# Variational truncations of singular integrals on spaces of homogeneous type

Pavel Zorin-Kranich  University of Bonn
Mathematical Institute

###### Abstract.

We prove sharp weighted estimates for $r$-variations of averages and truncated
singular integrals on suitable spaces of homogeneous type, including
homogeneous nilpotent Lie groups.

##  1\. Introduction

Throughout the article $(X,\rho,\mu)$ denotes an (Ahlfors–David) $D$-regular
space of homogeneous type (definitions of these and other terms are given in
Section 2). Variational estimates for the averaging operators

(1.1) $A_{t}f(x)=\mu(B(x,t))^{-1}\int_{B(x,t)}f(y)\mathrm{d}\mu(y)$

on $X=\mathbb{R}^{D}$ have been introduced by Bourgain [Bou89]. A
comprehensive theory covering the full range of $L^{p}$ spaces and variational
exponents $r$ both for averages (1.1) and truncated singular integrals

(1.2) $T_{t}f(x)=\int_{\rho(x,y)>t}K(x,y)f(y)\mathrm{d}\mu(y)$

has been developed by a number of authors [JRW03, Jon+98, JSW08]. Some of
their estimates have been extended to weighted $L^{p}$ spaces in [MTX15,
arxiv:1409.7120, MTX15a].

Sparse domination has been developed in [Ler13, arXiv:1501.05818, HRT17] in
order to simplify the proof of the $A_{2}$ theorem for Calderón–Zygmund (CZ)
operators [Hyt12]. In a short period of time since 2015 this idea has been
applied in many settings which go beyond CZ theory, and we are not going to
survey these developments. In the CZ setting it is by now well understood that
sparse domination follows from suitable localized non-tangentional endpoint
estimates; several abstract results formalizing this principle appeared in
[Ler16, dFZ16, Con+16]. These techniques have been applied to $r$-variational
estimates for truncated singular integrals in [HLP13] (smooth truncations) and
[dFZ16] (sharp truncations).

In this article we extend these $r$-variational estimates to a class of non-
convolution type singular integrals. We formulate our results on classes of
spaces of homogeneous type that include homogeneous nilpotent Lie groups. In
this setting we also obtain some sharp weighted inequalities for square
functions and $r$-variation of averages.

Weighted estimates for $r$-variation of averaging operators (1.1) have been
obtained in [arxiv:1409.7120]. While in retrospect the methods of that article
easily imply sparse domination of $r$-variations, the endpoint estimate for
the jump counting function below requires a new ingredient.

######  Theorem 1.3.

Suppose that the space $X$ has the small boundary property. Then the operators
$f\mapsto\lambda^{-1}\sqrt{\mathcal{J}_{\lambda}(A_{t}f:0<t<\infty)}$ are
pointwise controlled by sparse operators uniformly in $\lambda>0$. Moreover,
for every $r>2$ the operator $f\mapsto V^{r}(A_{t}f:0<t<\infty)$ is pointwise
controlled by sparse operators with constant $C_{X}r/(r-2)$.

Known estimates for sparse operators (see [HL18] for $p>1$ and [DLR16] for
$p=1$) yield the following consequences for the jump counting function (and
similar ones for $r$-variations).

######  Corollary 1.4.

With the hypotheses of Theorem 1.3

(1.5)
$\displaystyle\lVert\lambda^{-1}\sqrt{\mathcal{J}_{\lambda}(A_{t}(f\sigma):t>0)}\rVert_{L^{p}(w)}$
$\displaystyle\lesssim_{X}[w,\sigma]_{A_{p}}^{1/p}([w]_{A_{\infty}}^{1/p^{\prime}}+[\sigma]_{A_{\infty}}^{1/p})\lVert
f\rVert_{L^{p}(\sigma)},$ $\displaystyle 1<p<\infty,$ (1.6)
$\displaystyle\lVert\lambda^{-1}\sqrt{\mathcal{J}_{\lambda}(A_{t}(f\sigma):t>0)}\rVert_{L^{p,\infty}(w)}$
$\displaystyle\lesssim_{X}[w,\sigma]_{A_{p}}^{1/p}[w]_{A_{\infty}}^{1/p^{\prime}}\lVert
f\rVert_{L^{p}(\sigma)},$ $\displaystyle 1<p<\infty,$ (1.7)
$\displaystyle\lVert\lambda^{-1}\sqrt{\mathcal{J}_{\lambda}(A_{t}(f\sigma):t>0)}\rVert_{L^{1,\infty}(w)}$
$\displaystyle\lesssim_{X}\frac{1}{\alpha-1}\int_{X}\lvert f\rvert
M_{L\log\log L(\log\log\log L)^{\alpha}}w,$ $\displaystyle 1<\alpha<2.$

Finally, we obtain $r$-variational estimates for a class of non-convolution
type singular integral operators.

######  Theorem 1.8.

Suppose that the space $X$ has the small boundary property and the metric
$\rho$ satisfies the Hölder type condition

(1.9) $\lvert\rho(x,z)-\rho(y,z)\rvert\leq
C\max(\rho(x,z),\rho(y,z))^{1-\eta}\rho(x,y)^{\eta}$

for some $0<\eta\leq 1$.

