1.2. Proof of Metric Space Carleson, overview🔗

This section organizes the proof of Theorem 1.1.1.1 into sections Proof of Metric Space Carleson, Proof of Finitary Carleson, Proof of discrete Carleson, Proof of the Antichain Operator Proposition, Proof of the Forest Operator Proposition, Proof of the Holder cancellative condition, and Proof of Vitali covering and Hardy--Littlewood. These sections are mutually independent except for referring to the statements formulated in the present section. Proof of Metric Space Carleson proves the main Theorem 1.1.1.1, while sections Proof of Finitary Carleson, Proof of discrete Carleson, Proof of the Antichain Operator Proposition, Proof of the Forest Operator Proposition, Proof of the Holder cancellative condition, and Proof of Vitali covering and Hardy--Littlewood each prove one proposition that is stated in the present section. The present section also introduces all definitions used across these sections.

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

Let a, q be given as in Theorem 1.1.1.1.

Define D:= 2^{100 a^2}, \kappa:= 2^{-10a}, and Z := 2^{12a}. Let \psi:\R \to \R be the unique compactly supported, piece-wise linear, continuous function with corners precisely at \frac 1{4D}, \frac 1{2D}, \frac 14 and \frac 12 which satisfies \sum_{s\in \mathbb{Z}} \psi(D^{-s}x)=1 for all x>0. This function vanishes outside [\frac1{4D},\frac 12], is constant one on [\frac1{2D},\frac 14], and is Lipschitz with constant 4D.

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

Let a Borel measurable function \tQ:X\to \Mf with finite range be given. Let a one-sided Calderon-Zygmund kernel K on X be given so that for every \mfa\in \Mf the operator T_{\tQ}^{\mfa} defined above satisfies the assumption linnontanbound.

For s\in\mathbb{Z}, we define K_s(x,y):=K(x,y)\psi(D^{-s}\rho(x,y)), so that for each x,y\in X with x\neq y we have K(x,y)=\sum_{s\in\mathbb{Z}}K_s(x,y).

In Proof of Metric Space Carleson, we prove Theorem 1.1.1.1 and Theorem 1.1.1.2 from the finitary version, Theorem 1.2.1 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 Proof of Finitary Carleson, we prove Theorem 1.2.1 using a bound for a dyadic model formulated in Theorem 1.2.2 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 coverdyadic, dyadicproperty, subsetmaxcube, eq-vol-sp-cube, and eq-small-boundary 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<s(I) we have I\subset \bigcup_{J\in \mathcal {D}: s(J)=k}J. Any two non-disjoint dyadic cubes I,J with s(I)\le s(J) satisfy I\subset J. 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 J \subset I_0. For any dyadic cube I, c(I)\in B(c(I), \frac{1}{4} D^{s(I)}) \subset I \subset B(c(I), 4 D^{s(I)}). For any dyadic cube I and any t with tD^{s(I)} \ge D^{-S}, \mu(\{x \in I \ : \ \rho(x, X \setminus I) \leq t D^{s(I)}\}) \le 2 t^\kappa \mu(I).

A tile structure (\fP,\scI,\fc,\fcc,\pc,\ps) for a given grid structure (\mathcal{D}, c, s) is a finite set \fP of elements called tiles with five maps \scI\colon \fP\to {\mathcal{D}} \fc\colon \fP\to \mathcal{P}(\Mf) \fcc \colon \fP\to \tQ(X) \pc \colon \fP\to X \ps \colon \fP\to \mathbb{Z} with \scI surjective and \mathcal{P}(\Mf) denoting the power set of \Mf such that the five properties eq-dis-freq-cover, eq-freq-dyadic, eq-freq-comp-ball, tilecenter, and tilescale hold. For each dyadic cube I, the restriction of the map \Omega to the set \fP(I)=\{\fp: \scI(\fp) =I\} is injective and we have the disjoint covering property \tQ(X)\subset \dot{\bigcup}_{\fp\in \fP(I)}\fc(\fp). For any tiles \fp,\fq with \scI(\fp)\subset \scI(\fq) and \fc(\fp) \cap \fc(\fq) \neq \emptyset we have \fc(\fq)\subset \fc(\fp). For each tile \fp, \fcc(\fp)\in B_{\fp}(\fcc(\fp), 0.2) \subset \fc(\fp) \subset B_{\fp}(\fcc(\fp),1), where B_{\fp} (\mfa, R) := \{\mfb \in \Mf \, : \, d_{\fp}(\mfa, \mfb) < R\}, and d_{\fp} := d_{B(\pc(\fp),\frac 14 D^{\ps(\fp)})}. We have for each tile \fp \pc(\fp)=c(\scI(\fp)), \ps(\fp)=s(\scI(\fp)).

The proof of Theorem 1.2.2 is done in Proof of discrete Carleson by a reduction to two further propositions that we state below.

