend jointly on scale, frequency, and output point. Its
-% decisive hypothesis is a POINTWISE square-integral bound over
-% scales and frequencies. No sum of frequency suprema is
-% required.
-
-% Setup. Fix q>0, D>=1, and let Xi_D subset [-2D,2D] be
-% measurable with measure dxi/D, so mu(Xi_D)<=4. The frequency
-% windows and their convolutions are
-\begin{equation}\label{eq:frequency-windows}
- \ell_\xi(z)=D^{-1}\frac{\dd}{\dd z}
- \bigl(g_q(z)e^{i\xi z}\bigr),\qquad
- H_\xi=\ell_\xi*\ell_\xi*\ell_\xi .
-\end{equation}
-% The parameters q,D are suppressed in this notation.
-
-\begin{proposition}[Frequency-block estimate]\label{prop:frequency-block}
-Let $\mathcal I\subset\mathbb Z$ be finite and nonempty, and
-$L_k=2^k$. Let $A_0,A_1,A_2$ be finite arrays on the lattice
-$u_i=ih$ with the $L^3$ norms $n_v$ defined as in
-Section~\ref{sec:heat}. Suppose measurable coefficients
-$\mu_{ij,k}:\Xi_D\to\C$ satisfy
-\begin{equation}\label{eq:frequency-coefficient-condition}
- |\mu_{ij,k}(\xi)|\le C_1,\qquad
- \sum_{k\in\mathcal I}\int_{\Xi_D}
- |\mu_{ij,k}(\xi)|^2\,\dd\gamma(\xi)\le C_1^2,
- \quad \dd\gamma=\dd\xi/D,
-\end{equation}
-for every index pair $(i,j)$, the first inequality holding
+
+ Phi(R_+^* R_+) - Phi(R_-^* R_-)
+ = 3 int_0^1 Re < R(z) sqrt(R(z)^* R(z)), D >_{HS} dz.
+
+For every entry (i,j) of any rectangular R,
+
+ | (R sqrt(R^* R))_{i j} |
+ <= || row_i(R) ||_2 || col_j(R) ||_2 .
+
+Proof. For eta > 0, differentiable square roots on the support give
+
+ d/dz tr( (R(z)^* R(z) + eta I)^{3/2} )
+ = 3 Re < R(z) sqrt(R(z)^* R(z) + eta I), D >_{HS}.
+
+Spectral calculus gives || sqrt(R^* R + eta I) - sqrt(R^* R) ||_{op}
+<= sqrt(eta). The matrices R(z) stay bounded for 0 <= z <= 1, so
+the derivatives converge uniformly as eta -> 0, and the endpoint
+energies converge. Integrating first and passing to the limit
+produces the derivative formula. The entry bound is Cauchy-Schwarz
+in the matrix product: entry (i,j) is bounded by the norm of row i
+of R times the norm of column j of sqrt(R^*R); the j-th column of
+the square root has squared norm (R^* R)_{j j} = || col_j(R) ||^2.
+The general gradient formula for unitarily invariant matrix
+functions is given in Lewis (J. Convex Analysis 2, 1995), of which
+the present statement is a special case.
+
+ ./lewis-1995-jca02012-convex-analysis-unitarily-invariant.pdf
+
+The cyclic mixed-trace inequality is the whole point of collecting
+the two incident blocks into one row matrix. For Hermitian
+D_v : H_v -> H_v, put D_* = max_v ||D_v||_{op}. Then
+
+ | tr( D_0 W_0 D_1 W_1 D_2 W_2 ) |
+ <= (5/3) D_* sum_{v=0}^{2} b_{T_v}( R_v^* D_v R_v ).
+
+The constant 5/3 is dimension-free, and no Gram is assumed
+invertible.
+
+Proof. There are three stages.
+
+Stage one, one quadratic form. On H = H_0 + H_1 + H_2, set
+
+ M = [ 0 W_0 W_2^* ;
+ W_0^* 0 W_1 ;
+ W_2 W_1^* 0 ],
+ D = diag(D_0, D_1, D_2).
+
+Both are Hermitian and ||D||_{op} = D_*. Let P_v be the orthogonal
+projection of H onto H_v, and set A_v = M P_v M, K_v = M P_v D M.
+After permuting blocks and extending by zero, A_v = T_v and
+K_v = R_v^* D_v R_v. Because P_v commutes with D, K_v is
+Hermitian. Also ker A_v = ker(P_v M) is contained in ker K_v and
+0 <= A_v <= M^2. Lemma-order together with the inequality
+sum_v b(X_v) >= (1/3) b(sum_v X_v) for a positive quadratic form
+gives
+
+ E := sum_v b_{T_v}(R_v^* D_v R_v)
+ >= sum_v b_{M^2}(K_v)
+ >= (1/3) b_{M^2}(M D M).
+
+Stage two, cubic trace estimate. Choose an orthonormal eigenbasis
+m_i of M with q_i = |m_i| nonincreasing. The entry formula yields
+
+ E >= (1/2) Q, Q := sum_{i,j : q_i + q_j > 0}
+ q_i^2 q_j^2 |D_{ij}|^2 / (q_i + q_j).
+
+Set A_i = q_i^3 |D_{ii}|^2, E_i = q_i sum_{j > i} q_j^2 |D_{ij}|^2,
+A = sum A_i, E = sum E_i. Hermitian symmetry and the ordering
+q_j <= q_i for j > i give
+
+ Q = A/2 + 2 sum_{i < j, q_i + q_j > 0}
+ q_i^2 q_j^2 |D_{ij}|^2 / (q_i + q_j)
+ >= A/2 + E.
+
+Expand tr((D M)^3) = sum_{i,j,k} m_i m_j m_k D_{ij} D_{jk} D_{ki}
+and group the triples by the multiplicity of the smallest index i.
+If i occurs once, its three cyclic positions give
+3 m_i sum_{j,k > i} m_j m_k D_{ij} D_{jk} D_{ki}; the inner sum is
+x^* D_{>i} x where x = (m_j D_{ji})_{j > i} and D_{>i} is the
+compression of D, of operator norm at most D_*. Its modulus is at
+most 3 D_* E_i. If i occurs twice, the contribution is
+3 m_i^2 D_{ii} sum_{k > i} m_k |D_{ik}|^2; Cauchy-Schwarz and
+sum_k |D_{ik}|^2 = ||D e_i||^2 <= D_*^2 give modulus at most
+3 D_* sqrt(A_i E_i). If all three indices are i, the modulus is at
+most D_* A_i. Summing over i and using Cauchy-Schwarz,
+
+ |tr((D M)^3)|
+ <= D_* (A + 3 E + 3 sqrt(A E))
+ <= D_* (5/2 A + 9/2 E)
+ <= 5 D_* (A/2 + E) <= 5 D_* Q.
+
+Stage three, complex phase. Set tau = tr(D_0 W_0 D_1 W_1 D_2 W_2).
+M has zero diagonal blocks, so every nonzero closed three-step block
+product visits all three vertices. The three starting points in one
+orientation give tau, and the reverse orientation gives conjugate
+tau; hence tr((D M)^3) = 6 Re(tau). If tau is nonzero, choose a unit
+modulus z with z tau = |tau| and replace W_0 by z W_0. This right
+multiplies R_0 and R_1 by unitary phase diagonals, which conjugate
+their Gram and insertion by the same unitary; the costs and D_* are
+unchanged. Applying the bounds above to the modified matrices
+yields
+
+ 6 |tau| <= 5 D_* Q <= 10 D_* E.
+
+This proves the inequality. The case tau = 0 is immediate.
+
+Complex insertions. When D_v is complex diagonal, split
+D_v = D_v^{(0)} + i D_v^{(1)} with both parts diagonal and Hermitian
+of operator norm at most ||D_v||_{op}. The cyclic trace expands into
+eight products with Hermitian diagonals, each bounded above with D_*
+unchanged; the six costs of each part occur four times in the sum,
+giving
+
+ | tr(D_0 W_0 D_1 W_1 D_2 W_2) |
+ <= (20/3) D_* sum_{v=0}^{2} sum_{eps=0}^{1}
+ b_{T_v}( R_v^* D_v^{(eps)} R_v ).
+
+Every direction on the right stays Hermitian.
+
+The heat flow attaches Gaussian weights to the matrix entries and
+differentiates the resulting scale-dependent energy.
+
+Fix h > 0, N >= 0, and nodes u_i = i h for -N <= i <= N. Vertex
+indices v range over Z/3. For finite complex edge arrays A_v, put
+
+ n_v = ( sum_{i,j} h^2 |A_v(i,j)|^3 )^{1/3} .
+
+For centers p_v in R and variances sigma_v > 0, define
+
+ g_sigma(t) = (2 pi sigma)^{-1/2} exp( -t^2 / (2 sigma) ),
+ w_v(i) = h g_{sigma_v}(u_i - p_v),
+ W_v(i,j) = A_v(i,j) sqrt( w_v(i) w_{v+1}(j) ).
+
+The star at vertex v has two blocks:
+
+ C_{v,v+1} = W_v, C_{v,v-1} = W_{v-1}^*,
+ R_v = [ C_{v,v+1} C_{v,v-1} ] .
+
+The Gaussian scores and their insertions are
+
+ N_v = diag_i (u_i - p_v) / sigma_v,
+ U_v = R_v^* N_v R_v,
+ P_{v,w} = C_{v,w} N_w C_{v,w}^* for w != v.
+
+The three quadratic costs are
+
+ V_v = b_{T_v}(U_v),
+ Y_{v,w} = b_{S_v}(P_{v,w}),
+ Z_v = B_{S_v}( P_{v,v+1}, P_{v,v-1} ).
+
+Each insertion is supported on its own Gram, because a sand-
+wiched matrix B^* D B annihilates ker B on both sides.
+
+For d in R, the plane Pi_d = { p_0 + p_1 + p_2 = d } is equipped
+with the measure d pi_d obtained by pushing d p_0 d p_1 forward
+under the parametrisation p_2 = d - p_0 - p_1. Any two centers can
+serve as free coordinates; the change of chart has absolute
+determinant one. For positive rates alpha = (alpha_0, alpha_1,
+alpha_2), evaluate the weights at sigma_i = alpha_i s and put
+
+ J_v(d,s) = int_{Pi_d} e_v d pi_d,
+ J(d,s) = sum_v J_v(d,s),
+ Sigma_alpha = sum_i alpha_i.
+
+The arrays are frozen under every scale derivative.
+
+Heat identities. For fixed finite arrays and any positive rates,
+the energies e_v are smooth in the centers and in the positive
+variances. On Pi_d, for s > 0,
+
+ 2 d_s e_v = sum_i alpha_i d_{p_i}^2 e_v
+ - alpha_v V_v - sum_{w != v} alpha_w Y_{v,w} ,
+
+ d_{p_{v+1}} d_{p_{v-1}} e_v = Z_v .
+
+The plane integrals J_v(d, s) are finite, continuously different-
+iable in s, and twice continuously differentiable in d, with
+
+ 2 d_s J_v(d, s)
+ = Sigma_alpha d_d^2 J_v(d, s)
+ - int_{Pi_d} ( alpha_v V_v
+ + sum_{w != v} alpha_w Y_{v,w} ) d pi_d,
+
+ d_d^2 J_v(d, s) = int_{Pi_d} Z_v d pi_d .
+
+On compact positive scale intervals every plane integral of a
+fixed derivative grows at most polynomially in |d|.
+
+Proof. Each star factors as R_v = D_row R_v^0 D_col with positive
+invertible diagonal multipliers. Hence ker R_v, supp T_v, ran R_v,
+and supp S_v do not move locally as the centers and variances
+vary. Smoothness of e_v reduces to smoothness of the positive
+eigenvalues of Q'^{1/2} P' Q'^{1/2}, where P' = X'^* X' and
+Q' = Y'^* Y' are smooth positive definite matrices. A standard
+result on smooth families of positive matrices covers this case;
+the rank-zero edge case follows by a limiting argument.
+
+For a Gaussian density w = h g_sigma(u - p) and its score
+N = (u - p) / sigma,
+
+ d_p w = N w,
+ d_p^2 w = (N^2 - sigma^{-1}) w,
+ 2 d_sigma w = d_p^2 w .
+
+With the column parameters fixed, T_v is linear in the row density,
+so d_{p_v} T_v = U_v and 2 d_{sigma_v} T_v = d_{p_v}^2 T_v. The
+chain rule on the fixed support, together with the Hessian cost,
+gives
+
+ 2 d_{sigma_v} e_v = d_{p_v}^2 e_v - V_v .
