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
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 sqrt(n) factor comes from a normalization. Apply the bound to
A_0' = A_0 n^{-1/2} / ||A_0||, A_1' = A_1 / ||A_1||,
A_2' = A_2 / ||A_2||, whose L^3 norms are (n^{-1/2}, 1, 1). The
right side of the switch-dissipation inequality,
n(n^{-3/2} + 2n^{-1}) + n^{-3/2} + 2, is then bounded by an
absolute constant, and the initial energy sum_v n_v^3 becomes
n^{-3/2} + 2, also bounded. By trilinearity, the original
trilinear form is n^{1/2} ||A_0|| ||A_1|| ||A_2|| times the
trilinear form on the normalized arrays, which proves the
sqrt(n) factor in the final smooth-switches bound.
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
h^2 <= q min_k L_k^2 / (16 D^2) ,
then
| 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'). 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 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
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.
Hard-smooth comparison. Fix psi = -g_3''' and its primitive
P(v) = 2 int_0^v psi(u) d u (v >= 0), c_0 = P(infinite).
The function psi is odd, smooth, and Gaussian-tailed. The window
k_1(t) = t g_1(t) equals -g_1'(t); convolving three Gaussian
derivatives adds variances, so k_1 * k_1 * k_1 = -g_3''' = psi. The
primitive P is even, smooth, bounded, equal to c_0 outside a fixed
range, and vanishes to second order at v = 0. The identity
(hard-smooth) is the equality of three exact forms:
I_{eps,R}(t) = ( P(|t|/eps) - P(|t|/R) ) / t
= ( c_0 / t ) 1_{ eps < |t| < R }
+ E_eps(t) - E_R(t) ,
where
E_rho(t) = ( P(|t|/rho) - c_0 1_{ |t| > rho } ) / t .
The middle expression is the direct consequence of the substitution
z = |t| / sqrt(s) in the scale integral I_{eps,R}. The rightmost
expression is the rearrangement
(c_0/t)( 1_{|t| > eps} - 1_{|t| > R} )
+ ( P(|t|/eps) - c_0 1_{|t| > eps}
- P(|t|/R) + c_0 1_{|t| > R} ) / t
= ( P(|t|/eps) - P(|t|/R) ) / t ,
in which each boundary indicator c_0 1_{|t| > rho} appears twice
with opposite signs. Each E_rho is odd, smooth away from t = 0,
vanishes linearly at t = 0, and every derivative has Gaussian decay
in t uniformly for rho in any compact positive range.
Multiplying (hard-smooth) by the step choice and integrating,
c_0 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
= Lambda_{K_smooth}(G_0, G_1, G_2)
- Lambda_{E}(G_0, G_1, G_2) .
The smooth term is bounded by the smooth-switches proposition. The
endpoint term Lambda_{E}, in which E denotes the sum
sum_j beta_j (E_{eps_j} - E_{R_j}), is controlled 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
|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.