Let $K:X\times X\setminus\\{\mathrm{diagonal}\\}\to\mathbb{C}$ be a Hölder
continuous CZ kernel, that is, a function that satisfies the size condition

(1.10) $\sup_{x\neq y}\rho(x,y)^{D}\lvert K(x,y)\rvert\leq 1,$

the smoothness estimate

(1.11) $\lvert K(x,y)-K(x^{\prime},y)\rvert+\lvert
K(y,x)-K(y,x^{\prime})\rvert\leq\big{(}\frac{\rho(x,x^{\prime})}{\rho(x,y)}\big{)}^{\eta}\rho(x,y)^{-D}$

for all $x,y$ with $\rho(x,y)\geq 2\rho(x,x^{\prime})>0$, and the cancellation
condition

(1.12)
$\int_{r<\rho(x,y)<R}K(x,y)\mathrm{d}x=\int_{r<\rho(x,y)<R}K(x,y)\mathrm{d}y=0,\quad
0<r<R<\infty.$

Consider the associated truncated singular integral operators (1.2).

Then for every $r>2$ the operator $f\mapsto V^{r}(T_{t}f:0<t<\infty)$ is
pointwise controlled by sparse operators with norm $C_{X,\eta}r/(r-2)$.

######  Corollary 1.13.

With the hypotheses of Theorem 1.8

(1.14) $\displaystyle\lVert V^{r}(T_{t}(f\sigma):t>0)\rVert_{L^{p}(w)}$
$\displaystyle\lesssim_{X}\frac{r}{r-2}[w,\sigma]_{A_{p}}^{1/p}([w]_{A_{\infty}}^{1/p^{\prime}}+[\sigma]_{A_{\infty}}^{1/p})\lVert
f\rVert_{L^{p}(\sigma)},$ $\displaystyle 1<p<\infty,$ (1.15)
$\displaystyle\lVert V^{r}(T_{t}(f\sigma):t>0)\rVert_{L^{p,\infty}(w)}$
$\displaystyle\lesssim_{X}\frac{r}{r-2}[w,\sigma]_{A_{p}}^{1/p}[w]_{A_{\infty}}^{1/p^{\prime}}\lVert
f\rVert_{L^{p}(\sigma)},$ $\displaystyle 1<p<\infty,$ (1.16)
$\displaystyle\lVert V^{r}(T_{t}(f):t>0)\rVert_{L^{1,\infty}(w)}$
$\displaystyle\lesssim_{X}\frac{r}{r-2}\frac{1}{\alpha-1}\int_{X}\lvert
f\rvert M_{L\log\log L(\log\log\log L)^{\alpha}}w,$ $\displaystyle
1<\alpha<2.$

It is known [MS79, Theorem 2] that for every quasimetric $\rho$ there exist
$\alpha>0$ and a metric $\tilde{\rho}\sim\rho^{\alpha}$ with the property
(1.9). More practically, the property (1.9) holds for homogeneous metrics on
homogeneous nilpotent Lie groups.

The conclusion of Theorem 1.8 has been previously known for convolution type
CZ kernels on $\mathbb{R}^{D}$ by [arXiv:1512.07523, Theorem A.1] (unweighted
$L^{p}$ estimates), [Cam+03, Theorem B] (weak type $(1,1)$), and [dFZ16]
(reduction of weighted to unweighted estimates).

For rough homogeneous kernels on $\mathbb{R}^{D}$ variational estimates have
been proved in [Cam+03, Theorem A] (see also [JSW08] and [DHL17a]).
Quantiative weighted estimates for rough homogeneous kernels have been
recently obtained in [HRT17, Con+16] and extended to their maximal truncations
in [Ler17, DHL17]. It would be interesting to extend these results to
variational truncations and also homogeneous groups as in [Tao99].

##  2\. Notation and tools

###  2.1. Spaces of homogeneous type

######  Definition 2.1.

A _quasi-metric_ on a set $X$ is a function $\rho:X\times X\to[0,\infty)$ such
that $\rho(x,y)=0\iff x=y$ that is symmetric and satisfies the quasi-triangle
inequality

(2.2) $\rho(x,y)\leq A_{0}(\rho(x,z)+\rho(z,y))\quad\text{for all}\quad
x,y,z\in X$

with some $A_{0}<\infty$ independent of $x,y,z$.

A measure $\mu$ on a quasi-metric space $(X,\rho)$ is called _doubling_ if
there exists $A_{1}<\infty$ such that

(2.3) $\mu(B(x,2r))\leq A_{1}\mu(B(x,r))\quad\text{for all}\quad x\in X,r>0,$

A tuple $(X,\rho,\mu)$ consisting of a set $X$, a quasi-metric $\rho$, and a
doubling measure $\mu$ is called a _space of homogeneous type_.

A space of homogeneous type $(X,\rho,\mu)$ is called (Ahlfors–David) _$D$
-regular_, $D>0$, if there exist $0<c,C<\infty$ such that for all $x\in X$ and
$r>0$ we have

$cr^{D}\leq\mu(B(x,r))\leq Cr^{D}.$

A $D$-regular space necessarily has no atoms. We say that a family
$\mathcal{D}$ of subsets of $X$ has the _small boundary property_ if there
exist $\eta>0$ and $C_{3}<\infty$ such that for every $Q\in\mathcal{D}$ and
every $0<\tau\leq 1$

(2.4) $\mu(\partial_{\tau\mathrm{diam}(Q)}Q)\leq C_{3}\tau^{\eta}\mu(Q),$

where

(2.5) $\partial_{\tau}(Q)=\\{x\in Q:\mathrm{dist}(x,X\setminus
Q)\leq\tau\\}\cup\\{x\in X\setminus Q:\mathrm{dist}(x,Q)\leq\tau\\}.$

We say that $(X,\rho,\mu)$ has the small boundary property if the collection
of all metric balls has the small boundary property.

We denote the measure of a set $Q$ by $\lvert Q\rvert=\mu(Q)$ and the average
of a function $f$ over $Q$ by $\left\langle f\right\rangle_{Q}=\lvert
Q\rvert^{-1}\int_{Q}f\mathrm{d}\mu$.

