Bi-Lipschitz Coordinates at Regular Points
of Noncollapsed RCD Spaces
A Guided Tour for Young Mathematicians
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This is a guided tour of a research paper in metric geometry.
The paper proves that a "regular" point of a certain kind of
singular curved space always has a small open neighborhood that
looks, metrically and up to a fixed distortion, like an open
piece of ordinary flat n-dimensional space. The distortion is a
constant that depends only on the dimension n, never on the
particular space or point.
The tour is written for a motivated high-school reader. Every
technical term is introduced before it is used, the key ideas of
each argument are explained in plain language, and the precise
statements are also given so the reader can compare against the
original. It is honest about what the paper proves from scratch
and what it uses as a black box from earlier work.
The original paper has a short introduction and four numbered
sections:
Section 1. Uniform harmonic coordinates.
Section 2. Heat averages and cumulative matrix control.
Section 3. Summable directional pinching.
Section 4. Renormalization in the space of coordinate values.
The tour follows the paper's own order. A reading guide near the
end maps each section and each named lemma back to the tour. The
reader is encouraged to read the tour once for the plot and a
second time for the estimates.
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PART I. THE QUESTION AND THE MAIN THEOREM
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1. WHAT KIND OF SPACE IS THIS ABOUT?
A metric space is a set of points together with a distance
function d(x,z) that assigns a nonnegative number to each pair of
points, is zero exactly when the points are equal, is symmetric,
and obeys the triangle inequality
d(x,y) <= d(x,z) + d(z,y).
The most familiar example is ordinary flat n-dimensional space
R^n with d(x,z) = |x - z|. A ball of radius r about a point x is
B_r(x) = { y : d(x,y) < r }.
Throughout the paper the spaces are:
complete -- every Cauchy sequence converges (no missing
points, no open edge);
separable -- there is a countable dense set of points;
geodesic -- any two points are joined by a shortest path;
measured -- there is a reference measure m on the space,
finite on every bounded set, positive on every
nonempty open set ("full support").
The measure m is the tool that lets us average functions and talk
about volume. Concretely we will need: how much volume sits in a
ball, and how that volume grows with the radius.
2. WHERE THE CURVATURE COMES IN
On a smooth Riemannian manifold -- a smooth surface or higher-
dimensional curved shape whose distances come from a metric
tensor g, with a notion of length of tangent vectors -- there are
several notions of curvature. The one that controls volume is
Ricci curvature. The classical fact (Bishop-Gromov) is:
if a Riemannian n-manifold has Ricci curvature >= K, then the
volume ratio
r -> m(B_r(x)) / (omega_n r^n)
is nonincreasing in r, where omega_n is the volume of the unit
ball in R^n. In particular the ratio is at most 1.
So negative lower Ricci curvature makes balls grow faster than
Euclidean balls as the radius increases, and the excess can be
read off from the shrinking of that ratio at small scales. A
space in which the ratio is identically 1 near a point is locally
flat near that point.
The engine of the whole paper is the way this volume comparison
forces a singular, non-smooth space to look Euclidean on fine
enough scales, provided its curvature is bounded below.
3. METRIC MEASURE SPACES AND THE RCD CONDITION
The trouble is that many interesting limits of Riemannian
manifolds are not manifolds: they have cone points, edges, or
lower-dimensional strata. There is no metric tensor to
differentiate. We need a definition of "Ricci curvature bounded
below" that makes sense for such a singular metric measure space.
Two equivalent routes were developed:
(CD) Curvature-Dimension, Lott-Villani and Sturm. The space
satisfies CD(K,n) if a certain convexity holds for the entropy
functional along optimal-transport geodesics between
probability measures. The parameter K is the lower curvature
bound and n the dimension bound.
(Bochner / heat) The same content phrased through the heat
flow and a Bochner inequality, by Erbar-Kuwada-Sturm.
The CD condition also allows genuinely non-Riemannian spaces
(some Finsler-type structures) that behave unlike manifolds. The
fix is the RCD condition (Riemannian Curvature-Dimension),
introduced through the work of Ambrosio-Gigli-Savare:
RCD(K,n) = CD(K,n) plus the requirement that the Cheeger
energy is a quadratic form ("Riemannian" or "infinitesimally
Hilbertian"). Equivalently, the heat flow is linear and the
space admits a Hilbert-space-like first-order differential
calculus.
From now on, "the space" means a complete, separable, geodesic
metric measure space (X,d,m) satisfying RCD(K,n) for some real K
and some integer n >= 2. On such a space one still has, even
though there is no coordinate chart:
- a gradient and a Laplace operator (defined through the
quadratic Cheeger energy and its generator);
- a carre du champ Gamma(f,g) = <grad f, grad g>, the "dot
product of gradients";
- a Hessian for nice functions, and a Bochner inequality;
- a heat semigroup and a continuous heat kernel.
These are the only analytic tools the proof uses. None of them
needs a single coordinate chart; that is the whole point.
4. NONCOLLAPSED: THE MEASURE REALLY IS n-DIMENSIONAL
A subtle but central hypothesis is noncollapsedness. It says the
reference measure is exactly the n-dimensional Hausdorff measure
H^n of the metric, with the usual normalization. In words: the
space has honest dimension n measured by volume, not a lower
dimension. In particular the local volume density is at most 1
everywhere, matching the Bishop-Gromov bound above.
For a limit of Riemannian n-manifolds with lower Ricci bound, the
noncollapsed case is the one where the limit did not lose a
dimension: it does not suddenly become a surface or a line
segment sitting inside R^3. The collapsed case is real and
important, but this paper is about the noncollapsed case only.
A generous reward for assuming it: by De Philippis-Gigli (2018),
in a noncollapsed RCD(K,n) space the reference measure equals H^n,
volume convergence and volume rigidity hold, and the set of
"Euclidean points" has full measure. That theorem is an input,
not something reproved here.
5. TANGENT CONES AND REGULAR POINTS
To say a space "looks Euclidean near a point" we zoom in. Make the
distance smaller and smaller:
fix a basepoint p. For each radius r > 0, form the rescaled
space (X, d/r, m/r^n, p): all distances multiplied by 1/r, so
what used to be size r now has size 1. Let r tend to 0 and ask
what the rescaled spaces converge to.
"Convergence" here is pointed Gromov-Hausdorff (GH) convergence:
metric spaces can converge to a limit even without sitting in one
common larger space. The limit of a sequence of pointed metric
spaces is a new pointed metric space, together with
almost-isometries on larger and larger balls up to smaller and
smaller error. In the noncollapsed setting volume convergence
tracks the limit too.
Every limit obtained this way is called a tangent of X at p. For
a smooth Riemannian manifold every tangent is the Euclidean space
R^n with the flat metric. For a general RCD space a tangent can be
a cone, or a space with lower-dimensional pieces.