+
+For a column vertex w, use S_v, which is linear in that column
+density with fixed support and satisfies d_{p_w} S_v = P_{v,w}.
+The same argument gives
+
+ 2 d_{sigma_w} e_v = d_{p_w}^2 e_v - Y_{v,w} .
+
+Taking the variance derivative along sigma_i = alpha_i s gives the
+first heat identity. The two summands of
+S_v = C_{v,v+1} C_{v,v+1}^* + C_{v,v-1} C_{v,v-1}^* depend on
+different column densities, so their mixed derivative in those two
+centers vanishes; the Hessian term in the chain rule reduces to
+Z_v, producing the second identity.
+
+Before integrating, the domination needed for differentiation
+under the integral is verified. Lemma-insertion gives
+
+ V_v <= (3/4) ||N_v||_{op}^2 e_v,
+ Y_{v,w} <= (3/4) ||N_w||_{op}^2 e_v,
+ 2 |Z_v| <= Y_{v,v+1} + Y_{v,v-1} .
+
+The second inequality pads N_w by zero on the other column block.
+For any Hermitian B on the row space,
+
+ | tr( sqrt(T_v) R_v^* B R_v ) |
+ <= ||B||_{op} e_v ,
+
+because R_v sqrt(T_v) R_v^* is positive and has trace e_v. The
+column analogue uses S_v. Every first center derivative, pure
+second center derivative, mixed column derivative, and first scale
+derivative appearing above is therefore dominated by a poly-
+nomial in the centers times e_v, locally uniformly in s > 0.
+
+The energy itself satisfies
+
+ e_v <= ( ||W_v||_{HS}^2 + ||W_{v-1}||_{HS}^2 )^{3/2}
+ <= sqrt(2) ( ||W_v||_{HS}^3 + ||W_{v-1}||_{HS}^3 ) .
+
+Since the node set is finite, each edge norm cubed is bounded by a
+Gaussian in its two endpoint centers, locally uniformly in the
+variances. On Pi_d with |d| <= D, every triple of centers satisfies
+
+ p_0^2 + p_1^2 + p_2^2 <= 4 ( p_i^2 + p_j^2 ) + 3 D^2 .
+
+The bounds just established have integrable majorants on every
+chart, uniformly in (d, s) on compact sets, so differentiation
+under the plane integral is justified. Choosing p_i as the depen-
+dent coordinate gives d_d^2 J_v = int_{Pi_d} d_{p_i}^2 e_v d pi_d.
+Reparametrising with p_{v+1} dependent and differentiating once,
+then with p_{v-1} dependent and differentiating once more, gives
+
+ d_d^2 J_v
+ = int_{Pi_d} d_{p_{v+1}} d_{p_{v-1}} e_v d pi_d
+ = int_{Pi_d} Z_v d pi_d .
+
+Integrating the first heat identity gives the first plane identity.
+These comparisons concern integrated derivatives; pointwise
+equality of pure and mixed center derivatives is not asserted.
+
+Plane dissipation. For fixed finite arrays and any positive rates,
+
+ 2 d_s J(d, s)
+ <= int_{Pi_d} sum_w
+ ( -alpha_w + sqrt(2) max{ 0, Sigma_alpha/2 - alpha_w } )
+ V_w d pi_d .
+
+In particular, with lambda = 11/10, for each permutation of
+(1, lambda, lambda),
+
+ -d_s J(d, s) >= c_* int_{Pi_d} sum_w V_w d pi_d ,
+
+ c_* = (1/2) min{ 1 - (3/5) sqrt(2),
+ 11/10 - (1/2) sqrt(2) } > 0 .
+
+Proof. The neighbor comparison is
+
+ Y_{w+1, w} + Y_{w-1, w} <= sqrt(2) V_w .
+
+Write A = W_w, B = W_{w-1}^*, N = N_w. The elementary inequality
+||A z + B z'||^2 <= 2||A z||^2 + 2||B z'||^2 implies T_w <= 2Q
+with Q = diag(A^* A, B^* B). Lemma-order gives
+
+ V_w >= 2^{-1/2} b_Q( [A B]^* N [A B] ) .
+
+In a block-preserving eigenbasis of Q, discarding the off-diagonal
+blocks gives
+
+ b_Q( [A B]^* N [A B] )
+ >= b_{A^* A}( A^* N A ) + b_{B^* B}( B^* N B ) .
+
+Now A^* A <= S_{w+1} and B^* B <= S_{w-1}, with respective
+insertions P_{w+1, w} and P_{w-1, w}. Lemma-order with factor one
+completes the neighbour comparison. In the plane identity, replace
+2 Z_v by its upper bound Y_{v, v+1} + Y_{v, v-1} and sum over v.
+The two column costs carrying the score at vertex w have common
+coefficient Sigma_alpha/2 - alpha_w. Where this coefficient is
+negative, discard the terms; where positive, apply the neighbour
+comparison. This proves the first inequality. For the specific
+rates, Sigma_alpha = 16/5, and the two possible coefficients are
+-1 + (3/5) sqrt(2) and -11/10 + (1/2) sqrt(2); both are negative,
+which gives the strict dissipation.
+
+Initial energy. For all d in R and all variances sigma_v >= h^2,
+
+ int_{Pi_d} ||W_v||_{HS}^3 d pi_d <= M n_v^3 ,
+ M = 1 + sqrt(2 / pi) ,
+
+ int_{Pi_d} sum_v e_v d pi_d <= 2 sqrt(2) M sum_v n_v^3 .
+
+Both bounds continue to hold after replacing any entry of A_v by
+an entry of smaller modulus.
+
+Proof. The Gaussian grid mass satisfies
+
+ sum_i h g_sigma(u_i - p) <= 1 + 2 h / sqrt(2 pi sigma) .
+
+For u_i >= p + h the summand is comparable to the integral over the
+preceding interval of length h, and for u_i <= p - h, the integral
+over the succeeding interval. The intervals are disjoint, and at
+most two remaining nodes contribute at most 2h sup g_sigma. For
+sigma >= h^2 the right side is at most M. Weighted Holder gives
+
+ ||W_v||_{HS}^3
+ = ( sum_{i,j} |A_v(i,j)|^2 w_v(i) w_{v+1}(j) )^{3/2}
+ <= M sum_{i,j} |A_v(i,j)|^3 w_v(i) w_{v+1}(j) .
+
+Using p_v and p_{v+1} as free coordinates on Pi_d, each density
+integrates to h; this gives the first estimate. Each edge occurs
+in two stars, and summing e_v over v together with the elementary
+energy bound above gives the second.
+
+Single-edge energy. Fix v != w, keep the weighted block C = C_{v,w},
+and delete the other column block of the star at v.
+
+ e^e_{v,w} = Phi(CC^*) ,
+ V^e_{v,w} = b_{C^* C}( C^* N_v C ) ,
+ Y^e_{v,w} = b_{CC^*}( C N_w C^* ) ,
+ J^e_{v,w}(s) = int int e^e_{v,w} d p_v d p_w .
+
+The plane integrals of e^e, V^e, Y^e do not depend on d. For all
+positive rates,
+
+ -2 d_s J^e_{v,w}(s)
+ = int int ( alpha_v V^e_{v,w} + alpha_w Y^e_{v,w} )
+ d p_v d p_w .
+
+Let n_e be the discrete L^3 norm of the edge array. Whenever
+alpha_v s_0 >= h^2, alpha_w s_0 >= h^2, and s_1 > s_0,
+
+ int_{s_0}^{s_1} int int
+ ( alpha_v V^e_{v,w} + alpha_w Y^e_{v,w} )
+ d p_v d p_w d s <= 2 M n_e^3 .
+
+For the full two-block star, Y_{v,w} <= Y^e_{v,w}.
+
+Proof. The integrands depend only on the two endpoint centers, and
+those can serve as free coordinates. In the one-edge star the other
+column block is zero, so Z_v = 0. After permuting column blocks,
+
+ R = [C 0], T = diag(C^* C, 0),
+ U = diag(C^* N_v C, 0), S = CC^* .
+
+The row and column Hessian costs are exactly V^e_{v,w} and
+Y^e_{v,w}; the plane identity then yields the exact dissipation
+formula. Since e^e_{v,w} <= ||C||_{HS}^3, Lemma-initial-energy
+gives J^e_{v,w}(s_0) <= M n_e^3; integrating the dissipation and
+discarding the nonnegative terminal energy gives the integrated
+bound. Finally CC^* <= S_v and CN_w C^* is supported on CC^*, so
+Lemma-order with factor one gives Y_{v,w} <= Y^e_{v,w}.
+
+The heat annihilation above is for a single fixed list of annuli.
+Up to n annuli at every output point enter the count theorem, and
+the switch from one list to another costs something. The next
+construction shows that the switch cost is additive in n rather
+than in any higher power, and that a rescaling converts the
+additive cost into the sqrt(n) that the count theorem charges.
+
+Define the odd smooth kernel
+
+ k_s(t) = (t / sqrt(s)) g_s(t) , s > 0,
+
+and its scale integral over an annulus of squared radii,
+
+ I_{eps,R}(t) = int_{eps^2}^{R^2} (k_s * k_s * k_s)(t) ds / s .
+
+Its second derivative produces, up to constants, a rescaling of
+the truncated triangular Hilbert kernel. The exact comparison is
+the identity (hard-smooth) stated below.
+
+Smooth-switches proposition. Let G_v in C_c^infty(R^2) be
+complex. At each output pair (a, b), make a step choice of at most
+n >= 1 disjoint radius intervals (eps_j, R_j) and complex
+coefficients |beta_j| <= 1. For
+
+ K_{a,b}(t) = sum_j beta_j I_{eps_j, R_j}(t) ,
+
+one has
+
+ | Lambda_K(G_0, G_1, G_2) |
+ <= C sqrt(n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+The constant is absolute and does not depend on the step partition,
+the menu of choices, or the endpoints.
+
+The proof is on the finite grid of the heat section.
+
+For each index pair (i,j), let a finite list of disjoint annuli
+with complex coefficients be chosen independently. Abbreviate the
+masked dual array
+
+ A_0^s(i,j) = A_0(i,j) sum_l beta_l
+ 1_{ eps_l^2 < s < R_l^2 } .
+
+The sum runs through the list chosen at the particular index pair
+(i,j). The arrays A_1 and A_2 are held fixed. The mask satisfies
+
+ |A_0^s(i,j)| <= |A_0(i,j)| ,
+ sum_tau |A_0^{tau+}(i,j) - A_0^{tau-}(i,j)|
+ <= 2 n |A_0(i,j)| ,
+
+where the sum runs over the distinct switch scales in the chosen
+list, and one-sided values are used at each switch. Both
+inequalities hold for complex coefficients, including simultaneous
+openings and closings. The first is evident. The second is a
+statement about the total variation of the mask, which is bounded
+by 2n because at each scale at most n intervals can be open and
+each close or open changes the mask value by at most 1 at any
+coefficient.
+
+Switch dissipation. With the masked array, and with variance rates
+alpha a permutation of (1, 11/10, 11/10),
+
+ int_{s_-}^{s_+} int_{Pi_0} sum_v V_v d pi_0 d s
+ <= C ( sum_v n_v^3 + n ( n_0^3 + n_0^2 (n_1 + n_2) ) ) .
+
+Proof. Between consecutive switch scales the arrays are frozen, so
+the strict plane dissipation from the heat section gives
+
+ int_{s_-}^{s_+} int_{Pi_0} sum_v V_v d pi_0 d s
+ <= (1/c_*) ( J(0, s_-) - J(0, s_+) ) .
+
+The only new work is bounding the total absolute energy jump.
+
+Fix a switch tau, freeze the weights and set
+
+ dA(i,j) = A_0^{tau+}(i,j) - A_0^{tau-}(i,j) .
+
+Interpolate every changed entry linearly so that, along the
+interpolation, every coefficient is bounded in modulus by the
+fixed |A_0|. For fixed centers and one particular index, define
+
+ X^2 = sum_m |A_0(i,m)|^2 w_1(m) ,
+ Y^2 = sum_m |A_0(m,j)|^2 w_0(m) ,
+ H^2 = sum_m |A_2(m,i)|^2 w_2(m) ,
+ Q^2 = sum_m |A_1(j,m)|^2 w_2(m) .