###  2.2. Dyadic cubes

Filtrations on spaces of homogeneous type that closely resemble dyadic
filtrations on $\mathbb{R}^{D}$ have been first constructed by Christ [Chr90].
We recall their properties.

######  Definition 2.6.

Let $(X,\rho,\mu)$ be a space of homogeneous type. A _system of dyadic cubes_
$\mathcal{D}$ with constants $\kappa>1$, $a_{0}>0$, $C_{1}<\infty$ consists of
collections $\mathcal{D}_{k}$, $k\in\mathbb{Z}$, of open subsets of $X$ such
that and constants $\kappa>1$, $a_{0},\eta>0$, $C_{1},C_{2}<\infty$ with the
following properties.

  1. (1)

$\forall k\in\mathbb{Z}\quad\mu(X\setminus\cup_{Q\in\mathcal{D}_{k}}Q)=0$,

  2. (2)

If $l\geq k$, $Q\in\mathcal{D}_{l}$, $Q^{\prime}\in\mathcal{D}_{k}$, then
either $Q^{\prime}\subseteq Q$ or $Q^{\prime}\cap Q=\emptyset$,

  3. (3)

For every $l\geq k$ and $Q^{\prime}\in\mathcal{D}_{k}$ there exists a unique
$Q\in\mathcal{D}_{l}$ such that $Q\supseteq Q^{\prime}$,

  4. (4)

$\forall k\in\mathbb{Z},Q\in\mathcal{D}_{k}\quad\exists c_{Q}\in
X:B(c_{Q},a_{0}\kappa^{k})\subseteq Q\subseteq B(c_{Q},C_{1}\kappa^{k})$.

Abusing the notation we write $k(Q)=k$ if $Q\in\mathcal{D}_{k}$, so that each
cube remembers its scale although the same cube (as a set) may appear in other
$\mathcal{D}_{l}$. We use $\mathcal{D}$ to denote the disjoint union of
$\mathcal{D}_{k}$.

If in addition the collection $\mathcal{D}$ has the small boundary property
(2.4), then we call $\mathcal{D}$ a _Christ_ system of dyadic cubes.

######  Theorem 2.7 ([Chr90]).

Every space of homogeneous type admits a system of Christ dyadic cubes.

######  Remark 2.8.

It is known that for every quasimetric $\rho$ there exists $0<\alpha\leq 1$
and a metric $d$ such that $d(x,y)\sim\rho(x,y)^{\alpha}$, see [MS79] for the
first proof and [PS09] for the fact that one can choose
$(2C_{\eqref{qm}})^{\alpha}=2$ (a much more verbose statement of the latter
result also appears in [Mit+13]). It is easy to see that a system of (Christ)
dyadic cubes with respest to $d$ is also a system of (Christ) dyadic cubes
with respect to $\rho$ (with different constants). However, the small boundary
property of dyadic cubes seems to be substantially easier to achieve in the
metric setting using the Hardy–Littlewood maximal function on $(0,\infty)$ as
on [DM00, p. 146].

Throughout the article we fix a system of Christ dyadic cubes $\mathcal{D}$ on
$(X,\rho,\mu)$. We denote by $\mathbb{E}_{k}$ the conditional expectation
operator onto the $\sigma$-algebra generated by $\mathcal{D}_{k}$. The
martingale difference operator is denoted by
$\mathbb{D}_{k}=\mathbb{E}_{k}-\mathbb{E}_{k+1}$. The function
$\mathbb{D}_{k}f$ is constant on each cube of scale $k$ and has integral $0$
on each cube of scale $k+1$.

######  Definition 2.9.

A finite family $(\mathcal{D}^{\alpha})_{\alpha}$ of systems of dyadic cubes
is called _adjacent_ if all constants in their definitions coincide and there
exists a constant $C_{3}<\infty$ such that for every ball $B(x,r)\subset X$
there exists $\alpha$ and a cube $Q\in\mathcal{D}^{\alpha}$ such that
$B(x,r)\subseteq Q\subseteq B(x,C_{3}r)$.

######  Theorem 2.10 ([HK13]).

Every space of homogeneous type admits a finite collection of adjacent systems
of dyadic cubes.

######  Remark 2.11.

It is also possible to construct adjacent systems of Christ dyadic cubes.

######  Example 2.12.

Let $X=\mathbb{R}^{D}$ with the Euclidean distance and the Lebesgue measure.
For each $\alpha\in\\{0,1,2\\}^{D}$ the corresponding _shifted system of
dyadic cubes_ is given by

$\mathcal{D}^{\alpha}=\\{2^{-k}([0,1)^{D}+m+(-1)^{k}\frac{1}{3}\alpha),k\in\mathbb{Z},m\in\mathbb{Z}^{D}\\}.$

Then the systems $\mathcal{D}^{\alpha}$, $\alpha\in\\{0,1,2\\}^{D}$, are
adjacent. In fact, on $\mathbb{R}^{D}$ one can construct $D+1$ shifted systems
of dyadic cubes that are adjacent [Mei03].

###  2.3. Sparse collections and operators

######  Definition 2.13.

Let $\mathcal{D}$ be a system of dyadic cubes. A collection
$\mathcal{S}\subset\mathcal{D}$ is called

  1. (1)

_$\eta$ -sparse_ if there exist pairwise disjoint subsets $E(Q)\subset
Q\in\mathcal{S}$ with $\lvert E(Q)\rvert\geq\eta\lvert Q\rvert$ and

  2. (2)

_$\Lambda$ -Carleson_ if one has $\sum_{Q^{\prime}\subset
Q,Q^{\prime}\in\mathcal{S}}\mu(Q^{\prime})\leq\Lambda\mu(Q)$ for all
$Q\in\mathcal{D}$.

