2 Proof of Metric Space Carleson, overview

This section organizes the proof of Theorem 1.0.2 into sections 3, 4, 5, 6, 7, 8, and 9. These sections are mutually independent except for referring to the statements formulated in the present section. Chapter 3 proves the main Theorem 1.0.2, while sections 4, 5, 6, 7, 8, and 9 each prove one proposition that is stated in the present section. The present section also introduces all definitions used across these sections.

Section 2.1 proves some auxiliary lemmas that are used in more than one of the sections 3-9.

Let \(a, q\) be given as in Theorem 1.0.2.

Define

and

Let \(\psi :{\mathbb {R}}\to {\mathbb {R}}\) be the unique compactly supported, piece-wise linear, continuous function with corners precisely at \(\frac1{4D}\), \(\frac1{2D}\), \(\frac14\) and \(\frac12\) which satisfies

for all \(x{\gt}0\). This function vanishes outside \([\frac1{4D},\frac12]\), is constant one on \([\frac1{2D},\frac14]\), and is Lipschitz with constant \(4D\).

Let a doubling metric measure space \((X,\rho ,\mu , a)\) be given. Let a cancellative compatible collection \({\Theta }\) of functions on \(X\) be given. Let \(o\in X\) be a point such that \({\vartheta }(o)=0\) for all \({\vartheta }\in {\Theta }\).

Let a Borel measurable function \({Q}:X\to {\Theta }\) with finite range be given. Let a one-sided Calderón–Zygmund kernel \(K\) on \(X\) be given so that for every \({\vartheta }\in {\Theta }\) the operator \(T_{{Q}}^{{\vartheta }}\) defined in 1.0.21 satisfies 1.0.23.

For \(s\in \mathbb {Z}\), we define

so that for each \(x, y \in X\) with \(x\neq y\) we have

In Chapter 3, we prove Theorem 1.0.2 and Theorem 1.0.3 from the finitary version, Proposition 2.0.1 below. Recall that a function from a measure space to a finite set is measurable if the pre-image of each of the elements in the range is measurable.

Let measurable functions \({\sigma _1}\leq \sigma _2\colon X\to \mathbb {Z}\) with finite range be given. Let bounded Borel sets \(F,G\) in \(X\) be given. Let \(S\) be the smallest integer such that the ranges of \(\sigma _1\) and \(\sigma _2\) are contained in \([-S,S]\) and \(F\) and \(G\) are contained in the ball \(B(o, \frac{1}{4}D^S)\).

In Chapter 4, we prove Proposition 2.0.1 using a bound for a dyadic model formulated in Proposition 2.0.2 below.

A grid structure \((\mathcal{D}, c, s)\) on \(X\) consists of a finite collection \(\mathcal{D}\) of pairs \((I, k)\) of Borel sets in \(X\) and integers \(k \in [-S, S]\), the projection \(s\colon \mathcal{D}\to [-S, S], (I, k) \mapsto k\) to the second component which is assumed to be surjective and called scale function, and a function \(c:\mathcal{D}\to X\) called center function such that the five properties 2.0.7, 2.0.8, 2.0.9, 2.0.10, and 2.0.11 hold. We call the elements of \(\mathcal{D}\) dyadic cubes. By abuse of notation, we will usually write just \(I\) for the cube \((I,k)\), and we will write \(I \subset J\) to mean that for two cubes \((I,k), (J, l) \in \mathcal{D}\) we have \(I \subset J\) and \(k \le l\).

For each dyadic cube \(I\) and each \(-S\le k{\lt}s(I)\) we have

Any two non-disjoint dyadic cubes \(I,J\) with \(s(I)\le s(J)\) satisfy

There exists a \(I_0 \in \mathcal{D}\) with \(s(I_0) = S\) and \(c(I_0) = o\) and for all cubes \(J \in \mathcal{D}\), we have

For any dyadic cube \(I\),

For any dyadic cube \(I\) and any \(t\) with \(tD^{s(I)} \ge D^{-S}\),

A tile structure \(({\mathfrak P},{\mathcal{I}},{\Omega },{\mathcal{Q}},{\mathrm{c}},{\mathrm{s}})\) for a given grid structure \((\mathcal{D}, c, s)\) is a finite set \({\mathfrak P}\) of elements called tiles with five maps

with \({\mathcal{I}}\) surjective and \(\mathcal{P}({\Theta })\) denoting the power set of \({\Theta }\) such that the five properties 2.0.13, 2.0.14, 2.0.15, 2.0.18, and 2.0.19 hold. For each dyadic cube \(I\), the restriction of the map \(\Omega \) to the set

is injective and we have the disjoint covering property (we use the union symbol with dot on top to denote a disjoint union)

For any tiles \({\mathfrak p},{\mathfrak q}\) with \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak q})\) and \({\Omega }({\mathfrak p}) \cap {\Omega }({\mathfrak q}) \neq \emptyset \) we have

For each tile \({\mathfrak p}\),

where

and

We have for each tile \({\mathfrak p}\)

The proof of Proposition 2.0.2 is done in Chapter 5 by a reduction to two further propositions that we state below.