Fix a grid structure (\mathcal{D}, c, s) and a tile structure (\fP,\scI,\fc,\fcc,\pc,\ps) for this grid structure.

We define the relation \fp\le \fp' on \fP\times \fP meaning \scI(\fp)\subset \scI(\fp') and \Omega(\fp')\subset \Omega(\fp). We further define for \lambda,\lambda' >0 the relation \lambda \fp \lesssim \lambda' \fp' on \fP\times \fP meaning \scI(\fp)\subset \scI(\fp') and B_{\fp'}(\fcc(\fp'),\lambda') \subset B_{\fp}(\fcc(\fp),\lambda).

Define for a tile \fp and \lambda>0 E_1(\fp):=\{x\in \scI(\fp)\cap G: \tQ(x)\in \fc(\fp)\}, E_2(\lambda, \fp):=\{x\in \scI(\fp)\cap G: \tQ(x)\in B_{\fp}(\fcc(\fp), \lambda)\}.

Given a subset \fP' of \fP, we define \fP(\fP') to be the set of all \fp \in \fP such that there exist \fp' \in \fP' with \scI(\fp)\subset \scI(\fp'). Define the densities {\dens}_1(\fP') := \sup_{\fp'\in \fP'}\sup_{\lambda \geq 2} \lambda^{-a} \sup_{\fp \in \fP(\fP'), \lambda \fp' \lesssim \lambda \fp} \frac{\mu({E}_2(\lambda, \fp))}{\mu(\scI(\fp))}, {\dens}_2(\fP') := \sup_{\fp'\in \fP'} \sup_{r\ge 4D^{\ps(\fp)}} \frac{\mu(F\cap B(\pc(\fp),r))}{\mu(B(\pc(\fp),r))}.

An antichain is a subset \mathfrak{A} of \fP such that for any distinct \fp,\fq\in \mathfrak{A} we do not have \fp\le \fq.

The following proposition is proved in Proof of the Antichain Operator Proposition.

Let n\ge 0. An n-forest is a pair (\fU, \mathfrak{T}) where \fU is a subset of \fP and \mathfrak{T} is a map assigning to each \fu\in \fU a nonempty set \fT (\fu)\subset \fP called tree such that the following properties forest1, forest2, forest3, forest4, forest5, and forest6 hold.

For each \fu\in \fU and each \fp\in \fT(\fu) we have \scI(\fp) \ne \scI(\fu) and 4\fp\lesssim \fu. For each \fu \in \fU and each \fp,\fp''\in \fT(\fu) and \fp'\in \fP we have \fp, \fp'' \in \mathfrak{T}(\fu), \fp \leq \fp' \leq \fp'' \implies \fp' \in \mathfrak{T}(\fu). We have \|\sum_{\fu\in \fU} \mathbf{1}_{\scI(\fu)}\|_\infty \leq 2^n. We have for every \fu\in \fU \dens_1(\fT(\fu))\le 2^{4a + 1-n}. We have for \fu, \fu'\in \fU with \fu\neq \fu' and \fp\in \fT(\fu') with \scI(\fp)\subset \scI(\fu) that d_{\fp}(\fcc(\fp), \fcc(\fu))>2^{Z(n+1)}. We have for every \fu\in \fU and \fp\in \fT(\fu) that B(\pc(\fp), 8D^{\ps(\fp)})\subset \scI(\fu).

The following proposition is proved in Proof of the Forest Operator Proposition.

Theorem 1.1.1.1 is formulated at the level of generality for general kernels satisfying the mere Holder regularity condition eqkernel-y-smooth. On the other hand, the cancellative condition eq-vdc-cond is a testing condition against more regular, namely Lipschitz functions. To bridge the gap, we follow the literature to observe a variant of eq-vdc-cond that we formulate in the following proposition proved in Proof of the Holder cancellative condition.

Define \tau:=\frac 1a. Define for any open ball B of radius R in X the L^\infty-normalized \tau-Holder norm by \|\varphi\|_{C^\tau(B)} = \sup_{x \in B} |\varphi(x)| + R^\tau \sup_{x,y \in B, x \neq y} \frac{|\varphi(x) - \varphi(y)|}{\rho(x,y)^\tau}.

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 Proof of Vitali covering and Hardy--Littlewood. By a ball B we mean a set B(x,r) with x\in X and r>0 as defined in eq-define-ball. For a finite collection \mathcal{B} of balls in X and 1\le p< \infty define the measurable function M_{\mathcal{B},p}u on X by M_{\mathcal{B},p}u(x):=\left(\sup_{B\in \mathcal{B}} \frac{\mathbf{1}_{B}(x)}{\mu(B)}\int _{B} |u(y)|^p\, d\mu(y)\right)^\frac 1p, Define further M_{\mathcal{B}}:=M_{\mathcal{B},1}.

This completes the overview of the proof of Theorem 1.1.1.1.