It is known that a collection is $\eta$-sparse if and only if it is
$1/\eta$-Carleson [LN15, §6.1].

The corresponding _sparse operator_ is given by

(2.14)
$A_{\mathcal{S}}f=\sum_{Q\in\mathcal{S}}\mathbf{1}_{Q}\left\langle\lvert
f\rvert\right\rangle_{CQ}$

and _sparse square function_ by

(2.15)
$A_{\mathcal{S}}^{2}f=\Big{(}\sum_{Q\in\mathcal{S}}\mathbf{1}_{Q}\left\langle\lvert
f\rvert\right\rangle_{CQ}^{2}\Big{)}^{1/2}$

The sparse operators (2.14) and square functions (2.15) can be dominated by
finite linear combinations of similar sparse operators/square functions with
respect to adjacent dyadic grids in which the averages of $f$ are taken over
$Q$ instead of $CQ$, cf. [Ler16, Remark 4.3]. Hence the usual estimates for
sparse operators [HL18, DLR16] apply to (2.14) and (2.15).

We say that an operator $T$ is pointwise controlled by a sparse operator (or
square function) with constant $C<\infty$ if for every function $f$ there
exist $1/2$-sparse collections $\mathcal{S}^{n}\subset\mathcal{D}$,
$n\in\mathbb{N}$, such that

$\lvert Tf\rvert\leq
C\liminf_{n\to\infty}A_{\mathcal{S}^{n}}f\quad\text{or}\quad\lvert
Tf\rvert\leq C\liminf_{n\to\infty}A_{\mathcal{S}^{n}}^{2}f,$

respectively, holds pointwise almost everywhere.

We use a version of the sparse domination principle in [Ler16]. Given a dyadic
grid $\mathcal{D}$ denote by $\mathcal{D}^{2}_{\subseteq}$ the set of pairs
$(Q^{\prime},Q)\in\mathcal{D}\times\mathcal{D}$ with $Q^{\prime}\subseteq Q$.
For a function $F:\mathcal{D}^{2}_{\subseteq}\to[0,\infty]$ let

(2.16) $\mathcal{N}_{Q}F(x):=\sup_{x\in Q^{\prime}\subseteq
Q}F(Q^{\prime},Q),\quad\mathcal{N}F(x):=\sup_{x\in Q^{\prime}\subseteq
Q}F(Q^{\prime},Q).$

In the first definition above the supremum is taken over all $Q^{\prime}$ and
in the second over all $Q^{\prime},Q$. We use the convention that the supremum
of an empty subset of $[0,\infty]$ is $0$.

A simple stopping time argument using the doubling condition and starting at a
scale $k_{0}$ gives the following result.

######  Lemma 2.17.

Let $F:\mathcal{D}^{2}_{\subseteq}\to[0,\infty]$ be a function that is
monotonic in the sense that

$Q^{\prime\prime\prime}\subseteq Q^{\prime\prime}\subseteq Q^{\prime}\subseteq
Q\implies F(Q^{\prime\prime},Q^{\prime})\leq F(Q^{\prime\prime\prime},Q)$

and $\ell^{r}$-subadditive for some $0<r<\infty$ in the sense that

$Q^{\prime\prime}\subseteq Q^{\prime}\subseteq Q\implies
F(Q^{\prime\prime},Q)^{r}\leq
F(Q^{\prime\prime},Q^{\prime})^{r}+F(Q^{\prime},Q)^{r}.$

Suppose that for every dyadic cube $Q\in\mathcal{D}$ we have

(2.18) $\lVert\mathcal{N}_{Q}F\rVert_{1,\infty}\leq C\lVert
f\rVert_{L^{1}(CQ)}.$

Then there exist $1/2$-sparse collections
$\mathcal{S}^{k_{0}}\subset\mathcal{D}$ of cubes such that

(2.19) $\mathcal{N}F\lesssim
C\liminf_{k_{0}\to\infty}\Big{(}\sum_{Q\in\mathcal{S}^{k_{0}}}\mathbf{1}_{Q}\left\langle\lvert
f\rvert\right\rangle_{CQ}^{r}\Big{)}^{1/r}$

holds pointwise almost everywhere.

###  2.4. Bounded $r$-variation and jump counting

######  Definition 2.20.

Let $I$ be an ordered set and $(a_{t})_{t\in I}$ a family of complex numbers.
The _homogeneous $r$-variation seminorm_ is denoted by

$\dot{V}^{r}(a_{t}:t\in I):=\sup_{t_{0}<t_{1}<\dots<t_{J}\in
I}\big{(}\sum_{j=1}^{J}\lvert a_{t_{j}}-a_{t_{j-1}}\rvert^{r}\big{)}^{1/r}$

and the _inhomogeneous $r$-variation norm_ by

$V^{r}(a_{t}:t\in I):=\sup_{t\in I}\lvert a_{t}\rvert+\dot{V}^{r}(a_{t}:t\in
I).$

The _$\lambda$ -jump counting function_ $\mathcal{J}_{\lambda}(a_{t}:t\in I)$
is the supremum over all $J$ such that there exist $t_{0}<t_{1}<\dots<t_{J}$
with $\lvert a_{t_{j}}-a_{t_{j-1}}\rvert>\lambda$ for all $j=1,\dots,J$.

We refer to [JSW08] for the basic properties of the jump counting function and
its relation to $r$-variations.