Call p an n-regular point when:
every pointed GH tangent of X at p is pointed-isometric to
(R^n, |.|, 0), the flat Euclidean space based at the origin.
That is the cleanest possible "infinitesimally Euclidean" for a
point. De Philippis-Gigli proved that in a noncollapsed RCD(K,n)
space the set of n-regular points has full H^n measure. Full
measure does not mean every point, and it certainly does not mean
the regular set is open. Both of those gaps are exactly where the
difficulty of this paper lives.
6. WHAT A BI-LIPSCHITZ CHART IS
A map F : U -> R^n from a subset U of X is bi-Lipschitz with
constant L >= 1 when, for all x,z in U,
L^{-1} d(x,z) <= |F(x) - F(z)| <= L d(x,z).
The right inequality says F does not stretch distances by more
than L; the left says it does not crush them by more than L. A
bi-Lipschitz map is automatically continuous and injective, and so
is its inverse on the image, so it is a homeomorphism onto its
image. It preserves Hausdorff dimension, the metric's notion of
"size at every scale," and many coarse invariants.
A bi-Lipschitz chart on X is an open U inside X and a
bi-Lipschitz map F from U onto an open subset V of R^n. Such a
chart is the strongest "this piece is really n-dimensional and
metrically finite" statement one can make without any smooth
structure. The paper proves that each regular point sits inside
such a chart with L depending only on n.
Two strengths are deliberately NOT claimed:
- F is not claimed to be smooth, or even differentiable in any
classical sense. There is no smooth structure to compare to.
- the pullback of the Euclidean metric under F is not claimed
to be continuous, or Holder continuous, as a metric tensor.
Nearby results (Colding-Naber) show genuine examples where no
coordinates make the metric tensor Holder continuous, even
though Euclidean tangents exist. So demanding it would be
false. Bi-Lipschitz is the honest target.
The distances on both sides of the display are the ambient
distance d restricted to U. No intrinsic path distance along U is
substituted: the chart controls the actual distances in the
whole space between nearby points, which is a genuinely stronger
and more useful statement.
7. THE MAIN THEOREM
Here is the paper's theorem in full.
Theorem (main). For every integer n >= 2 there is a constant
L_n >= 1 with the following property. Let K be a real number and
let (X,d,H^n) be a noncollapsed RCD(K,n) space. Let p be a
point of X. If every pointed GH tangent of X at p is
pointed-isometric to (R^n, |.|, 0), then there exist:
- an open neighborhood U of p in X,
- an open set V in R^n,
- a homeomorphism F : U -> V,
such that for all x,z in U,
L_n^{-1} d(x,z) <= |F(x) - F(z)| <= L_n d(x,z).
The neighborhood U and the chart F may both depend on the space
and on p; the constant L_n depends only on the dimension n.
Read it as a sentence: "a single regular point always grows a
whole open bi-Lipschitz Euclidean patch around itself, with a
distortion that is uniform in the dimension and nothing else."
8. THE FULL-MEASURE ATLAS COROLLARY
Because regular points have full measure, the theorem immediately
gives a global structure statement.
Corollary. Under the theorem's hypotheses there is an open set
U' containing every n-regular point, with H^n(X minus U') = 0, and a
countable atlas of L_n-bi-Lipschitz charts from ambient-open
subsets of U' onto open subsets of R^n. Each chart uses the
restricted ambient distance, and any two overlapping charts have
a transition map that is L_n^2-bi-Lipschitz.
The proof is short. Take the union of all the chart domains from
the main theorem; it contains every regular point and has full
measure complement. The union is second countable (the space is
separable), so a countable subcover exists. On an overlap,
composing two charts and their inverses multiplies the two
Lipschitz constants, giving L_n^2. One honest warning: the open
region U' may contain points that are not themselves regular. The
corollary does not assert the regular set is open; it only
produces an open region containing it.
9. WHY THIS IS HARD, AND WHAT WAS KNOWN BEFORE
The conjecture is old. Cheeger and Colding, in their 1997 paper on
the structure of spaces with Ricci curvature bounded below, built
manifold neighborhoods that are bi-Holder-parametrized and
explicitly asked (their Theorem 5.14 and Remark 5.15) whether a
bi-Lipschitz upgrade is possible on quantitatively regular
Ricci-limit neighborhoods.
Since then:
- Mondino-Naber (2019) proved countable bi-Lipschitz
rectifiability: measurable pieces around almost every point
map to measurable Euclidean subsets. This is a measurable
almost-everywhere statement, not an ambient-open neighborhood
of a prescribed point.
- Kapovitch-Mondino (2021) proved the whole regular set is
contained in an open manifold region with bi-Holder charts.
Holder, not Lipschitz, and again not at a prescribed regular
point with a fixed distortion.
- Honda-Zhang (2026) recalled the regular-point bi-Lipschitz
conjecture explicitly, in work whose analytic criteria assume
a neighborhood-uniform positive lower bound for injectivity
radius or a Morrey-type control on Laplacians with a uniform
positive Gram determinant. This paper assumes neither.
The obstruction is real and conceptual, not technical. Knowing
that the tangent cone at p is R^n is an infinitesimal statement
at one point. A chart has to compare all pairs of points in a full
neighborhood, across infinitely many scales. A natural attempt is
to build coordinates from harmonic functions that approximate
Euclidean ones on finer and finer scales. That works up to a
point, but the matrix measuring the "size" of the coordinate
gradients can degenerate as the scale shrinks. Each rescaling uses
its own normalization matrix, those matrices need not commute, and
their size may grow without bound. The contribution of this paper
is a way to live with exactly that degeneration.
Two nearby results calibrate the strength. Colding-Naber (2013)
give bi-Lipschitz embeddings of bounded pieces of quantitative
Reifenberg loci into some Euclidean space, possibly of higher
dimension, and construct examples where Euclidean tangents occur
but no coordinates have Holder-continuous metric tensor. Neither
upsets the target: this paper proves an open chart into R^n and is
silent about Holder continuity of the metric tensor.
10. THE SHAPE OF THE PROOF: TWO ESTIMATES AND ONE ITERATION
The proof has three moving parts. The first two are estimates on
the singular space; the third is a purely Euclidean iteration in
the space of coordinate values.
Fix an n-regular point p. Rescale repeatedly by a sequence
s_1 > s_2 > ... tending to 0. The rescaled spaces become flatter
and flatter. On each we build a column u = (u_1,...,u_n) of
harmonic functions that approximate Euclidean coordinates.