+
+In the row matrix R_0 = [W_0 W_2^*], the changing entry lies in
+the W_0 block. Its row has norm at most sqrt(w_0(i)) (X + H), its
+column has norm sqrt(w_1(j)) Y, and its derivative under the
+mask motion has magnitude |dA(i,j)| sqrt(w_0(i) w_1(j)). The
+adjoint of the same entry lies in the W_0^* block of the row
+matrix R_1 = [W_1 W_0^*], with row and column bounds
+sqrt(w_1(j)) (Y + Q) and sqrt(w_0(i)) X. The star at vertex 2 is
+untouched. The rank-change derivative formula, Lemma-rank-change,
+gives
+
+ |Delta e_0| + |Delta e_1|
+ <= C sum_{i,j} w_0(i) w_1(j) |dA(i,j)|
+ ( 2 X Y + H Y + Q X ) .
+
+The measure h^{-2} w_0(i) w_1(j) d pi_0 is a probability measure;
+under it p_0, p_1 are independent normals with means u_i, u_j and
+variances sigma_0, sigma_1, and p_2 = -p_0 - p_1. Adding Gaussian
+variances gives
+
+ E w_0(m) = h g_{2 sigma_0}(u_m - u_i) ,
+ E w_1(m) = h g_{2 sigma_1}(u_m - u_j) ,
+ E w_2(m) = h g_{sigma_0 + sigma_1 + sigma_2}(u_m + u_i + u_j) .
+
+Lemma-line-maximal bounds the expectations of X^2, Y^2, H^2, Q^2
+by constant multiples of
+
+ X_*^2 = M_Z ( |A_0(i, .)|^2 )(j) ,
+ Y_*^2 = M_Z ( |A_0(., j)|^2 )(i) ,
+ H_*^2 = M_Z ( |A_2(., i)|^2 )(-i - j) ,
+ Q_*^2 = M_Z ( |A_1(j, .)|^2 )(-i - j) ,
+
+uniformly in the centers and in s. Cauchy-Schwarz under the
+measure gives E X Y <= X_* Y_* and the analogous bounds for the
+mixed terms. Weighted Holder with exponents (3/2, 3) on the sum
+over (i,j) and the l^{3/2} bounds for M_Z discussed in the
+preliminaries give
+
+ int_{Pi_0} ( |Delta e_0| + |Delta e_1| ) d pi_0
+ <= C |dA|_3 ( n_0 + n_1 + n_2 ) ( n_0 + n_1 + n_2 ) .
+
+Summing over the (at most 2n) switches of the chosen list and
+using the elementary l^3 triangle bound |dA|_3 <= 2 |A_0|_3 gives
+
+ sum_tau int_{Pi_0} ( |Delta e_0| + |Delta e_1| ) d pi_0
+ <= 2 n C sum_v n_v^2 ( sum_v n_v ) .
+
+The strict dissipation between switches and the initial energy
+bound at the first switch combine with the switch cost to produce
+the stated inequality. The factor n_0^2 (n_1 + n_2) is the
+extreme case where the changed edge is the third input of the
+star at vertex 0; the factor n_0^3 is absorbed into the initial
+energy; the factors n_1^3 and n_2^3 are the initial energy at the
+two fixed stars.
+
+From energy to smooth kernel. Continue on the same grid and choose
+variance rates alpha = (1, 11/10, 11/10).
+
+In Lemma-mixed-trace, insert at vertex v the diagonal matrix with
+entries
+
+ k_s(u_i - p_v) / g_{lambda s}(u_i - p_v) .
+
+This matrix is Hermitian and uniformly bounded in (i, s, p_v): the
+polynomial factor in k_s is absorbed by the extra Gaussian decay
+of g_{lambda s} relative to g_s.
+
+To pay the cost at vertex v, narrow only the row variance of that
+vertex to s. Let T_v^b be the base Gram at variance lambda s, and
+let T_v^n be the narrowed-row Gram at variance s. Then
+
+ ( R_v^b )^* D_v R_v^b = sqrt(s) U_v^n ,
+ T_v^n <= sqrt(lambda) T_v^b .
+
+Row scaling preserves support, so Lemma-order bounds the base cost
+by C s V_v^n. The narrowed variances are exactly one of the
+permutations allowed in Switch dissipation. Hence
+
+ int_{s_-}^{s_+} int_{Pi_0} ( cost at vertex v ) d pi_0 ds / s
+ <= C ( sum_v n_v^3 + n ( n_0^3 + n_0^2 (n_1 + n_2) ) ) .
+
+Now rescale the dual array: replace A_0 by A_0 / (n_0 + delta) and
+then multiply the resulting trilinear form by (n_0 + delta). The
+rescaled coefficients obey the same mask bounds with the same n,
+and trilinearity gives the desired bound
+
+ | Lambda_K(G_0, G_1, G_2) |
+ <= C sqrt(n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+The factor sqrt(n) arises as follows: the additive switch cost n
+in the dissipation is converted to a single-factor sqrt(n) after
+the A_0 rescaling, because the square root of n times the
+remaining factors n_1 + n_2 appears as (n_0^2 (n_1 + n_2))/n_0 =
+n_0 (n_1 + n_2); if all four norms are comparable, this is n^2
+divided by n_0 = n, which after rescaling gives the sqrt(n) loss
+in the final trilinear estimate.
+
+Passing to the continuous annulus kernel. The grid Riemann sums
+converge to the continuous scale integral I_{eps,R}. For fixed
+smooth compactly supported G_v, the discrete norms n_v converge
+to ||G_v||_3, the mask switches of the continuous annulus are
+finite in number, and the mesh can be chosen finer than the
+smallest gauge required by the frequency estimate. The smooth-
+switches proposition therefore holds with the same constant.
+
+The endpoint-error analysis depends on a frequency-block estimate
+whose coefficients are allowed to depend on the output point, the
+scale, and the frequency. The only requirement is a pointwise
+square-sum bound over those variables. This is the technical device
+that lets a linear pass over the endpoints cost only a sqrt(n)
+factor in the final count.
+
+Frequency-block estimate. Let I be a finite nonempty subset of Z,
+put L_k = 2^k, and let A_0, A_1, A_2 be finite arrays on the
+lattice u_i = i h with L^3 norms n_v defined as before. Let Xi_D
+be a measurable subset of [-2D, 2D] equipped with the measure
+d gamma(xi) = d xi / D, so gamma(Xi_D) <= 4. Suppose measurable
+coefficients mu_{ij,k} : Xi_D -> C satisfy
+
+ |mu_{ij,k}(xi)| <= C_1 ,
+ sum_k in I int_{Xi_D} |mu_{ij,k}(xi)|^2 d gamma(xi) <= C_1^2 ,
+
+for every output index pair (i,j), the first inequality holding
almost everywhere. If
-\begin{equation}\label{eq:frequency-mesh}
- h^2\le\frac{q\min_{k\in\mathcal I}L_k^2}{16D^2},
-\end{equation}
+
+ h^2 <= q min_k L_k^2 / (16 D^2) ,
+
then
-\begin{equation}\label{eq:frequency-finite-estimate}
- \Big|h^3\int_{\Xi_D}\sum_{k\in\mathcal I}\sum_{i,j,l}
- A_0(i,j)\mu_{ij,k}(\xi)A_1(j,l)A_2(l,i)
- \mathrm{Dil}_{L_k}H_\xi(u_i+u_j+u_l)\,\dd\gamma(\xi)\Big|
- \le C(q,C_1)\prod_{v=0}^2n_v .
-\end{equation}
-\end{proposition}
-
-% There are two reasons for refining one Gaussian variance in the
-% proof.
-% (1) A window oscillating at frequency D becomes an average
-% of score insertions at variance comparable to D^{-2}s.
-% (2) The boundary energy from changing the output array can
-% still be estimated uniformly at this small variance.
-% The refined row scores are controlled by the heat identity
-% through single-edge energies and changes of star energy at
-% slab boundaries. Such a change contains the square of the
-% changing edge norm times the norm of the unchanged edge; we
-% isolate a uniform estimate for that mixed term first.
-
-\begin{lemma}[Uniform mixed boundary estimate]
-\label{lem:frequency-boundary}
-Fix distinct $v,w\in\{0,1\}$, so $\{v,w\}=\{0,1\}$. The edge
-$v-w$ has array $B$, indexed in original order $(0,1)$, and the
-edge $v-2$ has array $E$, indexed in the order $(v,2)$. Put
-variances $(\theta s,s,s)$ with $0<\theta\le1/16$ and
-$h^2\le\theta s$. Let $\dd\nu_s(d)=g_{(1-\theta)s}(d)\dd d$.
-There is a nonnegative array $\mathcal H_E(i,j)$, independent of
-$(s,\theta)$, such that
-\begin{align}
- &\int_{\R}\int_{\Pi_d}
- \|C_{v,w}\|_{\HS}^2\|C_{v,2}\|_{\HS}\,\dd\pi_d\dd\nu_s(d)
- \le C\sum h^2|B(i,j)|^2\mathcal H_E(i,j),\label{eq:frequency-boundary-mixed}\\
- &\Big(\sum h^2\mathcal H_E(i,j)^3\Big)^{1/3}
- \le C\Big(\sum h^2|E(m,l)|^3\Big)^{1/3}.\label{eq:frequency-boundary-norm}
-\end{align}
-Here $(i,j)$ are in the original endpoint order of edge $0$;
-$i_v,i_w$ denote the same two indices in the order $(v,w)$. The
-arrays $B,E$ are extended by zero on $\mathbb Z^2$.
-\end{lemma}
-\begin{proof}
-Expand the first Hilbert--Schmidt squared. For each $(i,j)$,
-$h^{-2}w_v(i_v)w_w(i_w)\dd\pi_d\dd\nu_s(d)$ is a probability
-measure: with $p_v,p_w$ the free coordinates, it makes them
-independent normals with means $u_{i_v},u_{i_w}$ and variances
-$\theta s,s$, while $d\sim g_{(1-\theta)s}$ and $p_2=d-p_v-p_w$.
-
-Under this measure, $\mathbb E h^{-2}w_v(m)w_2(l)$ is a bivariate
-Gaussian density at $(u_m,u_l)$ with mean $(u_{i_v},-u_{i_v}-u_{i_w})$
-and covariance
-$\Sigma=s\big(\!\begin{smallmatrix}2\theta&-\theta\\-\theta&3\end{smallmatrix}\!\big)$.
-The bounds $\Sigma\le4s\operatorname{diag}(\theta,1)$,
-$\det\Sigma=\theta(6-\theta)s^2$ and the Gaussian density
-formula give
-\begin{equation}\label{eq:frequency-covariance-domination}
- \mathbb E[w_v(m)w_2(l)]
- \le Ch^2g_{C'\theta s}(u_m-u_{i_v})
- g_{C's}(u_l+u_{i_v}+u_{i_w})
-\end{equation}
-with $C'=4$ absolute.
-
-Let $M_{\mathbb Z}^{(1)},M_{\mathbb Z}^{(2)}$ be line maximal
-operators in the two coordinates. By Lemma~\ref{lem:line-maximal},
-take
-\begin{equation}\label{eq:frequency-boundary-majorant}
- \mathcal H_E(i,j)^2
- =C\bigl(M_{\mathbb Z}^{(1)}M_{\mathbb Z}^{(2)}|E|^2\bigr)
- (i_v,-i_v-i_w).
-\end{equation}
-Then $\mathbb E\|C_{v,2}\|_{\HS}^2\le\mathcal H_E(i,j)^2$ by
-(7.4) and $h^2\le\theta s$. Cauchy--Schwarz under the
-probability measure gives (7.2). The map
-$(i,j)\mapsto(i_v,-i_v-i_w)$ is a bijection $\mathbb Z^2$;
-the $\ell^{3/2}$ bounds for the two line maximal operators
-applied to $|E|^2$ give (7.3).
-\end{proof}
-
-
-\subsection{Proof of the frequency-block estimate}
-\label{subsec:freq-proof}
-
-\begin{proof}[Proof of Proposition~\ref{prop:frequency-block}]
-
-\paragraph{Reduction to pointwise bounds.}
-Discard a common null set of frequencies so that both bounds in
-\eqref{eq:frequency-coefficient-condition} hold pointwise.
-Set
-\[
- B_{0,k}^\xi(i,j)=A_0(i,j)\mu_{ij,k}(\xi),\quad
- a_k=qL_k^2,\quad \theta=\frac1{16D^2}.
-\]
-For each $\xi$, use $B_{0,k}^\xi$ on the open slab
-$2a_k<s<3a_k$, and zero outside; keep $A_1,A_2$ throughout.
-Slabs are disjoint because $a_{k+1}=4a_k$. The mesh hypothesis
-$h^2\le\theta s$ holds on the whole scale interval
-$[s_-,s_+]=[\min a_k,4\max a_k]$.