##  3\. Short variations

We will consider two types of short variation operators, both involving a non-
tangentional supremum component introduced in [Kra13]. Ergodic averages will
be compared with a dyadic martingale using the short variations

(3.1) $S_{k}f(x):=\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k}}V^{2}(A_{t}f(x^{\prime})-\mathbb{E}_{k}f(x^{\prime}):\kappa^{k}/C\leq
t\leq C\kappa^{k}).$

For a CZ kernel $K$ as in Theorem 1.8 (in fact we only need the cancellation
condition in $y$ and the size estimate, but not the smoothness condition) we
define

(3.2) $S_{k}f(x):=\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k}}V^{2}(\int_{\kappa^{k}/C<\rho(x^{\prime},y)<t}K(x^{\prime},y)f(y)\mathrm{d}y:\kappa^{k}/C\leq
t\leq C\kappa^{k}).$

The non-tangentional short variation square function is defined by

(3.3) $Sf:=(\sum_{k}(S_{k}f)^{2})^{1/2}.$

In the short variation we can in fact replace $2$-variation by $r$-variation
for any $r>1$, see [arxiv:1409.7120]. The following result has been
essentially proved in [arxiv:1409.7120] following the arguments in [Jon+98,
JRW03].

######  Theorem 3.4.

Let $S_{k}$ be given either by (3.1) or by (3.2). Then the operator $S$ given
by (3.3) has weak type $(1,1)$.

As in [Zor17], Theorem 3.4 immediately implies that $S$ is pointwise
controlled by a sparse square function. Using the results from [HL18] this
implies a sharp weighted refinement of the case $p>1$ of [arxiv:1409.7120,
Theorem 1.4]. Similarly as in [Zor17] one can also show $\lVert
Sf\rVert_{L^{2}(w)}\lesssim\lVert f\rVert_{L^{2}(Mw)}$ and $\lVert
Sf\rVert_{L^{1,\infty}(w)}\lesssim\lVert f\rVert_{L^{1}(Mw)}$.

###  3.1. A reverse Hölder inequality

######  Lemma 3.5 (cf. [JRW03, Lemma 4.2]).

Let $\mathcal{Q}\subset\mathcal{D}$ be a collection of disjoint dyadic cubes
of scale $\leq k$. For each $Q\in\mathcal{Q}$ let $b^{Q}$ be a scalar-valued
function supported on $Q$ with $\int b^{Q}=0$. Then for every
$\alpha>\frac{D-\eta}{2}$ we have

(3.6)
$S_{k}(\sum_{Q\in\mathcal{Q}}b^{Q}(x))^{2}\lesssim(\kappa^{k})^{-2D}\sum_{Q:\mathrm{dist}(x,Q)\lesssim\kappa^{k}}(\kappa^{k}/\mathrm{diam}(Q))^{2\alpha}\lVert
b^{Q}\rVert_{1}^{2}.$

###### Proof.

Clearly only cubes with $\mathrm{dist}(x,Q)\lesssim\kappa^{k}$ contribute to
the left-hand side, so we may remove all other cubes from $\mathcal{Q}$. Now
the right-hand side of the conclusion becomes independent of $x$, so it
suffices to estimate the variation at an arbitrary point, which we again call
$x$.

We consider only the homogeneous variation, in order to get the inhomogeneous
variation it suffices to additionally consider an arbitrary (but fixed) $t$,
which is similar but easier. For a suitable sequence $\kappa^{k}/C\leq
t_{1}<\dots<t_{J}\leq C\kappa^{k}$ we have

$S_{k}(\sum_{Q}b^{Q}(x))^{2}\lesssim\sum_{j}\lvert\sum_{Q}\mathcal{B}b^{Q}(x,t_{j},t_{j+1})\rvert^{2},$

where $\mathcal{B}$ is either a difference between two averages or an integral
over an annulus. We decompose $\mathcal{Q}=\cup_{i\leq 0}\mathcal{Q}_{i}$
according to scale: $\mathcal{Q}_{i}=\mathcal{Q}\cap\mathcal{D}_{k+i}$. Each
cube $Q$ can only contribute to the $j$-th summand non-trivially in two cases:

  1. (1)

if $Q\cap(\partial B(x,t))\neq\emptyset$ for $t=t_{j}$ or $t=t_{j+1}$

  2. (2)

or $Q\subset B(x,t_{j+1})\setminus B(x,t_{j})$ in the case (3.2).

The second case can only occur for one index $j$, and the contribution of this
term can be estimated even without the growing factor in (3.6). In the first
case, for a fixed scale $k+i$, $i\leq 0$, the small boundary property implies
that there are at most
$O(\kappa^{kD}\kappa^{i\eta}/\kappa^{(k+i)D})=O(\kappa^{i(\eta-D)})$ cubes of
this kind in $\mathcal{Q}_{i}$, call this collection $\mathcal{Q}_{i,j}$.
Splitting the collection $\mathcal{Q}$ into scales and applying Hölder’s
inequality we estimate

(3.7) $\sum_{j}\lvert\sum_{i\leq
0}\sum_{Q\in\mathcal{Q}_{i}}\mathcal{B}b^{Q}(x,t_{j},t_{j+1})\rvert^{2}\\\
\lesssim\sum_{j}\big{(}\sum_{i\leq
0}\sum_{Q\in\mathcal{Q}_{i,j}}\lvert\kappa^{-\alpha
i}\mathcal{B}b^{Q}(x,t_{j},t_{j+1})\rvert^{2}\big{)}\cdot\big{(}\sum_{i\leq
0}\sum_{Q\in\mathcal{Q}_{i,j}}\lvert\kappa^{\alpha i}\rvert^{2}\big{)}\\\
\lesssim\sum_{j}\sum_{i\leq 0}\sum_{Q\in\mathcal{Q}_{i}}\lvert\kappa^{-\alpha
i}\mathcal{B}b^{Q}(x,t_{j},t_{j+1})\rvert^{2}$

in view of the hypothesis on $\alpha$ and the fact that
$\lvert\mathcal{Q}_{i,j}\rvert\lesssim\kappa^{i(\eta-D)}$. Estimating the
$2$-variation norm by the $1$-variation norm we obtain