Let G = (Gamma(u_j,u_k)) be the n x n matrix of pairwise
gradient dot products, the Gram matrix of the coordinate
gradients. Let R_i(y) be the average of G over the ball of
radius r_i = 2^{-i} centered at y. Let
M = (1/2) Delta G + kappa G m
be the matrix-valued Bochner measure (the kappa corrects for the
curvature lower bound; the precise formula is below). For a
fixed enlargement L > 1 define the positive semidefinite matrix
D_{i,y} = R_i(y)^{-1} [ r_i^{2-n} M(B_{L r_i}(y)) ] R_i(y)^{-1}
This D_{i,y} is the local form of the Bochner measure, conjugated
twice by the ball Gram so that it lives in coordinate-value units
and becomes dimensionless. It is the fundamental small quantity
of the construction.
(Estimate A, cumulative matrix control.) Corrected heat averages
of G produce positive definite forms H_i(x), increasing with i.
These serve as the "current yardstick" at scale r_i. Section 2
proves that the accumulated bounded-overlap sum of the D_{i,y}
is comparable to H_i, and proves the crucial reverse comparison
for suitably weighted sums. This controls how much the
coordinates must be stretched over the whole infinite ladder of
scales. The accumulated total may be unbounded; that is allowed.
(Estimate B, summable directional pinching.) Section 3 proves
that, on the same local forms,
D_{i,y}[ u(x) - u(z) ] <= C r_i^2 ( E_i(x) + E_i(z) ),
where the defect functions E_i are nonnegative and their total
sum over all scales, sup over x, tends to zero:
sup_x sum_i E_i(x) = o(1). Here D[xi] = xi^T D xi is the
quadratic form. Crucially the sum of the defects, not the sum of
the square roots of the defects, is small. That single fact is
what lets the infinite iteration in Section 4 converge even
though individual per-scale errors need not be summable on their
own.
(The iteration.) Section 4 works on the open coordinate-value
set Omega = u(B_1(p)) in R^n. It builds smooth maps
h_0, h_1, h_2, ... : Omega -> R^n by
h_0(P) = P, h_{i+1}(P) = h_i(P) + Dh_i(P)^{-T} v_i(P)
for smooth covector fields v_i assembled from the D_{i,y}. The
leading change in the pullback metric at step i is the positive
sum of the local forms D_{i,y}. Differentiating the factor
Dh_i(P)^{-T} produces a diagonal second-derivative term; keeping
a recurrence for that term is what allows the errors to be
summed using sum_i E_i alone. A second estimate on jets of all
orders, with factorial growth, keeps the iterates under control
despite each step losing one derivative. The limit F = lim h_i
o u is the chart.
Four more remarks make the plan honest.
- The harmonic-coordinate Gram matrices R_i and the heat
matrices H_i are only compared up to a multiplicative factor
tending to one. They are never assumed bounded or convergent.
- The matrices at different scales do not commute. Section 2
includes a careful matrix-algebra lemma that extracts what is
needed (eigenvalue comparisons and summed quadratic-form
bounds) without ever commuting two matrices.
- Every enlargement of a ball is fixed before the late
rescaling is chosen. Uniformity statements are always
quantified in that order, and "o(1)" always means uniform over
the centers and the finitely many scales under discussion.
- The paper does not reprove the analytic background (Bochner,
heat kernel, Poisson regularity, volume rigidity). It cites
these and uses them. The list of cited inputs is collected in
the reading guide at the end.
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PART II. THE ANALYTIC TOOLBOX
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The proof uses no coordinates on X. It uses five analytic objects
that exist on every RCD space. Each is covered by a cited theorem,
used as a black box. This part lists them, defines every symbol,
and states without proof the inequalities the rest of the tour
relies on.
11. SOBOLEV FUNCTIONS, THE ENERGY, AND HARMONIC FUNCTIONS
On a metric measure space, the role of "functions with one square-
integrable derivative" is played by the Sobolev space W^{1,2}. The
Cheeger energy of a function f is
Ch(f) = (1/2) * inf limit of integral |grad f_k|^2 d m
taken over the standard approximation procedure; in the RCD case
it is a quadratic form, so it defines a bilinear energy and a
gradient. Write Gamma(f,g) = <grad f, grad g> for the carre du
champ (dot product of gradients) and Gamma(f) = Gamma(f,f).
The Laplace operator Delta is the generator of this energy: for
nice f and every compactly supported Lipschitz test function phi,
integral <grad f, grad phi> d m = - integral phi * Delta f d m.
A function w is harmonic on an open set if Delta w = 0 there. It
is a "constant-Laplacian" solution if Delta w = a * m for some
real constant a. Such functions exist and are as regular as the
space allows; in particular locally bounded constant-Laplacian
solutions are locally Lipschitz.
Harmonic functions are the natural candidates for coordinates on
a singular space, exactly as on a smooth manifold where a harmonic
coordinate chart is a chart whose coordinate functions are
harmonic. They are chosen because the Bochner inequality gives
strong control over the squared gradient Gamma(w) of a harmonic
function -- the "energy" of the coordinate.
The fundamental approximation statement (Ambrosio-Honda, local
spectral convergence, 2018) is: on a ball that is close enough to
a Euclidean ball and whose rescaled curvature is small enough,
harmonic functions that match prescribed boundary data are close,
in energy and locally uniformly, to the corresponding Euclidean
harmonic functions. In particular harmonic functions can be made
arbitrarily close to the affine coordinate functions x_1,...,x_n
of R^n on a small ball.
12. THE GRAM MATRIX, THE BOCHNER MEASURE, AND PINCHING
Fix a column u = (u_1,...,u_n) of harmonic functions. Its Gram
matrix is
G = ( Gamma(u_j,u_k) )_{j,k=1..n} ,
an n x n matrix of functions; G is positive semidefinite and it
records the squared lengths and pairwise angles of the coordinate
gradients. Its average over a ball is our scale-by-scale yardstick:
R_i(y) = average over B_{r_i}(y) of G d m , r_i = 2^{-i}.
If G were Identically the identity matrix we would already have
Euclidean coordinates. The whole difficulty is that G is only
close to the identity on small scales, and the closeness can
degrade as the scale shrinks; R_i(y) need not stay bounded above
or below.
The tool that measures the size of G is the dimensional Bochner
inequality, valid on every RCD(K,n) space for test functions. For
a constant-Laplacian solution w with Delta w = a, and with
kappa = max{-K, 0}, define the matrix measure
M(w) = (1/2) Delta Gamma(w) + kappa Gamma(w) m .
Its content: the Bochner inequality says M(w) is a nonnegative
measure, locally finite, and dominates the squared Hessian:
M(w) >= |Hess w|^2 m .
Here Hess w is the Hessian of w, defined through the second-order
calculus of Gigli; its trace is Delta w, because the space is
noncollapsed and has dimension exactly n almost everywhere. For a
constant b (a "shift" toward the metric tensor g) one has the
shifted inequality
M(w) - (2 b Delta w - n b^2) m >= |Hess w - b g|^2 m .