-
-\paragraph{Refining a row variance.}
-Fix a slab, put $L=L_k$, $a=a_k$, $\omega=\xi/L$. The dilated
-window is
-\[
- f_{\xi,k}(u)=\Dil_L\ell_\xi(u)
- =\frac LD\frac{\dd}{\dd u}\bigl(g_a(u)e^{i\omega u}\bigr).
-\]
-Initially all three vertices have variance $s$. The diagonal
-insertion with entries $f_{\xi,k}(u_i-p_v)/g_s(u_i-p_v)$ has
-operator norm at most $C(q)$: $|L\omega/D|\le2$, and Gaussian
-decay in $g_a/g_s$ kills the rest. Splitting into real and
-imaginary parts gives eight terms, each a Hermitian insertion of
-uniformly bounded norm (Lemma~\ref{lem:mixed-trace}).
-
-Consider the row cost at $v$. Replace row variance $s$ by
-$b'=\theta s$, keep column variances at $s$, and shift the row
-center $p_v\to p_v+d$. Denote the refined Gram and score insertion
-by $T_v^d,U_v^d$. Gaussian variance addition gives
-\[
- T_v=\int_\R T_v^d\dd\nu_s(d),\qquad
- \dd\nu_s(d)=g_{s-b'}(d)\dd d .
-\]
-All these Grams share a common support (positive row scaling on
-the kernel preserves the kernel of the row matrix).
+
+ | h^3 int_{Xi_D} sum_k in I sum_{i,j,l}
+ A_0(i,j) mu_{ij,k}(xi) A_1(j,l) A_2(l,i)
+ Dil_{L_k} H_xi(u_i + u_j + u_l) d gamma(xi) |
+ <= C(q, C_1) n_0 n_1 n_2 ,
+
+with the kernel
+
+ H_xi = ell_xi * ell_xi * ell_xi ,
+ ell_xi(z) = D^{-1} (d/dz) ( g_q(z) e^{i xi z} ) ,
+
+and Dil_L H(t) = L^{-1} H(t/L).
+
+Proof. Discard a common null set of frequencies so that both
+coefficient bounds hold pointwise. Put
+
+ B_{0,k}^xi(i,j) = A_0(i,j) mu_{ij,k}(xi) ,
+ a_k = q L_k^2 ,
+ theta = 1 / (16 D^2) .
+
+For each fixed xi and each k, use B_{0,k}^xi on the open scale slab
+2 a_k < s < 3 a_k and zero outside. The slabs are disjoint because
+a_{k+1} = 4 a_k. On the whole scale interval
+[s_-, s_+] = [min a_k, 4 max a_k] the mesh hypothesis is
+h^2 <= theta s.
+
+Fix a single slab and put L = L_k, a = a_k, omega = xi / L. The
+window at each vertex is
+
+ f_{xi,k}(u) = Dil_L ell_xi(u)
+ = (L / D) (d/du) ( g_a(u) e^{i omega u} ) .
+
+Initially all three vertices carry the variance s. The diagonal
+insertion with entries f_{xi,k}(u_i - p_v) / g_s(u_i - p_v) has
+operator norm at most C(q), uniformly for 2a < s < 3a. Indeed,
+|L omega / D| <= 2, and the remaining factor is bounded by the
+Gaussian decay of g_a relative to g_s. Splitting into real and
+imaginary parts before Lemma-mixed-trace gives eight terms, each
+an insertion of a uniformly bounded Hermitian diagonal.
+
+Consider the cost belonging to a single row star R_v. Replace its
+row variance s by b' = theta s, leave both column variances equal
+to s, and shift the row center from p_v to p_v + d. Denote the
+refined Gram and score insertion by T_v^d and U_v^d. Addition of
+Gaussian variances gives
+
+ T_v = int_R T_v^d d nu_s(d) ,
+ d nu_s(d) = g_{s - b'}(d) d d .
+
+All these Grams have a common support: changing a strictly
+positive row density amounts to invertible row scaling and leaves
+the kernel of the row matrix unchanged.
To represent the inserted Gram in the same way, put
-$\omega'=a\omega/(a-b')$. Since $b'<a/2$, completing the square
-gives the exact identity
-\[
- g_a(u)e^{i\omega u}
- =\exp\!\Big(\frac{\omega^2ab'}{2(a-b')}\Big)
- \int_\R g_{b'}(u-d)e^{i\omega'd}g_{a-b'}(d)\dd d .
-\]
-Differentiating in $u$, the derivative of $g_{b'}(u-d)$
-supplies the negative refined row score $-(u-d)/b'$. The
-prefactor is at most $C(q)$; also
-$g_{a-b'}(d)\le Cg_{s-b'}(d)$ and $L/D\le C(q)\sqrt{\theta s}$.
-Applying Lemma~\ref{lem:joint-convexity} and quadratic
-homogeneity to the resulting average gives
-\begin{equation}\label{eq:frequency-score-refinement}
- b_{T_v}(K_v)\le C(q)\theta s\int_\R
- V_v(p_0,\ldots,p_v+d,\ldots,p_2;s)\dd\nu_s(d).
-\end{equation}
-The refined variances are now $(\sigma_v,\sigma_w,\sigma_{v'})=(b',s,s)$
-with $b'<a/2$.
-
-\paragraph{Integrated refined scores.}
-Fix $v$; all stars, energies, costs refer to the slab edge $0$.
-For a center-dependent $Q$, abbreviate
-$\widetilde Q(s)=\int_\R\int_{\Pi_d}Q(p,s)\dd\pi_d\dd\nu_s(d)$.
-On any open scale interval with fixed arrays, the heat identities
-give
-\begin{equation}\label{eq:frequency-averaged-heat}
- 2\partial_s\widetilde J_v=3\widetilde Z_v-\theta\widetilde V_v
- -\sum_{w\ne v}\widetilde Y_{v,w}.
-\end{equation}
-The coefficient $3$ comes from the original second derivative
-$2+\theta$ in the plane, plus $1-\theta$ from differentiating
-$\nu_s$ and integrating by parts. Dominating by polynomials
-times $e_v$ and using Gaussian decay of $\nu_s$ justifies the
-$s$-differentiation and the two integrations by parts, and gives
-one-sided limits at slab boundaries.
-
-The metric Cauchy--Schwarz inequality $2Z_v\le\sum_{w\ne v}Y_{v,w}$
-gives
-\begin{equation}\label{eq:frequency-score-differential}
- \theta\widetilde V_v
- \le-2\partial_s\widetilde J_v+\tfrac12\sum_{w\ne v}\widetilde Y_{v,w}.
-\end{equation}
-Pay column costs by single edges: $S_v\ge C_{v,w}C_{v,w}^*$, so
-Lemma~\ref{lem:order} yields $Y_{v,w}\le Y^e_{v,w}$. The integrated
-single-edge bound (5.13) controls the total of $Y^e_{v,w}$. If the
-edge is a fixed one (say at vertices $1,2$), it costs at most
-$C(n_1^3+n_2^3)$; if it is the slab edge $0$, it costs at most
-$C\sum_k\sum_{i,j}h^2|B_{0,k}^\xi(i,j)|^3$, which after frequency
-integration by \eqref{eq:frequency-coefficient-condition} is at
-most $C(C_1)n_0^3$. All column terms are thus controlled, and the
-coefficient $1$ of the column cost makes this independent of the
-small variance ratio $\theta$.
-
-\paragraph{Summing slab boundaries.}
-Before the first slab, edge $0$ is zero, so the initial energy is
-at most $C(n_1^3+n_2^3)$ by Lemma~\ref{lem:initial-energy}. At a
-boundary of each slab, the change in the pointwise star energy at
-$v\in\{0,1\}$ is at most
-\[
- C\bigl(\|C_{\rm sw}\|_{\HS}^3+\|C_{\rm sw}\|_{\HS}^2\|C_{\rm fix}\|_{\HS}\bigr).
-\]
-The cubic term integrates by Lemma~\ref{lem:initial-energy} to at
-most $C\sum_{i,j}k\sum h^2|B_{0,k}^\xi|^3$, i.e.\ at most
-$C(C_1)n_0^3$ after frequency integration.
-
-For the mixed term, let $A^{(v,2)}$ be the fixed unweighted array
-on edge $v-2$, indexed in the order $(v,2)$. Apply
-Lemma~\ref{lem:frequency-boundary} with this array; its majorant
-$\mathcal H_v$ is independent of $(k,\xi,s)$, and its cube norm
-is at most $C(n_1+n_2)$. Both boundaries together contribute at
-most a constant times
-\[
- \sum_{i,j}h^2|A_0(i,j)|^2\mathcal H_v(i,j)
- \sum_k\int_{\Xi_D}|\mu_{ij,k}|^2\dd\gamma
- \le C(C_1)n_0^2(n_1+n_2)
-\]
-by \eqref{eq:frequency-coefficient-condition} and H\"older. This
-step is exactly where averaging coefficient squares over
-frequency matters; taking a separate frequency supremum per scale
-would not give the bound.
-
-\paragraph{Recovering the frequency-block form.}
-Integrate (7.7) on intervals between consecutive boundaries and
-telescope. Discard the terminal energy (nonnegative). The
-initial energy, the absolute jumps just estimated, and the
-single-edge costs give
-\begin{equation}\label{eq:frequency-integrated-score}
- \int_{\Xi_D}\int_{s_-}^{s_+}
- \theta\widetilde V_v\,\dd s\,\dd\gamma
- \le C(C_1)\sum_{j=0}^2n_j^3 .
-\end{equation}
-Values at the finitely many slab boundaries are immaterial.
-Measurability in $\xi$ follows from measurability of the arrays
-and Borel dependence of finite-matrix energies; Tonelli's theorem
-justifies every nonnegative frequency integration above.
-
-Return to equal-variance stars and apply Lemma~\ref{lem:mixed-trace}
-to the real and imaginary parts of the three window insertions.
-Integrate over base centers on $\Pi_0$. Shifting the row center in
-(7.6) maps this plane to $\Pi_d$; the two column centers are free
-coordinates, so its Jacobian is one. The cost at vertex $v$ is at
-most $C(q)\theta s\widetilde V_v$; integrate over every slab with
-$\dd s/s$ and over $\Xi_D$ with $\dd\gamma$; (7.8) bounds the sum.
+omega' = a omega / (a - b'). Completing the square gives the exact
+identity
+
+ g_a(u) e^{i omega u}
+ = exp( omega^2 a b' / (2 (a - b')) )
+ int_R g_{b'}(u - d) e^{i omega' d} g_{a - b'}(d) d d .
+
+The exponential prefactor is at most C(q), because
+omega^2 b' <= C(q). Also
+
+ g_{a - b'}(d) <= C g_{s - b'}(d) ,
+ L / D <= C(q) sqrt(theta s) .
+
+Differentiate the identity in u. The derivative of the first
+Gaussian supplies the negative refined row score
+-(u - d) / b'. Consequently, for either the real or the imaginary
+diagonal insertion, the associated inserted Gram K_v has the form
+
+ K_v = int_R alpha(d) U_v^d d nu_s(d) ,
+ alpha(d) = Re c(d) or Im c(d) ,
+
+with explicit scalar coefficient
+
+ c(d) = - (L / D)
+ exp( omega^2 a b' / (2 (a - b')) )
+ e^{i omega' d} g_{a - b'}(d) / g_{s - b'}(d) .
+
+The preceding bounds give |alpha(d)| <= C(q) sqrt(theta s). The
+averaged matrices are integrable, because the finite-array
+Gaussian weights dominate the polynomial score factors. Joint
+convexity and the quadratic homogeneity of b_T in its second
+argument give
+
+ b_{T_v}(K_v) <= C(q) theta s
+ int_R b_{T_v^d}(U_v^d) d nu_s(d) .
+
+Integrating over d, the s squared factor in the numerator and one
+in the denominator cancel, and the right side is theta times a
+quantity controlled by Lemma-mixed-trace.
+
+The pointwise refinement of the star cost is
+
+ theta V~_v <= -2 d_s J~_v + (1/2) sum_{w != v} Y~_{v,w} ,
+
+where tilde denotes the refined energies after integrating over
+the refining center d. Paying the column costs by single edges:
+S_v >= C_{v,w} C_{v,w}^*, so Lemma-order yields Y~_{v,w} <=
+Y^{e,~}_{v,w}. The integrated single-edge bound controls the total
+of Y^{e,~}_{v,w}. If the edge is a fixed one (at vertices 1 or 2),
+it costs at most C (n_1^3 + n_2^3). If it is the slab edge 0, it
+costs at most
+
+ C sum_k sum_{i,j} h^2 |B_{0,k}^xi(i,j)|^3 ,
+
+which, after frequency integration and using the coefficient
+condition, is at most C(C_1) n_0^3. All column terms are thereby
+controlled, and the coefficient 1 of the column cost makes this
+uniform in theta.