$\eqref{eq:V-bi-biQ:CS-single-scale}\lesssim\sum_{i\leq 0}\kappa^{-2\alpha
i}\sum_{Q\in\mathcal{Q}_{i}}\big{(}\sum_{j}\mathcal{B}b^{Q}(x,t_{j},t_{j+1})\big{)}^{2}\\\
\lesssim\kappa^{-2kD}\sum_{Q:\mathrm{dist}(Q,x)\lesssim\kappa^{k}}(\kappa^{k}/\mathrm{diam}(Q))^{2\alpha}\lVert
b^{Q}\rVert_{1}^{2}.$

Taking the supremum over sequences $(t_{j})_{j}$ we obtain (3.6). ∎

###  3.2. Strong type $(2,2)$

######  Lemma 3.8.

There exists an $\epsilon>0$ such that for all $k,j\in\mathbb{Z}$ we have

(3.9) $\lVert S_{k}(\mathbb{D}_{k+j}f)\rVert_{2}\lesssim 2^{-\epsilon\lvert
j\rvert}\lVert\mathbb{D}_{k+j}f\rVert_{2}.$

###### Proof.

Rescaling the metric we may assume $k=0$.

Consider first $j<0$. Then $\mathbb{E}_{0}\mathbb{D}_{j}=0$, and by Lemma 3.5
with $\mathcal{Q}=\mathcal{D}_{j+1}$ we obtain

$\displaystyle S_{0}\mathbb{D}_{j}f(x)$
$\displaystyle\lesssim\Big{(}\sum_{Q\in\mathcal{D}_{j+1}:\mathrm{dist}(x,Q)\lesssim
1}(\kappa^{-j})^{\alpha
2}\lVert\mathbf{1}_{Q}\mathbb{D}_{j}f\rVert_{1}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim(\kappa^{-j})^{\alpha}\Big{(}\sum_{Q\in\mathcal{D}_{j+1}:\mathrm{dist}(x,Q)\lesssim
1}\lVert\mathbf{1}_{Q}\mathbb{D}_{j}f\rVert_{2}^{2}\lvert
Q\rvert\Big{)}^{1/2}$
$\displaystyle\lesssim\kappa^{j(D/2-\alpha)}\Big{(}\int_{\mathrm{dist}(x,y)\lesssim
1}\lvert\mathbb{D}_{j}f(y)\rvert^{2}\Big{)}^{1/2},$

and the conclusion follows since the averaging operator of scale $1$ is
bounded on $L^{1}$ and we can choose $\frac{D-\eta}{2}<\alpha<\frac{D}{2}$.

Consider now $j\geq 0$. Then $S_{0}$ does not vanish only in $1$-neighborhoods
of boundaries of $Q\in\mathcal{D}_{j}$, and there we can estimate the
$2$-variation by the $1$-variation. In particular,

$\displaystyle
S_{0}\mathbb{D}_{j}f(x)\leq\sum_{Q\in\mathcal{D}_{j}}S_{0}(\mathbf{1}_{Q}\mathbb{D}_{j}f)(x)\lesssim\sum_{Q\in\mathcal{D}_{j}}\lVert\mathbf{1}_{Q}\mathbb{D}_{j}f\rVert_{1}\mathbf{1}_{\partial_{1}Q}(x).$

The latter characteristic functions have bounded overlap in view of the
doubling property of our measure space. Hence

$\lVert
S_{0}\mathbb{D}_{j}f\rVert_{2}^{2}\lesssim\sum_{Q\in\mathcal{D}_{j}}\lVert\mathbf{1}_{Q}\mathbb{D}_{j}f\rVert_{1}^{2}\lvert\partial_{1}Q\rvert\lesssim\sum_{Q\in\mathcal{D}_{j}}\lVert\mathbf{1}_{Q}\mathbb{D}_{j}f\rVert_{1}^{2}\lvert
Q\rvert\kappa^{-j\eta}=\kappa^{-j\eta}\lVert\mathbb{D}_{j}f\rVert_{2}^{2},$

where we have used the small boundary property (2.4) of the dyadic cubes. ∎

###  3.3. Weak type $(1,1)$

###### Proof of Theorem 3.4.

It follows from Lemma 3.8 that $S$ has strong type $(2,2)$.

By homogeneity it suffices to prove

$\lvert\\{x:Sf(x)>1\\}\rvert\lesssim\lVert f\rVert_{1}.$

We use the Calderón–Zygmund decomposition. Let $\mathcal{Q}\subset\mathcal{D}$
be disjoint cubes such that such that $\lVert
f\rVert_{L^{\infty}(X\setminus\cup\mathcal{Q})}\leq 1$,
$\sum_{Q\in\mathcal{Q}}\lvert Q\rvert\lesssim\lVert f\rVert_{1}$, and $\lVert
f\rVert_{L^{1}(Q)}\lesssim\lvert Q\rvert$ for all $Q\in\mathcal{Q}$. Let

$g(x)=\begin{cases}\lvert Q\rvert^{-1}\int_{Q}f,&x\in Q\in\mathcal{Q},\\\
f(x),&x\not\in\cup\mathcal{Q}\end{cases}$

and

$b=\sum_{Q\in\mathcal{Q}}b^{Q},\quad b^{Q}(x)=\begin{cases}f(x)-\lvert
Q\rvert^{-1}\int_{Q}f,&x\in Q,\\\ 0,&x\not\in Q.\end{cases}$

Then $\lVert g\rVert_{1}\leq\lVert f\rVert_{1}$ and $\lVert
g\rVert_{\infty}\lesssim 1$, so we get the required weak type bound for $g$
from the strong type $(2,2)$ bound.

