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

D := 2^{100 a^{2}},

2.0.1

κ := 2^{- 10 a},

2.0.2

and

Z := 2^{12 a} .

2.0.3

Let $ψ : R \to R$ be the unique compactly supported, piece-wise linear, continuous function with corners precisely at $\frac{1}{4 D}$ , $\frac{1}{2 D}$ , $\frac{1}{4}$ and $\frac{1}{2}$ which satisfies

\sum_{s \in Z} ψ (D^{- s} x) = 1

2.0.4

for all $x > 0$ . This function vanishes outside $[\frac{1}{4 D}, \frac{1}{2}]$ , is constant one on $[\frac{1}{2 D}, \frac{1}{4}]$ , and is Lipschitz with constant $4 D$ .

Let a doubling metric measure space $(X, ρ, μ, a)$ be given. Let a cancellative compatible collection $Θ$ of functions on $X$ be given. Let $o \in X$ be a point such that $ϑ (o) = 0$ for all $ϑ \in Θ$ .

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

For $s \in Z$ , we define

K_{s} (x, y) := K (x, y) ψ (D^{- s} ρ (x, y)),

2.0.5

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

K (x, y) = \sum_{s \in Z} K_{s} (x, y) .

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.

Proposition 2.0.1 finitary Carleson

✓

Let $σ_{1}, σ_{2} : X \to Z$ be measurable functions with finite range and $σ_{1} \leq σ_{2}$ . Let $F, G$ be bounded Borel sets in $X$ . Then there is a Borel set $G^{'}$ in $X$ with $2 μ (G^{'}) \leq μ (G)$ such that for all Borel functions $f : X \to C$ with $| f | \leq 1_{F}$ .

\int_{G ∖ G^{'}} | \sum_{s = σ_{1} (x)}^{σ_{2} (x)} \int K_{s} (x, y) f (y) e (Q (x) (y)) d μ (y) | d μ (x)

\leq \frac{2^{434 a^{3}}}{(q - 1)^{5}} μ (G)^{1 - \frac{1}{q}} μ (F)^{\frac{1}{q}} .

2.0.6

Let measurable functions $σ_{1} \leq σ_{2} : X \to Z$ 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 $σ_{1}$ and $σ_{2}$ 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 $(D, c, s)$ on $X$ consists of a finite collection $D$ of pairs $(I, k)$ of Borel sets in $X$ and integers $k \in [- S, S]$ , the projection $s : D \to [- S, S], (I, k) \mapsto k$ to the second component which is assumed to be surjective and called scale function, and a function $c : D \to X$ 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 $D$ 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 $(I, k), (J, l) \in D$ we have $I \subset J$ and $k \leq l$ .

For each dyadic cube $I$ and each $- S \leq k < s (I)$ we have

I \subset ⋃_{J \in D : s (J) = k} J .

2.0.7

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

I \subset J .

2.0.8

There exists a $I_{0} \in D$ with $s (I_{0}) = S$ and $c (I_{0}) = o$ and for all cubes $J \in D$ , we have

J \subset I_{0} .

2.0.9

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)}) .

2.0.10

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

μ ({x \in I : ρ (x, X ∖ I) \leq t D^{s (I)}}) \leq 2 t^{κ} μ (I) .

2.0.11

A tile structure $(P, I, Ω, Q, c, s)$ for a given grid structure $(D, c, s)$ is a finite set $P$ of elements called tiles with five maps

\begin{aligned} I & : P \to D \\ Ω & : P \to P (Θ) \\ Q & : P \to Q (X) \\ c & : P \to X \\ s & : P \to Z \end{aligned}

with $I$ surjective and $P (Θ)$ denoting the power set of $Θ$ 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 $Ω$ to the set

P (I) = {p : I (p) = I}

2.0.12

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

Q (X) \subset {⋃^{˙}}_{p \in P (I)} Ω (p) .

2.0.13

For any tiles $p, q$ with $I (p) \subset I (q)$ and $Ω (p) \cap Ω (q) \neq \emptyset$ we have

Ω (q) \subset Ω (p) .