+
+Before the first slab the edge-0 array is zero, so the initial
+energy is at most C (n_1^3 + n_2^3). At a boundary of a slab, the
+change in pointwise star energy at v in {0, 1} is at most
+
+ C ( ||C_sv||_{HS}^3 + ||C_sv||_{HS}^2 ||C_fixed||_{HS} ) ,
+
+where C_sv is the slab-edge matrix and C_fixed is the fixed
+adjacent matrix of the star. The cubic term integrates by the
+initial-energy lemma to at most
+
+ C sum_{i,j} sum_k h^2 |B_{0,k}^xi(i,j)|^3 ,
+
+and this is C(C_1) n_0^3 after frequency integration. The mixed
+term contains ||C_sv||_{HS}^2 ||C_fixed||_{HS}; Cauchy-Schwarz
+under the frequency integration and the l^3 bound for the fixed
+adjacent matrix give
+
+ sum_{i,j} h^2 |A_0(i,j)|^2 H_v(i,j)
+ sum_k int_{Xi_D} |mu_{ij,k}(xi)|^2 d gamma(xi) ,
+
+where H_v is the boundary majorant of the fixed matrix. Using the
+coefficient condition and Holder, this is at most
+C(C_1) n_0^2 (n_1 + n_2). Taking a separate supremum in frequency
+for each scale would not give this bound. Averaging coefficient
+squares over frequency is what enforces uniformity in D.
+
+Return to equal-variance stars and apply Lemma-mixed-trace to the
+real and imaginary parts of the three window insertions. Integrate
+over base centers on Pi_0. Shifting the row center maps Pi_0 to
+Pi_d by a map of Jacobian one. The cost at vertex v is at most
+C(q) theta s V~_v; integrating over each slab with d s / s and
+over Xi_D with d gamma gives a bounded total, and the inequality
+just proved bounds that total.
On the other side, multiplying the three weighted windows in the
-trace and integrating the centers gives exactly
-\[
- h^3\sum_{i,j,l}B_{0,k}^\xi(i,j)A_1(j,l)A_2(l,i)
- \Dil_{L_k}H_\xi(u_i+u_j+u_l),
-\]
-the convolution identity on $\Pi_0$: the three window arguments
-sum to $u_i+u_j+u_l$. The expression is independent of $s$
-within its slab, whose logarithmic length is $\log(3/2)$.
-
-This proves \eqref{eq:frequency-finite-estimate} with the right
-side replaced by $C(q,C_1)\sum_vn_v^3$. If all $n_v>0$, apply
-this to $A_v/n_v$; coefficients and mesh hypothesis are unchanged,
-and trilinearity gives the product bound. If any norm is zero,
-the form is zero.
-\end{proof}
-
-
-\subsection{Continuous version}
-
-\begin{corollary}[Continuous frequency-block estimate]
-\label{cor:frequency-continuum}
-Let $q,D,\Xi_D,\mathcal I$ be as in Proposition~\ref{prop:frequency-block}.
-Suppose $\mu_{a,b,k}(\xi)$ is constant in $(a,b)$ on every cell
-of one finite rectangular partition, independent of $\xi$ and
-common to all $k$. Suppose its values on cells are bounded
-measurable functions of $\xi$ and that
-\begin{equation}\label{eq:frequency-continuous-condition}
- |\mu_{a,b,k}(\xi)|\le C_1,\qquad
- \sum_k\int_{\Xi_D}|\mu_{a,b,k}(\xi)|^2\dd\gamma\le C_1^2 .
-\end{equation}
+trace and integrating the centers yields exactly
+
+ h^3 sum_{i,j,l} B_{0,k}^xi(i,j) A_1(j,l) A_2(l,i)
+ Dil_{L_k} H_xi(u_i + u_j + u_l) ,
+
+the convolution identity on Pi_0: the three window arguments sum
+to u_i + u_j + u_l. The expression is independent of s within its
+slab, whose logarithmic length is log(3/2).
+
+This proves the frequency-block estimate with the right side
+replaced by C(q, C_1) sum_v n_v^3. If all three n_v vanish-free,
+apply the estimate to A_v / n_v; the coefficients and mesh
+hypothesis are unchanged and trilinearity gives the product
+bound. If some n_v vanishes, the trilinear form itself vanishes by
+the same rescaling.
+
+Continuous frequency-block estimate. Let q, D, Xi_D, I be as in
+the frequency-block estimate. Suppose mu_{a,b,k}(xi) is constant
+in (a,b) on every cell of one finite rectangular partition,
+independent of xi and common to all k. Suppose its values on those
+cells are bounded measurable functions of xi and that
+
+ |mu_{a,b,k}(xi)| <= C_1 ,
+ sum_k in I int_{Xi_D} |mu_{a,b,k}(xi)|^2 d gamma(xi) <= C_1^2 .
+
Define
-\begin{equation}\label{eq:frequency-continuous-kernel}
- \mathcal K_{a,b}(t)=\int_{\Xi_D}\sum_{k\in\mathcal I}
- \mu_{a,b,k}(\xi)\Dil_{L_k}H_\xi(t)\dd\gamma(\xi).
-\end{equation}
-Then for complex $G_v\in L^3(\R^2)$,
-\begin{equation}\label{eq:frequency-continuous-estimate}
- |\Lambda_{\mathcal K}(G_0,G_1,G_2)|\le C(q,C_1)\prod_{v=0}^2\|G_v\|_3 .
-\end{equation}
-\end{corollary}
-\begin{proof}
-Take $G_v$ smooth compactly supported. On each step choice the
-kernel (7.10) is continuous in $t$ (bounded block, bounded
-measurable coefficients, Gaussian common majorant). After the
-frequency integration, the trilinear integral is the limit of
-its lattice Riemann sums. The finitely many rectangular step
-boundaries have measure zero. The mesh condition (7.2) holds
-for all sufficiently fine lattices, and the discrete norms
-converge to $\|G_v\|_3$. Proposition~\ref{prop:frequency-block}
-proves (7.11) for these inputs.
-
-For general $L^3$ inputs, the fixed finite block and scale set
-admit an integrable majorant $k_*(t)\ge|\mathcal K_{a,b}(t)|$,
-uniform in $(a,b)$, obtained as a finite sum of
-$\sup_{\xi\in\Xi_D}|H_\xi|$ dilates. For any integrable $k\ge0$,
-\[
- \iiint|G_0(a,b)G_1(b,c)G_2(c,a)|k(a+b+c)\dd a\dd b\dd c
- \le\|k\|_1\prod_v\|G_v\|_3 .
-\]
-Set $t=a+b+c$, apply Holder in $(a,b)$ for fixed $t$, then use
-the determinant-one changes of variables for the other two edge
-functions. Thus the defining integral is absolutely convergent,
-is continuous in the three $L^3$ inputs, and smooth
-approximation finishes the proof. The final constant is
-independent of $D$, of the number of scales, and of the step
-choice.
-\end{proof}
-
-% ============================================================
-\section{Rough kernels and endpoint errors}\label{sec:rough}
-% ============================================================
-
-% The comparison (6.3) leaves one error kernel at each annular
-% endpoint. Naively estimating them separately would cost the
-% number of endpoints. We instead collect endpoints at
-% comparable scales, so disjointness of the annuli gives a
-% Gaussian bound independent of the number of discontinuities.
-% The frequency-block estimate from the previous section
-% controls the groups with few discontinuities; the remaining
-% groups are few enough to estimate pointwise.
-
-\subsection{A pointwise Gaussian estimate}
-\label{subsec:gaussian-envelope}
-
-% For a locally integrable function on R^2, let M_x and M_y
-% denote the centered HL maximal operators in the two
-% coordinates. For F,G in L^3, put
-\begin{equation}\label{eq:rough-maximal-envelope}
- \mathcal M_{F,G}(x,y)
- =\bigl(M_x(|F|^{5/2})(x,y)\bigr)^{2/5}
- \bigl(M_y(|G|^{5/2})(x,y)\bigr)^{2/5}.
-\end{equation}
-% The one-dimensional maximal inequality on L^{6/5}, then
-% Hölder, gives
-\begin{equation}\label{eq:rough-maximal-norm}
- \|\mathcal M_{F,G}\|_{3/2}\le C\|F\|_3\|G\|_3 .
-\end{equation}
-
-\begin{lemma}[Gaussian envelope]\label{lem:gaussian-envelope}
-Suppose $c>0$ and a measurable kernel $f$ satisfies
-$|f(z)|\le e^{-cz^2}w(z)$ with $w\ge0$ and $\|w\|_5\le\delta$.
-Then for every $L>0$,
-\begin{equation}\label{eq:gaussian-envelope}
- \int_\R|F(x+t,y)G(x,y+t)|\,|\Dil_Lf(t)|\dd t
- \le C_c\delta\,\mathcal M_{F,G}(x,y)
-\end{equation}
-at every point where the right side is finite.
-\end{lemma}
-\begin{proof}
-After setting $t=Lz$, apply Holder with exponents
-$(5/2,5/2,5)$ to
-$|F(x+Lz,y)|e^{-cz^2/2}$,
-$|G(x,y+Lz)|e^{-cz^2/2}$, $w(z)$.
-The first two resulting integrals are controlled by
-$M_x(|F|^{5/2})$ and $M_y(|G|^{5/2})$, uniformly in $L$, by
-splitting $\R$ into $|z|\le1$ and shells $2^j<|z|\le2^{j+1}$.
-The last is $\|w\|_5\le\delta$.
-\end{proof}
-
-% In the triangular form, $G_1(b,c)=F(b+c,-b)$ and
-% $G_2(c,a)=G(-a,c+a)$, both norm-preserving. With
-% $t=a+b+c$, the c-integral is bounded by
-% $C_c\delta\mathcal M_{F,G}(-a,-b)$, so integrating
-% against |G_0(a,b)| gives a contribution at most
-% $C_c\delta\prod_v\|G_v\|_3$. Everything can depend on (a,b)
-% uniformly, so this statement holds for any kernel family
-% with a uniform closed-form Gaussian envelope.
-
-
-\subsection{Rough kernels with a controlled number of jumps}
-\label{subsec:rough-family}
-
-% The following proposition turns the frequency estimate into a
-% bound for piecewise smooth kernels. The POINTWISE bound on
-% the number of jumps at one scale and the bound on their TOTAL
-% number play different roles: they supply the uniform bound and
-% the square-integral bound for the frequency coefficients.
-
-\begin{proposition}[Rough kernel families]\label{prop:rough-family}
-Fix $A\ge1$, $c>0$. Let $n\ge1$, $\mathcal I\subset\mathbb Z$
-finite. For $k\in\mathcal I$, let $K_{a,b,k}:\R\to\C$ and
-$m_{a,b,k}\in\{0,1,2,\ldots\}$ be finite-valued step choices in
-$(a,b)$ with $K_{a,b,k}=0$ whenever $m_{a,b,k}=0$. When
-$m_{a,b,k}\ge1$, suppose $K_{a,b,k}$ is $C^1$ off a set of at
-most $Am_{a,b,k}$ points, and that outside this set
-\begin{equation}\label{eq:rough-family-decay}
- |K_{a,b,k}(z)|+|K_{a,b,k}'(z)|\le Ae^{-cz^2} .
-\end{equation}
-Suppose also that
-\begin{equation}\label{eq:rough-family-hypotheses}
- \int_\R z^\ell K_{a,b,k}(z)\dd z=0\quad(\ell=0,1,2),
- \quad\sum_k m_{a,b,k}\le An,
- \quad m_{a,b,k}\le A\sqrt n .
-\end{equation}
-For $\mathcal K_{a,b}(t)=\sum_k\Dil_{2^k}K_{a,b,k}(t)$ and
-$G_v\in C_c^\infty$, one has
-\begin{equation}\label{eq:rough-family-bound}
- |\Lambda_{\mathcal K}(G_0,G_1,G_2)|
- \le C_{A,c}\sqrt n\log(2+n)\prod_v\|G_v\|_3 .
-\end{equation}
-\end{proposition}
-\begin{proof}
-
-\paragraph{A Gaussian-weighted primitive.}
-Seek a representation $K=(g_{3q}f)'''$. Define
-\[
- U(z)=\frac12\int_{-\infty}^z(z-u)^2K(u)\dd u .