The left side is a nonnegative quadratic form in the pair (w,b).
Polarizing gives a nonnegative bilinear form M(v,w), and matrix
Cauchy-Schwarz for its density gives, for any Borel set E,
|M(v,w)|(E) <= sqrt( M[v](E) * M[w](E) ) .
These measure-level statements are the only way the proof uses
curvature. The point of the shift b is that it lets us center the
Hessian on a multiple of the metric, which is what "close to
Euclidean" means at second order.
13. THE HEAT SEMIGROUP AND THE TWO HEAT BOUNDS
Parallel to harmonic functions sits the heat semigroup
(P_t f)(x) = integral p_t(x,z) f(z) d m(z) ,
which solves the heat equation d/dt (P_t f) = Delta (P_t f) with
P_0 f = f. On an RCD space the heat kernel p_t(x,z) is a
continuous symmetric function. On a fixed bounded region, for
0 < t <= 1, the following hold (Jiang-Li-Zhang 2016, Theorem 1.2
and Corollary 1.2):
Gaussian upper bound:
p_t(x,z) <= C t^{-n/2} exp( - d(x,z)^2 / (C t) ),
first spatial gradient bound:
|grad_x p_t(x,z)| <= C t^{-(n+1)/2} exp( - d(x,z)^2 / (C t) ),
on-diagonal lower bound (when d(x,z) <= A sqrt(t)):
p_t(x,z) >= c_A t^{-n/2} .
The constants are uniform on the rescaled balls of the construction
and depend only on the dimension and the fixed local data.
Why heat instead of plain harmonic averages? Because a heat average
of the Gram field enjoys a monotonicity in the time variable t, and
t = r^2 is the scale. Section 2 turns this into matrices H_i(x)
that are monotone as the scale decreases (H_i <= H_{i+1}), which
plain ball averages R_i do not have. Monotone yardsticks are what
make an infinite telescoping sum possible without commuting the
matrices at different scales.
14. RESCALING AND THE SMALL PARAMETERS
The whole proof happens along a sequence of rescalings of a fixed
n-regular point p. We rescale by a sequence s_1 > s_2 > ... -> 0:
(X_nu, d_nu, m_nu, p_nu) with d_nu = d / s_nu, m_nu = m / s_nu^n .
Along this sequence, volumes of fixed-radius balls converge to
their Euclidean values and the lower curvature bound, rescaled,
tends to 0: kappa_nu -> 0. The spaces become flatter and flatter.
Two small sequences drive the estimates:
- kappa_nu, the rescaled curvature lower bound (tends to 0);
- o(1), a generic name for any quantity tending to zero, always
meant uniformly over the centers and scales under discussion.
Choosing the sequence "late enough" is the recurring order of
quantifiers: fix all enlargement constants first, then take the
rescaling late enough that every finiteness threshold holds. This
is not bookkeeping; it is the only way to keep the constant L_n
independent of the specific space.
15. WHY THE HARD PART IS "IN VALUE SPACE"
Suppose for a moment that the harmonic coordinates u were already
a bi-Lipschitz chart with constant 1. Then done. The problem is
the correction: at each scale the coordinates are off from
Euclidean by something measured by D_{i,y}, and the corrections at
different scales do not add up trivially.
The paper's discipline is to never correct the coordinates by
changing them on X. Instead it uses the harmonic map u to push
everything into the open set
Omega = u(B_1(p)) inside R^n,
called the space of coordinate values. On Omega the Euclidean
analysis tools -- ordinary derivatives, ordinary linear algebra,
invariance of domain -- are all available. The chart is built by
an explicit smooth iteration on Omega. The geometric content of
the space is imported only through the local matrices D_{i,y} and
the heat yardsticks H_i, which are the inputs to the iteration.
That separation is what allows the proof to survive the
degeneration of R_i: the iteration never differentiates the scale
metrics H_i and never inverts a pointwise differential of the
singular map u on X. It only differentiates smooth maps defined
on the flat model Omega.
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PART III. PAPER SECTION 1: UNIFORM HARMONIC COORDINATES
============================================================
Section 1 builds the open coordinate domain, then proves that the
harmonic coordinates approximate Euclidean coordinates on finer and
finer scales, and finally that the harmonic map u is a
homeomorphism onto an open subset of R^n. It is careful to let the
size of the coordinates degenerate as the scale shrinks; it only
insists that on each fixed scale the approximation is uniform in
all centers.
16. WHAT SECTION 1 DELIVERS
By the end of the section we have, on a large ball around p in a
late enough (very flat) rescaling:
- a column u = (u_1,...,u_n) of harmonic functions that behaves
like orthonormal Euclidean coordinates near every point, up to
errors tending to zero with the rescaling, uniformly in the
center;
- a positive definite ball-Gram matrix R_i(y) at every dyadic
scale r_i = 2^{-i} and center y, whose normalized harmonic
map U = R_i(y)^{-1/2}(u-u(y))/r_i is an arbitrarily good model
of R^n on a fixed enlargement of B_{r_i}(y);
- a homeomorphism statement: u is injective, its image is open,
and it separates points at a scale proportional to their
distance.
The unavoidable gap left open at the end is that R_i(y) may
degenerate as i grows. Sections 2-4 exist to deal with that gap.
Section 1 says nothing about a uniform lower bound on R_i; it
merely records exactly how R_i can vary from scale to scale.
17. LEMMA 1.1: THE LOCAL ANALYTIC FACTS
The first lemma is a clearance lemma: it lists every analytic
fact used later and justifies them from cited input.
Lemma 1.1 (local analytic facts). Let (Y,d,m) be a noncollapsed
RCD(-kappa,n) space and Omega an open subset. A locally bounded
solution w of Delta w = a in Omega with a constant is locally
Lipschitz and is a local test function; it has a local L^2
Hessian; for two such functions Gamma(w_1,w_2) is locally in
W^{1,2}, and the Hessian product rule holds locally. The measure
M(w) = 1/2 Delta Gamma(w) + kappa Gamma(w) m is locally finite
and nonnegative, and for every constant b,
M(w) - (2 b Delta w - n b^2) m >= |Hess w - b g|^2 m .
The left side is a nonnegative quadratic form in the pair (w,b)
with both entries scaled together.
Moreover, on fixed interior balls, bounded Lipschitz cutoffs with
bounded Laplacian exist, and there are a number d_* > 1 and a
constant C such that for every h in W^{1,2}_0(B),
||h||_{L^2} + ||h||_{L^{2 d_*}} <= C ||grad h||_{L^2} .
The constants are uniform on the rescaled balls in use.