With $E:=\cup_{Q\in\mathcal{Q}}CQ$, where $C$ is sufficiently large in terms
of the constants in the definition of $S_{k}$, it suffices to show

(3.10) $\lvert\\{Sb>1\\}\setminus E\rvert\lesssim\sum_{Q}\lvert Q\rvert.$

To this end it suffices to show

(3.11) $\int_{X\setminus E}\sum_{k}(S_{k}b)^{2}\lesssim\sum_{Q}\lvert
Q\rvert.$

Let $(D-\eta)/2<\alpha<D/2$. For $x\not\in E$ only cubes of scale $\leq k$
contribute to $S_{k}$. Thus by Lemma 3.5 we have

$(S_{k}b)^{2}(x)\lesssim(\kappa^{k})^{-2D}\sum_{Q:\mathrm{dist}(x,Q)\lesssim\kappa^{k}}(\kappa^{k}/\mathrm{diam}(Q))^{2\alpha}\lVert
b^{Q}\rVert_{1}^{2}.$

Hence the left-hand side of (3.11) can be estimated by

$\displaystyle\int_{X\setminus
E}\sum_{k}(\kappa^{k})^{-2D}\sum_{Q}\mathbf{1}_{\mathrm{dist}(x,Q)\lesssim\kappa^{k}}(\kappa^{k}/\mathrm{diam}(Q))^{2\alpha}\lVert
b^{Q}\rVert_{1}^{2}\mathrm{d}x$ $\displaystyle\lesssim\sum_{Q}\sum_{k\geq
k(Q)}(\kappa^{k})^{-2D}(\kappa^{k}/\mathrm{diam}(Q))^{2\alpha}\lvert
Q\rvert^{2}\int_{X\setminus
E}\mathbf{1}_{\mathrm{dist}(x,Q)\lesssim\kappa^{k}}\mathrm{d}x$
$\displaystyle\lesssim\sum_{Q}\mathrm{diam}(Q)^{-2\alpha}\lvert
Q\rvert^{2}\sum_{k\geq k(Q)}\kappa^{k(-D+2\alpha)}$
$\displaystyle\lesssim\sum_{Q}\lvert Q\rvert^{2}\mathrm{diam}(Q)^{-D}$
$\displaystyle\lesssim\sum_{Q}\lvert Q\rvert.\qed$

##  4\. Averages

The estimate for the jump counting function in Theorem 1.3 will follow from
Lemma 2.17 and the next result.

######  Proposition 4.1.

For every $f\in L^{1}(X)\cap L^{\infty}(X)$, every $\lambda>0$ and
$k_{\max}<\infty$ there exists a subadditive function $F_{\lambda}$ on
$\mathcal{D}^{2}_{\subseteq}$ such that

$\lambda\sqrt{\mathcal{J}_{\lambda}(A_{t}f(x):\kappa^{k(Q^{\prime})}\leq
t\leq\kappa^{k(Q)})}\leq F_{\lambda}(Q^{\prime},Q),\quad x\in
Q^{\prime}\subseteq Q,k(Q)\leq k_{\max},$

and

$\mu\\{\mathcal{N}_{Q}F_{\lambda}>\nu\\}\leq C\nu^{-1}\lVert
f\rVert_{L^{1}(CQ)}\quad\text{for all}\quad Q\in\mathcal{D},\nu>0.$

###### Proof.

Fix $\lambda>0$. Note first that, as in [JSW08, Lemma 1.3], we have

$\displaystyle\lambda\sqrt{\mathcal{J}_{\lambda}(A_{t}f(x):\kappa^{k(Q^{\prime})}\leq
t\leq\kappa^{k(Q)})}$
$\displaystyle\lesssim\Big{(}\sum_{k=k(Q^{\prime})}^{k(Q)-1}\sup_{x^{\prime}\in
Q_{k}(x)}S_{k}f(x^{\prime})^{2}\Big{)}^{1/2}$
$\displaystyle+\lambda\sqrt{\mathcal{J}_{\lambda/2}(\mathbb{E}_{k}f(x):k(Q^{\prime})\leq
k\leq k(Q))}.$

The first term on the right-hand side is a subadditive function $F$ of the
order interval $[Q^{\prime},Q]$, and its contribution is estimated by Theorem
3.4.

In order to estimate the second term we follow the proof of Lépingle’s
inequality for martingales. We construct the greedy stopping times with jump
size $\lambda/8$ starting with $l_{0}(x)\equiv k_{\max}$: given $l_{j}(x)$ let
$l_{j+1}(x)<l_{j}(x)$ be the largest number with
$\lvert(\mathbb{E}_{l_{j+1}}-\mathbb{E}_{l_{j}})f(x)\rvert>\lambda/8$ (or
$-\infty$ if no such number exists). Then

$\lambda\sqrt{\mathcal{J}_{\lambda/2}(\mathbb{E}_{k}f(x):k_{0}\leq k\leq
k_{1})}\lesssim\Big{(}\sum_{j}\lvert(\mathbb{E}_{t_{j+1}(x)}-\mathbb{E}_{t_{j}(x)})(\mathbb{E}_{k_{0}}-\mathbb{E}_{k_{1}})f(x)\rvert^{2}\Big{)}^{1/2}.$