2.0.14

For each tile $p$ ,

Q (p) \in B_{p} (Q (p), 0.2) \subset Ω (p) \subset B_{p} (Q (p), 1),

2.0.15

where

B_{p} (ϑ, R) := {θ \in Θ : d_{p} (ϑ, θ) < R},

2.0.16

and

d_{p} := d_{B (c (p), \frac{1}{4} D^{s (p)})} .

2.0.17

We have for each tile $p$

c (p) = c (I (p)),

2.0.18

s (p) = s (I (p)) .

2.0.19

Proposition 2.0.2 discrete Carleson

✓

Let $(D, c, s)$ be a grid structure and

(P, I, Ω, Q, c, s)

a tile structure for this grid structure. Define for $p \in P$

E (p) = {x \in I (p) : Q (x) \in Ω (p), σ_{1} (x) \leq s (p) \leq σ_{2} (x)}

2.0.20

and

T_{p} f (x) = 1_{E (p)} (x) \int K_{s (p)} (x, y) f (y) e (Q (x) (y) - Q (x) (x)) d μ (y) .

2.0.21

Then there exists a Borel set $G^{'}$ with $2 μ (G^{'}) \leq μ (G)$ such that for all Borel functions $f : X \to C$ with $| f | \leq 1_{F}$ we have

\int_{G ∖ G^{'}} | \sum_{p \in P} T_{p} f (x) | d μ (x) \leq \frac{2^{434 a^{3}}}{(q - 1)^{4}} μ (G)^{1 - \frac{1}{q}} μ (F)^{\frac{1}{q}} .

2.0.22

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 $(D, c, s)$ and a tile structure $(P, I, Ω, Q, c, s)$ for this grid structure.

We define the relation

p \leq p^{'}

2.0.23

on $P \times P$ meaning $I (p) \subset I (p^{'})$ and $Ω (p^{'}) \subset Ω (p)$ . We further define for $λ, λ^{'} > 0$ the relation

λ p ≲ λ^{'} p^{'}

2.0.24

on $P \times P$ meaning $I (p) \subset I (p^{'})$ and

B_{p^{'}} (Q (p^{'}), λ^{'}) \subset B_{p} (Q (p), λ) .

2.0.25

Define for a tile $p$ and $λ > 0$

E_{1} (p) := {x \in I (p) \cap G : Q (x) \in Ω (p)},

2.0.26

E_{2} (λ, p) := {x \in I (p) \cap G : Q (x) \in B_{p} (Q (p), λ)} .

2.0.27

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

{dens}_{1} (P^{'}) := sup_{p^{'} \in P^{'}} sup_{λ \geq 2} λ^{- a} sup_{p \in P (P^{'}), λ p^{'} ≲ λ p} \frac{μ (E_{2} (λ, p))}{μ (I (p))},

2.0.28

{dens}_{2} (P^{'}) := sup_{p^{'} \in P^{'}} sup_{r \geq 4 D^{s (p)}} \frac{μ (F \cap B (c (p), r))}{μ (B (c (p), r))} .

2.0.29

An antichain is a subset $A$ of $P$ such that for any distinct $p, q \in A$ we do not have $p \leq q$ .

The following proposition is proved in Chapter 6.

Proposition 2.0.3 antichain operator

For any antichain $A$ and for all $f : X \to C$ with $| f | \leq 1_{F}$ and all $g : X \to C$ with $| g | \leq 1_{G}$

| \int \overset{―}{g (x)} \sum_{p \in A} T_{p} f (x) d μ (x) |

2.0.30

\leq \frac{2^{150 a^{3}}}{q - 1} {dens}_{1} (A)^{\frac{q - 1}{8 a^{4}}} {dens}_{2} (A)^{\frac{1}{q} - \frac{1}{2}} ‖ f ‖_{2} ‖ g ‖_{2} .