In plain terms: constant-Laplacian solutions are as smooth as the
space allows, the Bochner measure M is a genuine nonnegative
measure even when Hess w is only an L^2 object, the shifted
inequality always centers correctly, and the Dirichlet boundary
problem has coercivity (a Sobolev inequality with exponent
2 d_* > 2). The proof assembles facts from Bacher-Sturm,
Kell, Jiang, De Philippis-Gigli, Han, Gigli, Connell et al.,
Mondino-Naber, and Hajlasz-Koskela; the paper makes a point of
handling dimension n = 2 as well by choosing d_* larger than both
n and 2.
Two subtleties worth noticing:
- The Hessian inequality for M is stated as an inequality of
measure-valued quadratic forms. There is no pointwise Hessian
to compare against; the "square" |Hess w - b g|^2 lives as a
density of a nonnegative measure through polarization. Section
2 will use this heavily.
- The cutoffs with bounded Laplacian exist only on fixed
interior scales. They cannot be made uniform over all radii.
18. LEMMA 1.2: AN EXPANDING ALMOST EUCLIDEAN REGIME
The second lemma sets up the actual sequence of spaces used
throughout. It is a compactness-and-rigidity statement.
Lemma 1.2 (expanding almost Euclidean regime). Fix an
n-regular point p of the space in the main theorem. There are
rescalings
(X_nu, d_nu, m_nu, p_nu) = (X, d/s_nu, m/s_nu^n, p),
numbers kappa_nu decreasing to 0, and radii T_nu increasing to
infinity with kappa_nu T_nu^2 -> 0, such that the following hold.
Each X_nu is RCD(-kappa_nu, n). For a sequence delta_nu -> 0,
every ball with center in B_{64 T_nu}(p_nu) and radius at most
64 T_nu satisfies
| m_nu(B_r(y)) / (omega_n r^n) - 1 | <= delta_nu ,
d_GH( closure B_r(y) , closure B_r(0^n) ) <= delta_nu r .
There are harmonic columns u_nu on B_{32 T_nu}(p_nu) converging
on every fixed ball locally uniformly and strongly in W^{1,2} to
Euclidean coordinates, and
|grad u_nu| <= C_n on B_{16 T_nu}(p_nu) .
Two things are happening, both worth stating plainly.
- Because p is n-regular, the rescalings converge to flat R^n.
Noncollapsed volume convergence then identifies the limit
measure with H^n. The convergence of volumes is uniform over
the center within the working ball; the proof is by
contradiction, using that pointwise convergence plus
compactness of centers yields a uniform rate.
- The radii T_nu are chosen to grow to infinity slowly (slowly
compared to how fast the rescalings flatten). So on the
expanding balls of radius T_nu, the space is very nearly
Euclidean with rapidly shrinking error.
The uniform almost-Euclidean statement is extracted from volume
rigidity (De Philippis-Gigli): tiny volume excess forces tiny
Gromov-Hausdorff distance. Bishop-Gromov then traps the volume
ratio between its model-maximal value and 1, giving the display.
Harmonic coordinate columns are delivered by the local spectral
convergence of Ambrosio-Honda, applied on fixed enlargements and
then combined by a diagonal subsequence. The gradient bound
|grad u_nu| <= C_n is dimensional: the coordinate functions have
size O(r) on a ball of radius r, and the interior gradient estimate
carries the matching factor r^{-1}.
From here on the index nu is suppressed. A statement like "for all
large enough rescalings" is part of the meaning of every o(1).
19. DYADIC RADII AND THE BALL-GRAM MATRIX
With the rescalings fixed, set
r_i = 2^{-i}, i = 0, 1, 2, ... ,
R_i(y) = average over B_{r_i}(y) of G d m ,
where G = (Gamma(u_j,u_k)) is the Gram matrix of the harmonic
column. R_i(y) is the ball Gram matrix; it is the yardstick at
scale r_i and center y. Its inverse R_i(y)^{-1} shows up in the
"normalized" coordinate map
U_{i,y} = R_i(y)^{-1/2} (u - u(y)) / r_i .
Roughly: subtract the coordinate value at the center, rescale the
ball of radius r_i to unit size, and correct the target by the
inverse square root of the average Gram matrix. U_{i,y} is the
coordinate map that "would be Euclidean if the space were flat and
R_i were perfectly Euclidean". Lemma 1.3 makes this precise.
20. LEMMA 1.3: UNIFORM HARMONIC NORMALIZATION
This is the technical heart of Section 1. It says the normalized
harmonics are almost Euclidean on every fixed enlargement of every
ball, uniformly in the center, at every dyadic scale.
Lemma 1.3 (uniform harmonic normalization). For every pair of
fixed constants A,C >= 1, the matrices R_i(y) are positive
definite for y in B_A(p) and all i >= 0. In the rescaled ball
B_{C r_i}(y) the map U_{i,y} is, after an orthogonal change of
target coordinates, an o(1) approximation to Euclidean
coordinates, uniformly and strongly in interior W^{1,2}. Its
gradients are uniformly bounded on each fixed interior
enlargement, and its Gram matrix converges strongly in every
finite L^p on that enlargement to the identity Id.
For every fixed C >= 1, if d(y,z) <= C r_i and
C^{-1} r_i <= r_j <= C r_i, then
(1 - o(1)) R_i(y) <= R_j(z) <= (1 + o(1)) R_i(y) .
For every a > 0, after choosing a sufficiently late rescaling
and a constant C_a depending only on a,
C_a^{-1} 2^{-2a|i-j|} R_j(y) <= R_i(y) <= C_a 2^{2a|i-j|} R_j(y) .
In particular, since R_0(y) = Id + o(1) on fixed balls,
c_a r_i^{2a} Id <= R_i(y) <= C Id .
The first sentence gives positivity. The second is the core:
U_{i,y}, viewed on a fixed enlargement of its own ball of radius
r_i after rescaling to unit size, looks like the identity
coordinate map. The word "uniformly" is essential: the error tends
to zero at once over every center y and every scale i in a fixed
ball. The third sentence (nearby-R) says that Gram averages on
nearby balls of comparable radius agree up to a factor 1 + o(1).
The fourth (slow-R) says that between two dyadic scales i and j,
R can grow or shrink by at most a controlled exponential factor
2^{2a|i-j|}, with arbitrary prescribed a > 0 once the rescaling is
late enough. The final line tracks R_0(y) = Id + o(1): the scale-0
Gram average is almost Euclidean, and back-propagating using slow-R
gives a lower bound r_i^{2a} Id on R_i.
The proof is a compactness induction. Saying it is "almost
Euclidean at scale r_i" means: after normalizing by an invertible
matrix T, the rescaled map T(u - u(y))/r_i lies within a small
sup distance of the standard Euclidean coordinates on the ball of
radius 2. The proof shows this property improves strictly under
passing from scale r to a fixed fraction b = 2^{-m} of r:
- On an exactly Euclidean ball, applying interior harmonic
derivative estimates to V - x, one gets that the rescaled
normalized map is strictly closer to Euclidean after
contracting by a dyadic factor b chosen small enough -- its
derivative at the center is within O(tau) of the identity, and
its Taylor remainder is controlled by the Hessian estimate.