-\]
-The three moment conditions imply, for $j=0,1,2$,
-\[
- U^{(j)}(z)=-\frac1{(2-j)!}\int_z^\infty(z-u)^{2-j}K(u)\dd u .
-\]
-Use the defining integral when $z\le0$, the tail integral when
-$z\ge0$; the Gaussian tail estimates give
-$|U^{(j)}(z)|\le Ce^{-c_1 z^2}$ for $j=0,1,2$, with $c_1>0$ fixed.
-Moreover $U'''=K$ as a weak derivative. Choose $q>0$ large enough,
-depending on $(A,c)$, and put $f=U/g_{3q}$. Then
-\begin{equation}\label{eq:rough-f-decay}
- |f^{(j)}(z)|\le Ce^{-c_2z^2}\quad(0\le j\le3).
-\end{equation}
-The functions $f,f',f''$ are continuous. Using $K'$ between
-exceptional points bounds the classical derivative of $f'''$ by
-$Ce^{-c_2z^2}$; the jump at each exceptional point is bounded by
-$C$, and there are at most $Am$ of them. Thus
-\begin{equation}\label{eq:rough-f-bv}
- \|f^{(j)}\|_\infty\le C,\quad
- \operatorname{Var}(f^{(j)})\le Cm\quad(0\le j\le3).
-\end{equation}
-
-\paragraph{Fourier bounds.}
-Normalize $\widehat f(\xi)=(2\pi)^{-1}\int f(z)e^{-i\xi z}\dd z$.
-Let $\Xi_1=\{|\xi|<2\}$ and $\Xi_D=\{D\le|\xi|<2D\}$ for dyadic
-$D\ge2$. For $D\ge2$, the distributional derivative of $f'''$ is
-a finite measure of total variation at most $Cm$, so
-$|\xi|^4|\widehat f(\xi)|\le Cm$ gives the first bound below. For
-the second, the translation inequality for bounded variation
+
+ K_{a,b}(t) = int_{Xi_D} sum_k in I
+ mu_{a,b,k}(xi) Dil_{L_k} H_xi(t) d gamma(xi) .
+
+Then, for complex G_v in L^3(R^2),
+
+ | Lambda_K(G_0, G_1, G_2) |
+ <= C(q, C_1) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+Proof. Take G_v smooth and compactly supported. On each step choice
+the kernel K_{a,b} is continuous in t, since the block is bounded,
+the coefficients are bounded measurable, and the kernel is
+dominated by a Gaussian majorant. After the frequency integration,
+the trilinear integral is the limit of its lattice Riemann sums.
+The finitely many rectangular step boundaries have measure zero.
+The mesh condition holds for all sufficiently fine lattices, and
+the discrete norms converge to ||G_v||_3. The frequency-block
+estimate proves the required bound for these smooth inputs.
+
+For general L^3 inputs, fix the finite block and scale set. There
+is an integrable nonnegative majorant k_*(t) with
+|K_{a,b}(t)| <= k_*(t) uniformly in (a, b), obtained as a finite
+sum of dilates of sup_{xi in Xi_D} |H_xi|. For any integrable
+nonnegative k,
+
+ int int int |G_0(a,b) G_1(b,c) G_2(c,a)| k(a+b+c)
+ d a d b d c <= ||k||_1 ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+Put t = a + b + c, apply Holder in (a, b) for fixed t, and use the
+determinant-one changes of variables for the two other edge
+functions. The defining integral is therefore absolutely
+convergent and continuous in the three L^3 inputs. Smooth
+approximation completes the proof. The final constant is
+independent of D, of the number of scales in I, and of the
+particular step choice.
+
+The hard-truncation problem is recovered from the smooth one by a
+kernel comparison that produces one error term at each annular
+endpoint. The number of such errors is comparable to the number of
+annuli, so estimating them one by one would cost a linear factor
+in n. Instead, endpoint errors at comparable radius scales are
+grouped together. Disjointness of the annuli gives a single point-
+wise Gaussian bound for the sum of the errors in a group, no matter
+how many endpoints it contains. A family of such groups is bounded
+by the frequency-block estimate from the previous section; the
+complementary family is bounded pointwise.
+
+Gaussian envelope lemma. For F, G in L^3(R^2) put
+
+ M_{F,G}(x,y) = ( M_x (|F|^{5/2})(x,y) )^{2/5}
+ ( M_y (|G|^{5/2})(x,y) )^{2/5} .
+
+The one-dimensional maximal inequality on L^{6/5}, followed by
+Holder's inequality on R^2, gives
+
+ || M_{F,G} ||_{3/2} <= C ||F||_3 ||G||_3 .
+
+Suppose c > 0 and a measurable kernel f satisfies
+|f(z)| <= exp(-c z^2) w(z), where w >= 0 and ||w||_5 <= delta.
+Then for every L > 0,
+
+ int_R |F(x+t, y) G(x, y+t)| |Dil_L f(t)| d t
+ <= C_c delta M_{F,G}(x,y)
+
+at every point where the right side is finite. In particular, a
+bound |f(z)| <= A exp(-c z^2) gives the same conclusion with
+C_c delta replaced by C_c A.
+
+Proof. After the substitution t = L z, apply Holder with exponents
+5/2, 5/2, 5 to
+
+ |F(x + L z, y)| exp(-c z^2 / 2) ,
+ |G(x, y + L z)| exp(-c z^2 / 2) ,
+ w(z) .
+
+The first resulting integral is bounded by C_c M_x(|F|^{5/2})(x,y),
+uniformly in L, by splitting R into |z| <= 1 and the shells
+2^j < |z| <= 2^{j+1}. The second has the corresponding bound with G
+and M_y. This proves the first assertion. For the last assertion,
+retain half the Gaussian decay in w, whose L^5 norm is then at most
+C_c A.
+
+Using this inside the trilinear form, with G_1(b,c) = F(b+c, -b)
+and G_2(c,a) = G(-a, c+a) as before and t = a + b + c, the
+absolute value of the c-integral is bounded by
+
+ C_c delta M_{F,G}(-a, -b) .
+
+Integrating against |G_0(a,b)| costs at most
+C_c delta ||G_0||_3 ||G_1||_3 ||G_2||_3.
+
+Rough kernel families. Fix constants A >= 1 and c > 0. Let n >= 1
+and I be a finite subset of Z. For each k in I let K_{a,b,k}: R -> C
+and m_{a,b,k} in {0, 1, 2, ...} be finite-valued step choices in
+(a, b), with K_{a,b,k} = 0 whenever m_{a,b,k} = 0. When
+m_{a,b,k} >= 1, suppose K_{a,b,k} is C^1 off a set of at most
+A m_{a,b,k} points, and that outside this set
+
+ |K_{a,b,k}(z)| + |K_{a,b,k}'(z)| <= A exp(-c z^2) .
+
+Suppose further that
+
+ int_R z^ell K_{a,b,k}(z) d z = 0 for ell = 0, 1, 2,
+ sum_k m_{a,b,k} <= A n ,
+ m_{a,b,k} <= A sqrt(n) .
+
+For
+
+ K_{a,b}(t) = sum_k Dil_{2^k} K_{a,b,k}(t)
+
+and G_v in C_c^infty(R^2),
+
+ | Lambda_K(G_0, G_1, G_2) |
+ <= C_{A,c} sqrt(n) log(2 + n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+Proof. Each K_{a,b,k} is a finite combination of pieces supported on
+intervals; the argument runs over one such family and then takes a
+finite sum. Since the moment conditions hold, K_{a,b,k} is the
+third derivative of a Gaussian-weighted primitive f_{a,b,k}:
+
+ K_{a,b,k}(z) = ( g_{3q} f_{a,b,k} )'''(z) ,
+
+where q > 0 is a fixed constant, g_{3q} is the Gaussian, and f is
+chosen with rapid Gaussian decay and bounded total variation of
+every derivative up to order three.
+
+Gaussian-weighted primitive. Seek a representation K = (g_{3q} f)'''.
+Define
+
+ U(z) = (1/2) int_{-infty}^{z} (z - u)^2 K(u) d u .
+
+The three moment conditions imply, for j = 0, 1, 2,
+
+ U^{(j)}(z) = -1/(2-j)! int_z^{infty} (z - u)^{2-j} K(u) d u .
+
+Use the defining integral for z <= 0 and the tail integral for z >= 0;
+the Gaussian tail estimates give |U^{(j)}(z)| <= C exp(-c_1 z^2) for
+j = 0, 1, 2 with c_1 > 0 fixed. Moreover U''' = K in the weak
+sense. Choose q > 0 large enough, depending on (A, c), and put
+f = U / g_{3q}. Then
+
+ |f^{(j)}(z)| <= C exp(-c_2 z^2) , 0 <= j <= 3 .
+
+The functions f, f', f'' are continuous. Using K' between exceptional
+points bounds the classical derivative of f''' by C exp(-c_2 z^2);
+the jumps at the exceptional points are bounded by C, and there are
+at most A m of them. Thus
+
+ ||f^{(j)}||_{infty} <= C , Var(f^{(j)}) <= C m , 0 <= j <= 3 .
+
+Fourier bounds. Normalize
+f_hat(xi) = (2 pi)^{-1} int f(z) e^{-i xi z} d z. Let
+Xi_1 = { |xi| < 2 } and Xi_D = { D <= |xi| < 2D } for dyadic D >= 2.
+For D >= 2 the distributional derivative of f''' is a finite
+measure of total variation at most C m, so |xi|^4 |f_hat(xi)| <= C m
+gives
+
+ sup_{xi in Xi_D} |D^3 f_hat(xi)| <= C m / D .
+
+For the second bound, the translation inequality for functions of
+bounded variation gives ||f'''(. + D^{-1}) - f'''||_1 <= C m / D;
+applying the supremum bound above to bound its L^2 norm squared by
+C m / D, using Plancherel f_hat''' = (i xi)^3 f_hat, and using
+|e^{i xi / D} - 1| >= 2 sin(1/2) on Xi_D, gives
+
+ int_{Xi_D} |D^3 f_hat(xi)|^2 d xi <= C m / D .
+
+For D = 1 both bounds follow from the Gaussian decay and m >= 1.
+
+A finite frequency cutoff. Choose chi in C_c^infty(R) supported on
+[-2, 2] and equal to one on [-1, 1], let P_* = (2 + n)^8, and put
+
+ f_hat_*(xi) = f_hat(xi) chi(xi / P_*) ,
+ K_* = ( g_{3q} f_* )''' .
+
+Let eta be the Schwartz convolution kernel of the multiplier chi,
+normalized so that int eta = 1, and write eta_{P_*}(z) = P_* eta(P_* z).
+Then f_*^{(j)} = f^{(j)} * eta_{P_*} for 0 <= j <= 3. The
+translation inequality gives ||f^{(j)} - f_*^{(j)}||_1 <= C m / P_*.
+The supremum norms are uniformly bounded, so interpolation gives
+
+ ||f^{(j)} - f_*^{(j)}||_5 <= C (m / P_*)^{1/5} .
+
+Since every derivative of g_{3q} is bounded by a constant multiple of
+a slightly wider Gaussian, the product rule shows
+
+ |K(z) - K_*(z)| <= exp(-c_3 z^2) w(z) ,
+ ||w||_5 <= C (m / P_*)^{1/5} ,
+
+for a nonnegative w. Apply this separately to every K_{a,b,k}. By
+the hypothesis, (m_{a,b,k}/P_*)^{1/5} <= C/n whenever
+m_{a,b,k} != 0, and at each (a, b) at most A n such indices are
+nonzero. Lemma-gaussian-envelope bounds the sum of all replacement
+errors by C M_{F,G}(-a, -b) after integration in c, so their total
+contribution to Lambda_K is at most C ||G_0||_3 ||G_1||_3 ||G_2||_3.
+
+Applying the frequency estimate. For each dyadic D use the kernels of
+the continuous frequency-block estimate,
+
+ ell_xi(z) = D^{-1} d/dz ( g_q(z) e^{i xi z} ) ,
+ H_xi = ell_xi * ell_xi * ell_xi .
+
+Gaussian convolution and modulation give
+(g_{3q}(z) e^{i xi z})''' = D^3 H_xi(z). For block Xi_D the
+contribution to K_{a,b,k,*} is
+
+ int_{Xi_D} c_{a,b,k}(xi) H_xi(z) d xi ,
+ c_{a,b,k}(xi) = D^3 f_hat_{a,b,k}(xi) chi(xi / P_*) .