2.0.31

Let $n \geq 0$ . An $n$ -forest is a pair $(U, T)$ where $U$ is a subset of $P$ and $T$ is a map assigning to each $u \in U$ a nonempty set $T (u) \subset P$ 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 $u \in U$ and each $p \in T (u)$ we have $I (p) \neq I (u)$ and

4 p ≲ u .

2.0.32

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

p, p^{″} \in T (u), p \leq p^{'} \leq p^{″} ⟹ p^{'} \in T (u) .

2.0.33

We have

‖ \sum_{u \in U} 1_{I (u)} ‖_{\infty} \leq 2^{n} .

2.0.34

We have for every $u \in U$

{dens}_{1} (T (u)) \leq 2^{4 a + 1 - n} .

2.0.35

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

d_{p} (Q (p), Q (u)) > 2^{Z (n + 1)} .

2.0.36

We have for every $u \in U$ and $p \in T (u)$ that

B (c (p), 8 D^{s (p)}) \subset I (u) .

2.0.37

The following proposition is proved in Chapter 7.

Proposition 2.0.4 forest operator

For any $n \geq 0$ and any $n$ -forest $(U, T)$ we have for all $f : X \to C$ with $| f | \leq 1_{F}$ and all bounded $g$ with bounded support

| \int \overset{―}{g (x)} \sum_{u \in U} \sum_{p \in T (u)} T_{p} f (x) d μ (x) |

\leq 2^{432 a^{3}} 2^{- \frac{q - 1}{q} n} {dens}_{2} {(⋃_{u \in U} T (u))}^{\frac{1}{q} - \frac{1}{2}} ‖ f ‖_{2} ‖ g ‖_{2} .

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

τ := \frac{1}{a} .

2.0.38

Define for any open ball $B$ of radius $R$ in $X$ the $L^{\infty}$ -normalized $τ$ -Hölder norm by

‖ φ ‖_{C^{τ} (B)} = sup_{x \in B} | φ (x) | + R^{τ} sup_{x, y \in B, x \neq y} \frac{| φ (x) - φ (y) |}{ρ (x, y)^{τ}} .

2.0.39

Proposition 2.0.5 Holder van der Corput

Let $z \in X$ and $R > 0$ and set $B = B (z, R)$ . Let $φ : X \to C$ by supported on $B$ and satisfy $‖ φ ‖_{C^{τ} (B)} < \infty$ . Let $ϑ, θ \in Θ$ . Then

| \int e (ϑ (x) - θ (x)) φ (x) d x | \leq 2^{8 a} μ (B) ‖ φ ‖_{C^{τ} (B)} (1 + d_{B} (ϑ, θ))^{- \frac{1}{2 a^{2} + a^{3}}} .

2.0.40

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 $r > 0$ as defined in 1.0.6. For a finite collection $B$ of balls in $X$ and $1 \leq p < \infty$ define the measurable function $M_{B, p} u$ on $X$ by

M_{B, p} u (x) := {(sup_{B \in B} \frac{1_{B} (x)}{μ (B)} \int_{B} | u (y) |^{p} d μ (y))}^{\frac{1}{p}} .

2.0.41

Define further $M_{B} := M_{B, 1}$ .

Proposition 2.0.6 Hardy–Littlewood

Let $B$ be a finite collection of balls in $X$ . If for some $λ > 0$ and some measurable function $u : X \to [0, \infty)$ we have

\int_{B} u (x) d μ (x) \geq λ μ (B)

2.0.42

for each $B \in B$ , then

λ μ (⋃ B) \leq 2^{2 a} \int_{X} u (x) d μ (x) .

2.0.43

For every measurable function $v$ and $1 \leq p_{1} < p_{2}$ we have

‖ M_{B, p_{1}} v ‖_{p_{2}} \leq 2^{2 a} \frac{p_{2}}{p_{2} - p_{1}} ‖ v ‖_{p_{2}} .