- On an almost-Euclidean ball in the sequence, the same
improvement propagates by contradiction. If it failed, rescale
the failing ball to unit size; compactness of rescaled
harmonic maps on almost-Euclidean spaces (Ambrosio-Honda)
yields convergence of the normalized harmonic to a Euclidean
harmonic column; that limit has the strict improvement
property; contradiction.
- The induction runs on the finitely many dyadic residue
classes modulo m, so every dyadic scale is reached.
The "mean Gram matrix of a normalized column is Id + O(tau)" step
is what lets the proof replace an inductively chosen correction
matrix T with the canonical R_i(y)^{-1/2}; the two differ by an
orthogonal matrix times an O(tau) error, which is absorbed.
Fixed interior enlargements are handled by fixing the largest
enlargement first and running the induction on an "ancestor" ball
large enough to contain it; the number of contraction steps between
the ancestor and any descendant is bounded by a fixed integer, so
all constants survive.
The nearby-R estimate is the same argument run on a single
enlargement containing both balls. The slow-R estimate is
nearby-R iterated along adjacent scales, doubling the tolerance
once per step and choosing the per-step tolerance so that the
factor after |i-j| steps is at most 2^{2a|i-j|}.
21. LEMMA 1.4: SMALL NORMALIZED BOCHNER MASS
Lemma 1.4 (small normalized Bochner mass). Uniformly for y in
B_A(p) and all i >= 0, for fixed A,C >= 1,
r_i^{2-n} M( B_{C r_i}(y) ) <= o(1) R_i(y) .
The Bochner measure of a ball of radius comparable to r_i, scaled
by r_i^{2-n} and conjugated by R_i(y)^{-1}, is small compared to
the identity.
The proof is a one-step rescaling exercise with the local
calculus, and it is short enough to keep in view.
- In units r_i, apply Lemma 1.1 to a scalar combination of
U = U_{i,y}. The rescaled Bochner matrix is
M_{r_i}(U) = r_i^{2-n} R_i(y)^{-1/2} M(u) R_i(y)^{-1/2},
with curvature term kappa r_i^2.
- Pick a nonnegative cutoff phi, equal to 1 on B_C and supported
on a larger fixed ball, with uniformly bounded gradient and
Laplacian.
- Integrate the definition of M against phi. The curvature term
is kappa r_i^2 * (integral of phi Gamma(b.U)), and the
harmonic-energy term is (1/2) integral (Gamma(b.U) - |b|^2)
Laplacian(phi). Both terms tend to zero uniformly by Lemma
1.3 (normalized Gram -> Id; rescaled curvature -> 0).
- Positivity of the Bochner measure bounds its mass on the
inner ball by what the cutoff saw. Polarizing over the fixed
finite dimension gives the matrix inequality.
There is no subtlety about singular parts: the proof only uses M
on Lipschitz cutoffs, and M is a nonnegative measure there.
22. LEMMA 1.5: OPEN HARMONIC COORDINATES
Lemma 1.5 (open harmonic coordinates). For every fixed A >= 1,
the map u is a homeomorphism from B_A(p) onto an open subset of
R^n once the rescaling is late enough. There is a fixed c > 0
such that, whenever the balls involved stay in a fixed interior
region,
{ P : |P - u(x)|_{R_i(x)^{-1}} < c r_i } is contained in
u( B_{r_i/2}(x) ) .
For every fixed D >= 1, if D^{-1} r_i <= d(x,z) <= D r_i, then
| |u(z) - u(x)|_{R_i(x)^{-1}} - d(x,z) | <= o_D(1) r_i .
The first display is the crucial "ellipsoid containment": the image
of a ball of radius r_i/2 contains a small ellipsoid, in the
R_i(x)^{-1}-norm on coordinate values, of radius c r_i. The second
display is the "separation-scale" estimate: for two points at a
comparable dyadic distance, the coordinate difference in the
R_i(x)^{-1} norm equals the true distance up to a relative error
tending to zero.
The proofs are pleasingly geometric.
- Separation-scale is just Lemma 1.3: at a dyadic scale
comparable to d(x,z), U_{i,x} is close to Euclidean
coordinates, and comparing z to x through it gives the
display.
- Injectivity follows immediately: two distinct points in
B_A(p) at dyadic scale comparable to their distance have
coordinate values distinct, because the error term is a
fraction of the distance.
- Ellipsoid containment is proved by an iterative preimage
argument on the space X. Start at x_0 = x with a target P in
the ellipsoid, and a dyadic radius rho_0 = r_i.
Approximate surjectivity of the normalized Euclidean
coordinate map provides x_{j+1} within (c + eta) rho_j of
x_j, with the normalized distance of P from u(x_{j+1}) at most
eta rho_j; then rho_{j+1} = rho_j / 4, and the induction
continues. The total walked distance is at most (c + eta) *
(sum of rho_j) <= rho_0 / 2, so all centers stay in working
range. Completeness gives a limit point x_* with u(x_*) = P,
and the upper bound R_i <= C Id converts the
R_i(x)^{-1}-norm error into an ordinary coordinate residual
going to zero.
- Open image: apply ellipsoid containment locally at each point
to get an open image. Homeomorphism: a continuous injection
onto an open subset of a proper space is a local homeomorphism,
and local injectivity plus continuity of the inverse (from the
separation estimate) makes it global.
Two remarks round out Section 1.
- No openness of the set of exact regular points is used. The
proof runs entirely on the level of a fixed point p and the
local flatness around it; it produces an open ball around p
after the fact.
- The scale of the working region is expanding: it grows with
T_nu. That is why Section 2-4 can afford fixed enlargements
and still demand uniform constants.
============================================================
PART IV. PAPER SECTION 2: HEAT AVERAGES AND CUMULATIVE
MATRIX CONTROL
============================================================
Section 1 left open the possibility that the ball Gram matrices
R_i degenerate. Section 2 replaces them, for the purpose of the
proof, with heat averages that are monotone in the scale, and then
proves a cumulative comparison: a bounded-overlap sum of the local
Bochner matrices D_{i,y} is comparable to the heat yardstick Q_i,
and -- crucially -- the heat yardstick is bounded by a weighted
sum of those same matrices. That reverse direction is what Section
4 needs to conclude that the iteration is stable.
23. WHAT SECTION 2 DELIVERS
Three matrices appear and must not be confused:
- R_i(x): the plain ball average of the Gram matrix G at scale
r_i. No monotonicity in i.