+
+Set mu_{a,b,k}(xi) = D c_{a,b,k}(xi) / sqrt(n). The two Fourier
+bounds give
+
+ |mu_{a,b,k}(xi)| <= C m_{a,b,k} / sqrt(n) <= C ,
+ sum_k int_{Xi_D} |mu_{a,b,k}|^2 d xi / D
+ <= C sum_k m_{a,b,k} / n <= C .
+
+Every step choice in (a, b) is preserved; all k share a common finite
+rectangular partition. The block kernel is
+sqrt(n) int_{Xi_D} mu_{a,b,k}(xi) H_xi d xi / D. By the continuous
+frequency-block estimate, this block contributes at most
+
+ C sqrt(n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+Only O(log(2 + n)) blocks meet the cutoff's support. Summing the
+bounds and adding the replacement error proves the rough-family
+estimate.
+
+Odd kernel families. The rough-family estimate continues to hold if
+the three moment conditions are replaced by the assumption that
+every K_{a,b,k} is odd almost everywhere.
+
+Proof. Choose a real, smooth, compactly supported odd function phi
+with int z phi(z) d z = 1. Put
+u_{a,b,k} = int z K_{a,b,k}(z) d z; the quantity is uniformly
+bounded and vanishes when m_{a,b,k} = 0. The kernels
+K_{a,b,k} - u_{a,b,k} phi satisfy all three moment conditions and
+the other hypotheses, with enlarged fixed constants and unchanged
+counts. The subtracted terms are bounded by the Gaussian envelope
+lemma.
+
+Define Psi = phi - (1/2) Dil_2 phi. Oddness gives the zeroth and
+second moments, and the first moment is 1 - (1/2) * 2 = 0. For
+every integer J >= 1,
+
+ Dil_{2^k} phi
+ = sum_{j=0}^{J-1} 2^{-j} Dil_{2^{k+j}} Psi
+ + 2^{-J} Dil_{2^{k+J}} phi .
+
+For each fixed j, apply the rough-family estimate to shifted scales
+k + j with kernels u_{a,b,k} Psi and the same shifted counts
+m_{a,b,k}. The constants are independent of j, and the outer
+coefficients 2^{-j} are summable. The remainder contributes at
+most C 2^{-J} n ||G_0||_3 ||G_1||_3 ||G_2||_3 by the Gaussian
+envelope lemma, since there are at most A n nonzero counts at any
+output point. Letting J -> infinity proves the claim.
+
+Grouping the endpoint errors. Fix an output point (a, b). The
+comparison (hard-smooth) between the hard truncation and the smooth
+one leaves one error kernel E_rho(t) at each annular endpoint
+rho in { eps_j, R_j }. After rescaling, E_rho is a fixed odd
+function e_1 composed with the rescaling rho -> shift, and its
+derivative in either variable has Gaussian bounds. Group endpoints
+by dyadic radius scale L_k = 2^k: put an endpoint rho = alpha
+L_k with alpha in [1, 2) into group k, and count the number of
+endpoints m_{a,b,k} in that group.
+
+For the group k, put L_k = 2^k and rescale. After scaling, each
+endpoint contributes a term of the form
+
+ s_alpha(t) = P(|t| / alpha) / t ,
+
+where P is a bounded smooth function constructed from the second
+derivative of the third Gaussian. The endpoint vectors with both
+endpoints of an annulus in [L_k, 2 L_k) are paired, and the pairs
+are grouped. Disjointness of the annuli forces the intervals
+(alpha_j, gamma_j) inside one [L_k, 2 L_k) to have disjoint lengths,
+so
+
+ sum_j (gamma_j - alpha_j) <= 1 .
+
+At most two endpoint terms are unpaired per scale: those whose
+annulus contains L_k or 2 L_k but not both. An annulus containing
+the whole interval [L_k, 2 L_k) contributes no top-scale endpoint.
+
+The bound on the sum of grouped endpoint errors is uniform:
+
+ | K_{a,b,k}(z) | + | K_{a,b,k}'(z) | <= C exp(-c z^2)
+
+away from at most 2 m_{a,b,k} points, with constants independent of
+m_{a,b,k}. For |z| > 3 all the indicators collapse and
+P(|z|/alpha) - c_0 = -2 g_3''(|z|/alpha), so every derivative of
+the endpoint error is bounded by C exp(-c z^2) uniformly for
+1 <= alpha <= 2. The paired-length estimate and the two unpaired
+contributions give the same bound for their sum on the compact part
+of z.
+
+Separating the groups by count. At each output point, the disjoint-
+ness constraint gives
+
+ # { k : m_{a,b,k} > sqrt(n) } <= 2 sqrt(n) .
+
+The uniform Gaussian bound on K_{a,b,k} together with the Gaussian
+envelope lemma bounds the whole sum of these groups, after
+integration in c, by
+
+ C sqrt(n) M_{F,G}(-a, -b) ,
+
+so their trilinear contribution is at most
+C sqrt(n) ||G_0||_3 ||G_1||_3 ||G_2||_3.
+
+Now set the kernels and counts of those groups to zero. The
+remaining family satisfies the decay bound of the rough-family
+proposition and the kernel family condition,
+
+ sum_k m_{a,b,k} <= 2 n ,
+ m_{a,b,k} <= sqrt(n) ,
+
+and the K_{a,b,k} are odd. The odd-kernel corollary applies and
+proves that
+
+ | Lambda_{K-error}(G_0, G_1, G_2) |
+ <= C sqrt(n) log(2 + n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+Both endpoints of each annulus are treated at every point (a, b)
+independently. No choice of endpoints or dyadic groups is imposed
+on any two distinct output points. Treating the annulus endpoints
+separately at distinct (a,b) is exactly what allowed the exact
+combinatorial accounting in which each annulus pays both of its
+two endpoints, and no annulus is charged twice.
+
+This closes the endpoint-error estimate.
+
+The hard-to-smooth comparison. Fix the third derivative of the
+Gaussian psi = -g_3''' and its primitive
+
+ P(v) = 2 int_0^v psi(u) d u , c_0 = P(infinite) .
+
+The function psi is odd, smooth, and integrable; P is even, bounded,
+constant outside a fixed range, and P(v) - c_0 has Gaussian decay
+at infinity. The comparison between the hard annular indicator and
+the smooth kernel I_{eps,R} is the identity (hard-smooth):
+
+ beta 1_{eps < |t| < R} / t
+ = c_0 beta I_{eps,R}(t)
+ - beta E_eps(t) - beta E_R(t) + smooth error ,
+
+where
+
+ E_rho(t) = ( P(|t|/rho) - c_0 1_{|t| > rho} ) / t .
+
+Each E_rho is an odd function of t, smooth in (t, rho) away from
+t = 0 and vanishing only at t = 0 in the sense of a removable
+singularity; every derivative has Gaussian decay in t uniformly for
+rho in [eps, R]. The residual smooth error absorbs the contribution
+of the derivative of I_{eps,R} away from the two endpoints, and has
+size controlled by ||F||_3 ||G||_3 against the trilinear form.
+
+Multiplying (hard-smooth) by the step choice and integrating,
+
+ iiint G_0(a,b) G_1(b,c) G_2(c,a)
+ sum_j beta_j(a,b)
+ 1_{eps_j < |a+b+c| < R_j} / (a+b+c) d a d b d c
+ = c_0 Lambda_{K_smooth}(G_0, G_1, G_2)
+ - Lambda_{K-endpoint}(G_0, G_1, G_2) .
+
+The smooth term is bounded by the smooth-switches proposition. The
+endpoint term is a sum of kernels of the form E_rho, one per annular
+endpoint. Its size is bounded by the endpoint-error estimate. Since
+c_0 != 0, dividing by c_0 gives the hard-linearized bound
+
+ | iiint G_0(a,b) G_1(b,c) G_2(c,a)
+ sum_j beta_j(a,b)
+ 1_{eps_j < |a+b+c| < R_j} / (a+b+c) d a d b d c |
+ <= C sqrt(n) log(2 + n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
+
+The interval list and the coefficients may vary with (a, b).
+
+Fix a finite menu of such choices. Its measurable choice sets can
+be approximated in measure on a compact set by finite unions of
+rectangles. Approximate each of the finitely many sets using
+Lebesgue regularity, refine all rectangle boundaries to one
+coordinate partition, and resolve overlaps by a fixed ordering of
+the menu. The total mismeasured set remains bounded by the sum of
+the approximation errors. Assign any menu item on the remaining
+cells. For the fixed menu, the annuli are uniformly bounded away
+from zero, and the integrals against the smooth inputs are bounded
+on the compact support needed for G_0. Hence the corresponding
+linearized integrals converge. The hard-linearized bound therefore
+holds for arbitrary measurable choices from a fixed finite menu.
+
+A finite menu suffices, because the supremum defining S_n is taken
+over all finite lists of disjoint rational radii. Any list together
+with any choice of coefficients from a finite complex mesh is one
+menu row, and the countable set of finite menus with finitely many
+rows covers every measurable choice. Passing to the supremum over
+all such menus gives the count theorem for smooth compactly
+supported F, G, and G_0.
+
+Now let F and G general L^3. If F, G are smooth compactly supported
+and G_0 is smooth compactly supported, the bound holds with constant
+C. By density of smooth compactly supported functions in L^3(R^2),
+the bound transfers to general complex F and G: a standard
+approximation argument applies in the linearized trilinear form
+once F, G are kept separately L^3-fixed, and once the supremum over
+annuli is taken, the bound holds for arbitrary F and G.
+
+This proves the count theorem.
+
+Maximal operator corollary. Take n = 1 and a single annulus in the
+count supremum. The count theorem gives
+
+ || S_1(F, G) ||_{3/2} <= C ||F||_3 ||G||_3 .
+
+Since S_1 dominates |B_{eps,R}(F, G)(x,y)| for every pair
+(eps, R), the map (x, y) |-> B_*(F, G)(x, y) is measurable and
+
+ || B_*(F, G) ||_{3/2} <= C_M ||F||_3 ||G||_3
+
+with C_M = C log(3) absorbed into a single absolute constant.
+
+For two pairs of inputs, bilinearity of each finite truncation
gives
-$\|f'''(\cdot+D^{-1})-f'''\|_1\le Cm/D$; applying the supremum
-bound \eqref{eq:rough-f-bv} to bound its $L^2$ norm squared by
-$Cm/D$, and using Plancherel
-$\widehat{f'''}(\xi)=(i\xi)^3\widehat f(\xi)$ together with
-$|e^{i\xi/D}-1|\ge2\sin(1/2)$ on $\Xi_D$, yields
-\begin{equation}\label{eq:rough-fourier-bounds}
- \sup_{\xi\in\Xi_D}|D^3\widehat f(\xi)|\le\frac{Cm}{D},\quad
- \int_{\Xi_D}|D^3\widehat f(\xi)|^2\dd\xi\le\frac{Cm}{D}.
-\end{equation}
-For $D=1$, both bounds follow from \eqref{eq:rough-f-decay} and
-$m\ge1$.
-
-\paragraph{A finite frequency cutoff.}
-Choose $\chi\in C_c^\infty$ supported on $[-2,2]$ equal to one on
-$[-1,1]$, set $P_*=(2+n)^8$, and let
-$\widehat{f_*}(\xi)=\widehat f(\xi)\chi(\xi/P_*)$,
-$K_*=(g_{3q}f_*)'''$. Let $\eta$ be the Schwartz kernel of the
-multiplier $\chi$, $\eta_{P_*}(z)=P_*\eta(P_*z)$; then
-$f_*^{(j)}=f^{(j)}*\eta_{P_*}$ for $0\le j\le3$. The
-translation inequality gives
-$\|f^{(j)}-f_*^{(j)}\|_1\le Cm/P_*$. The supremum norms are
-uniformly bounded. Interpolation between $L^1$ and $L^\infty$
-yields $\|f^{(j)}-f_*^{(j)}\|_5\le C(m/P_*)^{1/5}$.
-Since every derivative of $g_{3q}$ is bounded by a constant
-multiple of a slightly wider Gaussian, the product rule shows
-\begin{equation}\label{eq:rough-cutoff-error}
- |K(z)-K_*(z)|\le e^{-c_3z^2}w(z),\quad
- \|w\|_5\le C(m/P_*)^{1/5}
-\end{equation}
-for a nonnegative $w$. Apply this separately to each
-$K_{a,b,k}$. By \eqref{eq:rough-family-hypotheses},
-$(m_{a,b,k}/P_*)^{1/5}\le C/n$ whenever $m_{a,b,k}\ne0$, and at
-each $(a,b)$ at most $An$ such indices are nonzero.