2.0.44

Moreover, given any measurable bounded function $w : X \to C$ there exists a measurable function $M w : X \to [0, \infty)$ such that the following 2.0.45 and 2.0.46 hold. For each ball $B \subset X$ and each $x \in B$

\frac{1}{μ (B)} \int_{B} | w (y) | d μ (y) \leq M w (x)

2.0.45

and for all $1 \leq p_{1} < p_{2} \leq \infty$

‖ M (w^{p_{1}})^{\frac{1}{p_{1}}} ‖_{p_{2}} \leq 2^{4 a} \frac{p_{2}}{p_{2} - p_{1}} ‖ w ‖_{p_{2}} .

2.0.46

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 $Θ$ equipped with any of the metrics $d_{B}$ , following from the doubling property 1.0.11.

Lemma 2.1.1 ball metric entropy

✓

Let $B^{'} \subset X$ be a ball. Let $r > 0$ , $ϑ \in Θ$ and $k \in N$ . Suppose that $Z \subset B_{B^{'}} (ϑ, r 2^{k})$ satisfies that ${B_{B^{'}} (z, r) ∣ z \in Z}$ is a collection of pairwise disjoint sets. Then

| Z | \leq 2^{k a} .

Proof ▶

By applying property 1.0.11 $k$ times, we obtain a collection $Z^{'} \subset Θ$ with $| Z^{'} | = 2^{k a}$ and

B_{B^{'}} (ϑ, r 2^{k}) \subset ⋃_{z^{'} \in Z^{'}} B_{B^{'}} (z^{'}, \frac{r}{2}) .

Then each $z \in Z$ is contained in one of the balls $B (z^{'}, \frac{r}{2})$ , but by the separation assumption no such ball contains more than one element of $Z$ . Thus $| Z | \leq | Z^{'} | = 2^{k a}$ .

The next lemma concerns monotonicity of the metrics $d_{B (c (I), \frac{1}{4} D^{s (I)})}$ with respect to inclusion of cubes $I$ in a grid.

Lemma 2.1.2 monotone cube metrics

✓

Let $(D, c, s)$ be a grid structure. Denote for cubes $I \in D$

I^{\circ} := B (c (I), \frac{1}{4} D^{s (I)}) .

Let $I, J \in D$ with $I \subset J$ . Then for all $ϑ, θ \in Θ$ we have

d_{I^{\circ}} (ϑ, θ) \leq d_{J^{\circ}} (ϑ, θ),

and if $I \neq J$ then we have

d_{I^{\circ}} (ϑ, θ) \leq 2^{- 95 a} d_{J^{\circ}} (ϑ, θ) .

Proof ▶

If $s (I) \geq s (J)$ then 2.0.8 and the assumption $I \subset J$ imply $I = J$ . Then the lemma holds by reflexivity.

If $s (J) \geq s (I) + 1$ , then using the monotonicity property 1.0.9, 2.0.1 and 1.0.10, we get

d_{I^{\circ}} (ϑ, θ) \leq d_{B (c (I), 4 D^{s (I)})} (ϑ, θ) \leq 2^{- 100 a} d_{B (c (I), 4 D^{s (J)})} (ϑ, θ) .

2.1.1

Using 2.0.10, together with the inclusion $I \subset J$ , we obtain

c (I) \in I \subset J \subset B (c (J), 4 D^{s (J)})

and consequently by the triangle inequality

B (c (I), 4 D^{s (J)}) \subset B (c (J), 8 D^{s (J)}) .

Using this together with the monotonicity property 1.0.9 and 1.0.8 in 2.1.1, we obtain

\begin{aligned} d_{I^{\circ}} (ϑ, θ) & \leq 2^{- 100 a} d_{B (c (J), 8 D^{s (J)})} (ϑ, θ) \\ \leq 2^{- 100 a + 5 a} d_{B (c (J), \frac{1}{4} D^{s (J)})} (ϑ, θ) \\ = 2^{- 95 a} d_{J^{\circ}} (ϑ, θ) . \end{aligned}

This proves the second inequality claimed in the Lemma, from which the first follows since $a \geq 4$ and hence $2^{- 95 a} \leq 1$ .

We also record the following basic estimates for the kernels $K_{s}$ .