- Q_i(x): the heat average of G at time r_i^2, plus a small
correction. Positive definite. Monotone: Q_i >= Q_{i+1}.
- H_i(x) = Q_i(x)^{-1}: the inverse. Monotone in the other
direction: H_i <= H_{i+1}.
At the end of Section 2 we have a constant C_* (depending only on
n and on the fixed enlargement constants) such that for every
testing center x and every integer k >= 0:
Id + sum_{i<k} Z_i(x) <= C_* H_k(x),
H_k(x) <= C_* ( Id + sum_{i<k} S_i(x) ),
where Z_i is the unweighted sum of the local D-forms at scale i
(over the active centers of the patch) and S_i is a weighted sum
with weights in [0,1] at least one of which equals one and has a
covering ball. The second inequality is the deep one; without it
nothing stops H_k from running off to infinity faster than the
local mass accounts for.
Five smaller lemmas feed into these two inequalities. The order
below is the paper's order.
24. THE HEAT YARDSTICK, BUILT CAREFULLY
Fix a cutoff chi with the following properties:
chi = 1 on B_T(p) ,
supp chi inside B_{2T}(p) ,
|grad chi| <= C / T ,
|Delta chi| <= C / T^2 .
Here T = T_nu is the expanding radius from Lemma 1.2. Extend chi G
by zero outside its support. The distributional Laplacian of chi G
is then the compactly supported matrix measure
Delta(chi G) = 2 chi M - 2 kappa chi G m + E_cut m ,
E_cut = (Delta chi) G + 2 Gamma(chi, G) .
The first two terms on the right are the "true Bochner" content of
the chip cutoff; E_cut is a cutoff error supported in the annulus
between radii T and 2T. Because the cutoff lives far from any
testing center (distance at least T/2), heat averaging will make
E_cut negligible. That is the point of the expanding domain T_nu.
Given a parameter lambda > 0 that will tend to 0, define
Q_i(x) = P_{r_i^2}(chi G)(x) + lambda r_i^2 Id ,
H_i(x) = Q_i(x)^{-1} .
Q_i is positive definite because the heat average P_{r_i^2}(chi G)
is a positive semidefinite matrix and the lambda r_i^2 Id term is
strictly positive. H_i is also positive definite.
The correction term lambda r_i^2 Id is a technical necessity: it
absorbs the curvature contribution and the cutoff error uniformly,
without which Q_i need not be positive definite in the strict
sense that later matrix inequalities need.
25. LEMMA 2.1: KERNEL DIFFERENCES AND MEASURE SOURCES
The proof needs two comparisons of heat kernels: a short-time
bound on the average of the kernel over a ball, and a long-time
bound on how much that average differs from a point evaluation.
Lemma 2.1 (kernel differences and measure sources). Fix
C_0 >= 1, points x,y in a fixed testing region, and
d(x,y) <= C_0 r with 0 < r <= 1. Put
K_t(z) = average over B_r(y) of p_t(a,z) d m(a) ,
D = d(x,z) .
After enlarging fixed constants:
K_t(z) <= C r^{-n} exp( - c D^2 / r^2 ) for 0 < t <= r^2 ,
| K_t(z) - p_t(x,z) | <=
C r t^{-(n+1)/2} exp( - c D^2 / t ) for r^2 <= t <= 1 .
If f in W^{1,2} cap L^inf has compact support and its Laplacian
is a finite signed measure mu of compact support, then
P_t f - P_s f = integral_s^t P_tau mu d tau , 0 < s < t .
The lemma is two-sided and the proof is worth summarizing because
the whole section's difficulty is concentrated here.
Short-time bound. If z is close to the averaging ball, symmetry and
total heat mass bound the average of the kernel by one; dividing by
volume gives C r^{-n}. If z is far, the Gaussian upper bound alone
gives C t^{-n/2} exp(-cD^2/t); for t <= r^2 and D/r bounded below,
its small-time power is absorbed in the exponential.
Long-time bound. For a in B_r(y), d(a,x) <= (C_0+1) r. Integrate
the first spatial gradient estimate along a minimizing curve from
x to a; the inequality d(b,z)^2 >= D^2/2 - d(b,x)^2 absorbs the
variation of the Gaussian along the curve. This gives the bound
pointwise for almost every fixed source pole z. To extend to every
pole, use symmetry and continuity of the heat kernel: the full-
measure set of "good" poles is dense, and passing to the limit
with relaxed Gaussian constants gives the inequality at every z.
No gradient representative is ever paired with a singular measure:
all the integration-by-parts is done against the smooth heat
kernel at positive time.
Semigroup identity against a measure. Write the distributional
Laplacian of f as mu; test against positive-time heat regularizations
of compactly supported test functions, integrate by parts using
symmetry, and use the local W^{1,2} regularity of p_t at positive
time. The result is the displayed identity, valid in distributions
and then pointwise by continuity of both sides.
26. LEMMA 2.2: MONOTONE HEAT GRAM MATRICES
Lemma 2.2 (monotone heat Gram matrices). There is a real
lambda = lambda_nu > 0, tending to 0 along the rescalings, such
that at all fixed testing centers and for 0 < t <= 1,
P_t(chi M) <= d/dt( P_t(chi G) + lambda t Id )
<= 3 P_t(chi M) + C Id .
With Q_i and H_i defined as above,
(1 - o(1)) R_i(x) <= Q_i(x) <= (1 + o(1)) R_i(x) ,
Q_0(x) = Id + o(1) ,
H_i(x) <= H_{i+1}(x) <= (1 + o(1)) H_i(x) .
The nearby-center and slow-growth estimates for R_i (Lemma 1.3)
hold for Q_i as well, and for every prescribed a > 0, once the
rescaling is late enough,
Q_j(x) <= C_a 2^{2a(i-j)} Q_i(x) (j <= i) ,
|| H_i(x) || <= C_a 2^{2a i} .
The derivative estimate is the analytical content. Differentiating
P_t(chi G) and using the semigroup identity gives
d/dt P_t(chi G) = 2 P_t(chi M) - 2 kappa P_t(chi G)
+ P_t E_cut .
Adding lambda Id and using the chip cutoff estimates:
- The cutoff error P_t E_cut is uniformly small: E_cut has total
variation at most C T^{n-2}, its support is a distance at least
T/2 from fixed testing centers (for T large along the
expansion), and the Gaussian estimate gives
sup_{0<t<=1} || P_t E_cut(x) ||
<= C T^{n-2} sup_{0<t<=1} t^{-n/2} exp(-c T^2 / t) = o(1) .
- The curvature contribution has norm at most 2 kappa ||G||_inf
on the fixed testing region.
- Choose lambda >= (that small bound) + 2 kappa ||G||_inf + a
positive number tending to zero. Then the sum of the last three
terms on the right is nonnegative and bounded above, giving
the two-sided estimate displayed.