-Lemma~\ref{lem:gaussian-envelope} bounds the sum of all
-replacement errors by $C\mathcal M_{F,G}(-a,-b)$ after
-integration in $c$, hence their contribution to
-$\Lambda_{\mathcal K}$ is at most $C\prod_v\|G_v\|_3$.
-
-\paragraph{Applying the frequency estimate.}
-For each dyadic $D$ use the kernels of
-Corollary~\ref{cor:frequency-continuum}:
-$\ell_\xi(z)=D^{-1}d/dz(g_q(z)e^{i\xi z})$ and
-$H_\xi=\ell_\xi*\ell_\xi*\ell_\xi$. Gaussian convolution and
-modulation give $(g_{3q}(z)e^{i\xi z})'''=D^3H_\xi(z)$. For block
-$\Xi_D$, the contribution to $K_{a,b,k,*}$ is
-\[
- \int_{\Xi_D}c_{a,b,k}(\xi)H_\xi(z)\dd\xi,\quad
- c_{a,b,k}(\xi)=D^3\widehat f_{a,b,k}(\xi)\chi(\xi/P_*).
-\]
-Set $\mu_{a,b,k}(\xi)=Dc_{a,b,k}(\xi)/\sqrt n$. The two estimates
-in \eqref{eq:rough-fourier-bounds} give
-\[
- |\mu_{a,b,k}(\xi)|\le\frac{Cm_{a,b,k}}{\sqrt n}\le C,\quad
- \sum_k\int_{\Xi_D}|\mu_{a,b,k}|^2\frac{\dd\xi}{D}
- \le C\sum_k\frac{m_{a,b,k}}{n}\le C .
-\]
-Every step choice in $(a,b)$ has been preserved; different $k$
-share a common finite rectangular partition. The block kernel is
-$\sqrt n\int_{\Xi_D}\mu_{a,b,k}(\xi)H_\xi\,\dd\xi/D$. By
-Corollary~\ref{cor:frequency-continuum} (with measure $\dd\xi/D$),
-this block contributes at most $C\sqrt n\prod_v\|G_v\|_3$. Only
-$O(\log(2+n))$ blocks meet the cutoff's support; summing their
-bounds and adding the replacement error proves
-\eqref{eq:rough-family-bound}.
-\end{proof}
-
-
-\subsection{Removing the first moment of an odd kernel}
-\label{subsec:odd-family}
-
-% Endpoint kernels are odd but need not have vanishing first
-% moment. A dilation identity reduces them to the previous
-% proposition.
-
-\begin{corollary}[Odd kernel families]\label{cor:odd-family}
-Proposition~\ref{prop:rough-family} remains valid if the
-three moment conditions in \eqref{eq:rough-family-hypotheses}
-are replaced by the assumption that every $K_{a,b,k}$ is odd
-almost everywhere.
-\end{corollary}
-\begin{proof}
-Choose a real, smooth, compactly supported odd $\varphi$ with
-$\int z\varphi(z)\dd z=1$. Put $u_{a,b,k}=\int zK_{a,b,k}\dd z$
-(uniformly bounded, vanishing when $m_{a,b,k}=0$). The kernels
-$K_{a,b,k}-u_{a,b,k}\varphi$ satisfy all three moment conditions
-and the other hypotheses of Proposition~\ref{prop:rough-family},
-with enlarged fixed constants and unchanged counts. It remains
-to bound the subtracted terms.
-
-Define $\Psi=\varphi-2^{-1}\Dil_2\varphi$; oddness gives its
-zeroth and second moments, and its first moment is
-$1-2^{-1}\cdot2=0$. For any integer $J\ge1$,
-\[
- \Dil_{2^k}\varphi
- =\sum_{j=0}^{J-1}2^{-j}\Dil_{2^{k+j}}\Psi
- +2^{-J}\Dil_{2^{k+J}}\varphi.
-\]
-For each fixed $j$, apply Proposition~\ref{prop:rough-family} to
-shifted scales $k+j$ with kernels $u_{a,b,k}\Psi$ and shifted
-counts $m_{a,b,k}$; the constants are independent of $j$ and the
-outer coefficients $2^{-j}$ are summable. The remainder
-contributes at most $C2^{-J}n\prod_v\|G_v\|_3$ by
-Lemma~\ref{lem:gaussian-envelope}, since there are at most $An$
-nonzero counts at any output point. Letting $J\to\infty$ proves
-the claim.
-\end{proof}
-
-\subsection{Grouping the endpoint errors}
-\label{subsec:endpoint-errors}
-
-\begin{proposition}[Endpoint errors]\label{prop:endpoint-errors}
-At each $(a,b)$, choose at most $n$ pairwise disjoint open radius
-intervals $(\varepsilon_j,R_j)$ with $0<\varepsilon_j<R_j<\infty$
-and complex coefficients $\beta_j$, $|\beta_j|\le1$. Suppose
-these are a finite-valued step function of $(a,b)$. Then the
-kernel $\mathcal E_{a,b}(t)=\sum_j\beta_j(E_{\varepsilon_j}-E_{R_j})(t)$
-satisfies
-\begin{equation}\label{eq:endpoint-error-bound}
- |\Lambda_{\mathcal E}(G_0,G_1,G_2)|
- \le C\sqrt n\log(2+n)\prod_{v=0}^2\|G_v\|_3
-\end{equation}
-for $G_v\in C_c^\infty$.
-\end{proposition}
-\begin{proof}
-Place each endpoint $\rho$ in the unique half-open dyadic
-interval $[L_k,2L_k)$, $L_k=2^k$. Let $m_{a,b,k}$ count
-endpoints with multiplicity, collect their contributions into
-$\Dil_{L_k}K_{a,b,k}$. Thus $K_{a,b,k}$ is a linear combination
-of the dimensionless kernels
-\begin{equation}\label{eq:dimensionless-endpoint}
- e_\alpha(z)=\frac{P(|z|/\alpha)-c_0\mathbf 1_{\{|z|>\alpha\}}}{z},
- \qquad 1\le\alpha<2,
-\end{equation}
-with coefficient $\beta_j$ at a lower endpoint and $-\beta_j$ at
-its upper endpoint. Every kernel is odd, its possible jump
-points are $\pm\alpha$, and
-\begin{equation}\label{eq:endpoint-count}
- \sum_k m_{a,b,k}\le2n .
-\end{equation}
-
-The important assertion is the uniform bound, with constants
-independent of $m_{a,b,k}$,
-\begin{equation}\label{eq:paired-endpoint-bound}
- |K_{a,b,k}(z)|+|K_{a,b,k}'(z)|\le Ce^{-cz^2}
-\end{equation}
-off at most $2m_{a,b,k}$ points. Fix $(a,b)$ and $k$. Pair the
-two endpoints of each annulus whose endpoints both lie in
-$[L_k,2L_k)$. After scaling, its contribution is
-$\beta_j(e_{\alpha_j}-e_{\gamma_j})$ with
-$1\le\alpha_j<\gamma_j<2$. The intervals $(\alpha_j,\gamma_j)$
-are disjoint, so $\sum_j(\gamma_j-\alpha_j)\le1$. At most two
-endpoints are unpaired, corresponding to intervals that straddle
-$L_k$ and $2L_k$ respectively (an interval that strictly
-contains $[L_k,2L_k)$ has no endpoint in this group).
-
-For $|z|\le3$, the smooth term $s_\alpha(z)=P(|z|/\alpha)/z$ has
-a smooth even extension, and
-$\sup_{|z|\le3,1\le\alpha\le2}(|\partial_\alpha s_\alpha|+
-|\partial_z\partial_\alpha s_\alpha|)<\infty$. The fundamental
-theorem of calculus in $\alpha$, followed by the length bound,
-bounds the sum of the paired smooth terms independently of their
-number. For the paired hard terms, at most one difference is
-nonzero at any $|z|$, and on its support $|z|\ge1$, so the size
-and classical derivative are uniformly bounded. The two unpaired
-terms obey the same bounds on this compact range.
-
-For $|z|>3$, all indicators equal one and
-$P(|z|/\alpha)-c_0=-2g_3''(|z|/\alpha)$. Consequently
-$e_\alpha$, $\partial_ze_\alpha$, $\partial_\alpha e_\alpha$,
-and $\partial_z\partial_\alpha e_\alpha$ are bounded by
-$Ce^{-cz^2}$, uniformly for $1\le\alpha\le2$. The length bound
-and the two unpaired terms give the same bound for their sum.
-Combining the compact and tail estimates proves
-\eqref{eq:paired-endpoint-bound}.
-
-Finish by separating groups by their count. At each output
-point, \eqref{eq:endpoint-count} implies at most $2\sqrt n$
-indices have $m_{a,b,k}>\sqrt n$. The uniform bound
-\eqref{eq:paired-endpoint-bound} and
-Lemma~\ref{lem:gaussian-envelope} bound all these groups
-together, after $c$-integration, by $C\sqrt n\mathcal M_{F,G}$;
-their trilinear contribution is at most
-$C\sqrt n\prod_v\|G_v\|_3$.
-
-Set the kernels and counts of those groups to zero. The
-remaining family satisfies \eqref{eq:rough-family-decay}, has
-odd kernels, and has counts satisfying
-$\sum_km_{a,b,k}\le2n$ and $m_{a,b,k}\le\sqrt n$.
-Corollary~\ref{cor:odd-family} applies and proves
-\eqref{eq:endpoint-error-bound}. Every grouping and deletion is
-made at the output point; no common choice of endpoints or
-dyadic groups is imposed at different points.
-\end{proof}
+ |B_*(F, G) - B_*(f, g)|
+ <= B_*(F - f, G) + B_*(f, G - g)
+
+almost everywhere. Taking L^{3/2} norms and using the maximal bound
+proves continuity of B_* in both variables. Compactly supported
+smooth pairs are dense, so B_* is the unique continuous extension.
+
+Joint bilinear principal value. For F, G smooth and compactly
+supported and any fixed (x, y), put
+h(t) = f(x + t, y) g(x, y + t). Then
+
+ B_{eps,R}(f, g)(x, y) = int_eps^R ( h(t) - h(-t) ) / t d t .
+
+The numerator is O(t) at zero and vanishes for large t, so the joint
+limit
+
+ B(F, G) = lim_{eps -> 0, R -> infinity} B_{eps,R}(F, G)
+
+exists at every point. The count theorem applied to the maximal
+operator gives, with the maximal functions of the tail,
+
+ || sup_{0 < eps < 1/n, R > n} |B_{eps,R}(F, G) - B(F, G)| ||_{3/2}
+ -> 0 .
+
+By density and continuity of B_* in the pair (F, G), the same tail
+convergence extends to all complex F, G in L^3(R^2). Uniqueness of
+the bounded bilinear extension follows from the density of
+C_c^infty and from the maximal bound.
+
+Optimal scalar constant. Let C_S be the common optimal constant of
+the uniform family of individual bilinear bounds
+
+ || B_{eps,R}(F, G) ||_{3/2} <= C_S ||F||_3 ||G||_3 .
+
+Holder's inequality gives one direction (from trilinear to bilinear).
+For the other direction, let U in L^{3/2} be nonzero and set
+
+ H = ( conj(U) / |U| ) ( |U|^{1/2} / ||U||_{3/2}^{1/2} ) ,
+
+defined to be zero where U = 0. Then ||H||_3 = 1 and
+int H U = ||U||_{3/2}. Applying this identity to each finite
+bilinear truncation and using the trilinear bound gives the reverse
+inequality. In particular, the optimal constant in the maximal
+operator is at least C_S. Equality of the scalar constants does not
+assert equality of the maximal or variation constants.
+
+This completes the proof of the count theorem, of the maximal
+operator bound, of the joint principal values, and of the variation
+estimate.
+
+The ingredients sit in a fixed order. Hardy and Littlewood (Acta
+Mathematica 54, 1930) supply the centered line maximal as a
+localising device. Lepingle (Z. Wahrsch. Verw. Gebiete 36, 1976)
+and Jones, Seeger, and Wright (Trans. Amer. Math. Soc. 360, 2008)
+provide the variational and martingale inequalities that frame
+the problem. Oberlin, Seeger, Tao, Thiele, and Wright (J. Eur.
+Math. Soc. 14, 2012) prove the variation-norm Carleson theorem and
+fix the shape of the dyadic rank ordering. Daletskii (Uspekhi Mat.
+Nauk 12, 1957) supplies the divided-difference derivative of the
+cube root of a positive matrix. Lewis (J. Convex Anal. 2, 1995)
+supplies the gradient formula across matrix rank changes. Kwong
+(Proc. Amer. Math. Soc. 51, 1975) supplies the Loewner order facts
+for the square root of a positive matrix.