Lemma 2.1.3 kernel summand

✓

Let $- S \leq s \leq S$ and $x, y, y^{'} \in X$ . If $K_{s} (x, y) \neq 0$ , then we have

\frac{1}{4} D^{s - 1} \leq ρ (x, y) \leq \frac{1}{2} D^{s} .

2.1.2

We have

| K_{s} (x, y) | \leq \frac{2^{102 a^{3}}}{μ (B (x, D^{s}))}

2.1.3

and

| K_{s} (x, y) - K_{s} (x, y^{'}) | \leq \frac{2^{150 a^{3}}}{μ (B (x, D^{s}))} {(\frac{ρ (y, y^{'})}{D^{s}})}^{\frac{1}{a}} .

2.1.4

Proof ▶

By Definition 2.0.5, the function $K_{s}$ is the product of $K$ with a function which is supported in the set of all $x, y$ satisfying 2.1.2. This proves 2.1.2.

Using 1.0.14 and the lower bound in 2.1.2 we obtain

| K_{s} (x, y) | \leq | K (x, y) | \leq \frac{2^{a^{3}}}{μ (B (x, \frac{1}{4} D^{s - 1}))}

2.1.5

Using $D = 2^{100 a^{2}}$ and the doubling property 1.0.5 $2 + 100 a^{2}$ times estimates the last display by

\leq \frac{2^{2 a + 101 a^{3}}}{μ (B (x, D^{s}))} .

2.1.6

Using $a \geq 4$ proves 2.1.3.

To prove 2.1.4 when $2 ρ (y, y^{'}) > ρ (x, y)$ , use the lower bound in 2.1.2, $2 ρ (y, y^{'}) > \frac{1}{4} D^{s - 1}$ . Then 2.1.4 follows from the triangle inequality, 2.1.3 and $a \geq 4$ .

If $2 ρ (y, y^{'}) \leq ρ (x, y)$ , we rewrite $| K_{s} (x, y) - K_{s} (x, y^{'}) |$ as

| (K (x, y) - K (x, y^{'})) ψ (D^{- s} ρ (x, y)) + K (x, y) (ψ (D^{- s} ρ (x, y)) - ψ (D^{- s} ρ (x, y^{'}))) | .

2.1.7

An upper bound for $| K (x, y) - K (x, y^{'}) |$ is obtained similarly to the proof of 2.1.3, using 1.0.15 and the lower bound in 2.1.2

| K (x, y) - K (x, y^{'}) | \leq \frac{2^{a^{3}}}{μ (B (x, \frac{1}{4} D^{s - 1}))} {(\frac{ρ (y, y^{'})}{\frac{1}{4} D^{s - 1}})}^{\frac{1}{a}} .

2.1.8

As above, this is estimated by

\leq \frac{4 D 2^{2 a + 101 a^{3}}}{μ (B (x, D^{s}))} {(\frac{ρ (y, y^{'})}{D^{s}})}^{\frac{1}{a}} = \frac{2^{2 + 2 a + 100 a^{2} + 101 a^{3}}}{μ (B (x, D^{s}))} {(\frac{ρ (y, y^{'})}{D^{s}})}^{\frac{1}{a}} .

2.1.9

We have the trivial bound $| ψ (D^{- s} ρ (x, y)) | \leq 1$ , and 2.1.6 provides a bound for $| K (x, y) |$ . Finally, we show that

| ψ (D^{- s} ρ (x, y)) - ψ (D^{- s} ρ (x, y^{'})) | \leq 4 D {(\frac{ρ (y, y^{'})}{D^{s}})}^{\frac{1}{a}}

2.1.10

by considering separately the cases $ρ (y, y^{'}) / D^{s} \geq 1$ and $ρ (y, y^{'}) / D^{s} < 1$ . In the former case, the inequality is trivial; in the latter case, it follows from the fact that $ψ$ is Lipschitz with constant $4 D$ .

Combining the above bounds and using $a \geq 4$ proves 2.1.4 in the case $2 ρ (y, y^{'}) \leq ρ (x, y)$ . □