Fix a grid structure \((\mathcal{D}, c, s)\) and a tile structure \(({\mathfrak P},{\mathcal{I}},{\Omega },{\mathcal{Q}},{\mathrm{c}},{\mathrm{s}})\) for this grid structure.

We define the relation

on \({\mathfrak P}\times {\mathfrak P}\) meaning \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak p}')\) and \(\Omega ({\mathfrak p}')\subset \Omega ({\mathfrak p})\). We further define for \(\lambda ,\lambda ' {\gt}0\) the relation

on \({\mathfrak P}\times {\mathfrak P}\) meaning \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak p}')\) and

Define for a tile \({\mathfrak p}\) and \(\lambda {\gt}0\)

Given a subset \({\mathfrak P}'\) of \({\mathfrak P}\), we define \({\mathfrak P}({\mathfrak P}')\) to be the set of all \({\mathfrak p}\in {\mathfrak P}\) such that there exist \({\mathfrak p}' \in {\mathfrak P}'\) with \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak p}')\). Define the densities

An antichain is a subset \(\mathfrak {A}\) of \({\mathfrak P}\) such that for any distinct \({\mathfrak p},{\mathfrak q}\in \mathfrak {A}\) we do not have have \({\mathfrak p}\le {\mathfrak q}\).

The following proposition is proved in Chapter 6.

Let \(n\ge 0\). An \(n\)-forest is a pair \(({\mathfrak U}, \mathfrak {T})\) where \({\mathfrak U}\) is a subset of \({\mathfrak P}\) and \(\mathfrak {T}\) is a map assigning to each \({\mathfrak u}\in {\mathfrak U}\) a nonempty set \({\mathfrak T}({\mathfrak u})\subset {\mathfrak P}\) called tree such that the following properties 2.0.32, 2.0.33, 2.0.34, 2.0.35, 2.0.36, and 2.0.37 hold.

For each \({\mathfrak u}\in {\mathfrak U}\) and each \({\mathfrak p}\in {\mathfrak T}({\mathfrak u})\) we have \({\mathcal{I}}({\mathfrak p}) \ne {\mathcal{I}}({\mathfrak u})\) and

For each \({\mathfrak u}\in {\mathfrak U}\) and each \({\mathfrak p},{\mathfrak p}''\in {\mathfrak T}({\mathfrak u})\) and \({\mathfrak p}'\in {\mathfrak P}\) we have

We have

We have for every \({\mathfrak u}\in {\mathfrak U}\)

We have for \({\mathfrak u}, {\mathfrak u}'\in {\mathfrak U}\) with \({\mathfrak u}\neq {\mathfrak u}'\) and \({\mathfrak p}\in {\mathfrak T}({\mathfrak u}')\) with \({\mathcal{I}}({\mathfrak p})\subset {\mathcal{I}}({\mathfrak u})\) that

We have for every \({\mathfrak u}\in {\mathfrak U}\) and \({\mathfrak p}\in {\mathfrak T}({\mathfrak u})\) that

The following proposition is proved in Chapter 7.

Theorem 1.0.2 is formulated at the level of generality for general kernels satisfying the mere Hölder regularity condition 1.0.15. On the other hand, the cancellative condition 1.0.12 is a testing condition against more regular, namely Lipschitz functions. To bridge the gap, we follow [ to observe a variant of 1.0.12 that we formulate in the following proposition proved in Chapter 8.

Define

Define for any open ball \(B\) of radius \(R\) in \(X\) the \(L^\infty \)-normalized \(\tau \)-Hölder norm by

We further formulate a classical Vitali covering result and maximal function estimate that we need throughout several sections. This following proposition will typically be applied to the absolute value of a complex valued function and be proved in Chapter 9. By a ball \(B\) we mean a set \(B(x,r)\) with \(x\in X\) and \(r{\gt}0\) as defined in 1.0.6. For a finite collection \(\mathcal{B}\) of balls in \(X\) and \(1\le p{\lt} \infty \) define the measurable function \(M_{\mathcal{B},p}u\) on \(X\) by

Define further \(M_{\mathcal{B}}:=M_{\mathcal{B},1}\).

This completes the overview of the proof of Theorem 1.0.2.

2.1 Auxiliary lemmas

We close this section by recording some auxiliary lemmas about the objects defined in Chapter 2, which will be used in multiple sections to follow.

First, we record an estimate for the metrical entropy numbers of balls in the space \({\Theta }\) equipped with any of the metrics \(d_B\), following from the doubling property 1.0.11.

The next lemma concerns monotonicity of the metrics \(d_{B(c(I), \frac14 D^{s(I)})}\) with respect to inclusion of cubes \(I\) in a grid.

We also record the following basic estimates for the kernels \(K_s\).