The comparison (1-o(1))R_i <= Q_i <= (1+o(1))R_i is a comparison of
relative, not just absolute, sizes. Fix x and i, conjugate by
R_i(x)^{-1/2}: on every fixed ball B_{L r_i}(x) the conjugated
Gram field converges in normalized L^1 to Id (Lemma 1.3), so the
heat-kernel-weighted error tends to zero. The tails -- the shell
with radius 2^m r_i up to a distance and beyond -- are computed by
slow growth of the conjugated average (bounding it by C 2^{2am})
against the Gaussian decay of the kernel; both contributions are
summable in m uniformly in i. The missing-heat-mass correction
from (1 - chi) is bounded the same way using total heat mass one.
Then lambda r_i^2 R_i(x)^{-1} is bounded by C_a lambda r_i^{2-2a}
Id = o(1) Id, which is why a is chosen below 1.
Monotonicity: from the derivative estimate, d/dt(P_t(chi G) +
lambda t Id) >= 0, so Q_i is decreasing in i (larger i = smaller
r_i^2 = earlier time). Therefore H_i = Q_i^{-1} is increasing in
i. Relative comparison with R_i and nearby-R of Lemma 1.3 propagate
the adjacent-scale upper bound.
The slow-growth estimate for Q_i and H_i follows by iterating the
adjacent-scale estimate, with the per-step tolerance chosen after
the prescribed a. Note ||H_i|| grows like 2^{2 a i} -- this is
allowed, and Section 4 never says otherwise. The reverse comparison
in Lemma 2.6 is exactly what quantifies how bad this growth can
possibly be.
27. LEMMA 2.3: THE DYADIC HEAT COMPARISON
This lemma produces the small series F_i that will appear
everywhere afterwards and states two inequalities: a decrement
estimate for Q_i - Q_{i+1} and a bilinear one relating R_i - Q_i
to F_i. Both are indispensable for the linear algebra in Section
2.6.
Lemma 2.3 (dyadic heat comparison). Fix A, C_0 >= 1 with
d(x,y) / r_i <= C_0, and fix a bounded region for x. There is a
fixed exponent gamma > 0 (taken to be 1/2), with the following
properties for all sufficiently late rescalings. Set
F_i(x) = 2^{-gamma i} Id + sum_{j=0}^{i}
2^{-gamma(i-j)} N_j(x) ,
N_j(x) = r_j^{2-n} M( B_{r_j}(x) ) .
For the prescribed enlargement A,
c_A r_i^{2-n} M( B_{A r_i}(x) ) <= Q_i(x) - Q_{i+1}(x)
<= C F_i(x) .
For d(x,y) <= C_0 r_i and v,w in R^n,
| v^T ( R_i(y) - Q_i(x) ) w |
<= C |v|_{Q_i(x)} sqrt( F_i(x)[w] ) .
The series F_i(x) is a damped sum of the local Bochner masses
N_0,...,N_i at all coarser scales, with the dyadic ratio 2^{-gamma}
as damping, plus an explicit background term 2^{-gamma i} Id to
guarantee positivity at every scale. Section 3 will show that the
individual N_j are summable in a strong sense; Section 4 only
needs to know that such an F_i exists.
The content of the lemma, phrase by phrase:
- The left side of the decrement estimate says that the decrement
Q_i - Q_{i+1} is at least as large as the local Bochner mass
A r_i inside the ball of radius A r_i. This is the lower bound
that lets the sum of decrements dominate the sum of local
masses later.
- The right side of the decrement estimate says the decrement
cannot be much bigger than F_i. Small F_i means the yardstick
barely changes between consecutive scales.
- The bilinear estimate says that the ball average R_i(y) at a
nearby center differs from the heat yardstick Q_i(x) only by
something of size |v|_{Q_i} sqrt(F_i). This is the "relative
comparison" -- magnitudes of operators are compared, not just
norms -- and it is what makes the linear-algebra estimates of
Section 2.4-2.6 work.
Proof sketch. Write r = r_i, suppress x, and let A f denote the
average of f over B_r(y). The exact semigroup identity for the
chip cutoff reads
R_i(y) - Q_i(x) = A P_1 f - P_1 f(x) - lambda r^2 Id
- integral_0^{r^2} (A P_t S) d t
- integral_{r^2}^{1} ( A P_t S - P_t S(x) ) d t ,
with f = chi G and S = Delta f. The first two lines are "endpoint
heat regularizations"; the third is "short-time", where the
kernel-based approximations do all the work; the fourth is
"long-time", where the difference between average and point
evaluation is controlled by the first spatial gradient of the
kernel.
The local source chi M is decomposed into contributions by time
block I_q = [r_{q+1}^2, r_q^2]. On each block, integrate the two
kernel comparisons of Lemma 2.1 across the ball and against the
measure M, and observe that the Gaussian at scale r_q is constant
on shells of radius between 2^{h-1} r_q and 2^h r_q, with a
coefficient k_h = 2^{(n-2)h} exp(-c 4^{h-1}) that is summable
against 2^h. The sum over blocks and shells reverses into a single
sum that gives both decrement estimates.
The exterior of B_1(x) is controlled by the raw-mass-growth
estimate
M(B_s(x))[v] <= C (s^{n-2} + kappa s^n) |v|^2 ,
applied shell by shell, and by the exponentially decaying
integrated kernel bounds outside B_1(x). The curvature source is
bounded by total heat mass one at short time and by
integral_r^2^1 r t^{-1/2} d t at long time; both together are
bounded by C r. The cutoff source E_cut is handled by the Gaussian
decay far from x.
With all those contributions collected into the raw bilinear
estimate
|v^T(R_i(y)-Q_i(x))w| <= C r |v||w|
+ C sum_{j=0}^i 2^{-(i-j)} sqrt( N_j[v] N_j[w] ) ,
the normalized bilinear comparison follows from slow growth of
N_j relative to Q_i and weighted Cauchy-Schwarz. The rest of
Section 2 only ever uses the normalized form.
28. A SMALL INDEXING REMARK
Sometimes the mass is taken on B_{A r_i}(y) for a fixed A > 1,
not the unit ball. The paper records how to write such an enlarged
mass back in terms of the notation, losing only a dimension-
dependent factor:
r_j^{2-n} M(B_{A r_j}(x)) <= 2^{h(n-2)} N_{j-h}(x) ,
h = ceil( log_2 A ) ,
with the h < small-index cases absorbed in the initial-scale bound
C_A Id. A center shift bounded by C_0 r_j only increases A by C_0.
So fixed enlargements do not change the summability structure of
F_i. A separate remark adds: the matrices D_{i,y} below carry two
Gram inverses; their comparison requires the argument of the next
three lemmas, not just this bookkeeping.