Let $F_{\lambda}^{\prime}(Q^{\prime},Q)$ denote the right-hand side when $x\in
Q^{\prime}\subset Q$, $k(Q^{\prime})=k_{0}$, and $k(Q)=k_{1}$. Then
$F_{\lambda}^{\prime}(Q^{\prime},Q)$ is clearly a subadditive function of the
order interval $[Q^{\prime},Q]$. Moreover,

$\mathcal{N}F\lesssim\Big{(}\sum_{j}\lvert
M_{\mathcal{D}}(\mathbb{E}_{t_{j+1}}-\mathbb{E}_{t_{j}})f\rvert^{2}\Big{)}^{1/2},$

where $M_{\mathcal{D}}$ is the dyadic maximal operator. Hence the operator
$f\mapsto\mathcal{N}F_{\lambda}^{\prime}$ has strong type $(2,2)$ by
orthogonality of the operators $\mathbb{E}_{t_{j+1}}-\mathbb{E}_{t_{j}}$ and
the martingale maximal inequality. Standard CZ theory can be applied to show
that it also has weak type $(1,1)$ because the filtration $\mathcal{D}$ is
doubling. We obtain the claim with $F_{\lambda}=F+F_{\lambda}^{\prime}$. ∎

In order to pass to $r$-variations we need the fact that the $p$-th power of
the $L^{p,\infty}$ quasimetric is equivalent to a metric for $0<p<1$.

######  Lemma 4.2 (cf. [SW69, Lemma 2.3]).

Let $0<p<1$. Then

$\lVert\sum_{j}g_{j}\rVert_{p,\infty}^{p}\leq
2^{p}\Big{(}1+\frac{1}{1-p}\Big{)}\sum_{j}\lVert g_{j}\rVert_{p,\infty}^{p}.$

###### Proof.

It suffices to consider $g_{j}\geq 0$. Moreover, by homogeneity we may assume

$\sum_{j}c_{j}=1,\quad c_{j}=\lVert g_{j}\rVert_{p,\infty}^{p}.$

With this normalization we have to show

$\lvert\\{\sum_{j}g_{j}>\lambda\\}\rvert\leq
2^{p}\Big{(}1+\frac{1}{1-p}\Big{)}\lambda^{-p}.$

To this end decompose $g_{j}=l_{j}+m_{j}+u_{j}$, where
$u_{j}=g_{j}\mathbf{1}_{g_{j}>\lambda/2}$ and
$l_{j}=\mathbf{1}_{g_{j}\leq\lambda/2}\min(g_{j},c_{j}\lambda/2)$. Then

$\sum_{j}\lvert\mathrm{supp}u_{j}\rvert=\sum_{j}\lvert\\{g_{j}>\lambda/2\\}\rvert\leq\sum_{j}c_{j}(\lambda/2)^{-p}=(\lambda/2)^{-p}$

and

$\sum_{j}l_{j}\leq\sum_{j}c_{j}\lambda/2\leq\lambda/2.$

Hence it remains to estimate

$\lvert\\{\sum_{j}m_{j}>\lambda/2\\}\rvert\leq(\lambda/2)^{-1}\int\sum_{j}m_{j}\leq(\lambda/2)^{-1}\sum_{j}\int(g_{j}-c_{j}\lambda/2)\mathbf{1}_{c_{j}\lambda/2\leq
g_{j}\leq\lambda/2}\\\
=(\lambda/2)^{-1}\sum_{j}\int_{t=c_{j}\lambda/2}^{\lambda/2}\lvert\\{g_{j}>t\\}\rvert\mathrm{d}t\leq(\lambda/2)^{-1}\sum_{j}\int_{t=c_{j}\lambda/2}^{\lambda/2}c_{j}t^{-p}\mathrm{d}t\\\
\leq(1-p)^{-1}(\lambda/2)^{-1}\sum_{j}c_{j}(\lambda/2)^{1-p}\leq\frac{2^{p}}{1-p}\lambda^{-p}.\qed$

######  Corollary 4.3.

There exists $C<\infty$ such that for all $r>2$ we have

$\mu\\{\mathcal{N}_{Q}F>\nu\\}\leq C\frac{r}{r-2}\nu^{-1}\lVert
f\rVert_{L^{1}(CQ)}\quad\text{for all}\quad Q\in\mathcal{D},$

where

$F(Q^{\prime},Q)=\sup_{x^{\prime}\in
Q^{\prime}}\dot{V}^{r}(A_{t}f(x^{\prime}):\kappa^{k(Q^{\prime})}\leq
t\leq\kappa^{k(Q)}).$

Corollary 4.3 and Lemma 2.17 imply the estimate for $r$-variations in Theorem
1.3.

###### Proof.

By homogeneity we may normalize $\nu=1$. Let $F_{\lambda}$ be the functions
from Proposition 4.1.

Suppose $F(Q,Q^{\prime})>1$. Then either $F_{1}(Q,Q^{\prime})>1$ or all jumps
involved in the definition of $r$-variation in $F(Q,Q^{\prime})$ are bounded
by $1$. In the latter case

$1<F(Q,Q^{\prime})^{r}\lesssim\sum_{l\geq 0}2^{-lr}\mathcal{J}_{2^{-l}},$

so that in either case we obtain

$\sum_{l\geq 0}2^{-l(r-2)}F_{2^{-l}}(Q^{\prime},Q)^{2}\gtrsim 1.$

Hence, using Lemma 4.2, we obtain

$\displaystyle\mu\\{\mathcal{N}_{Q}F>1\\}$
$\displaystyle\leq\mu\\{\mathcal{N}_{Q}\big{(}\sum_{l\geq
0}2^{-l(r-2)}F_{2^{-l}}^{2}\big{)}\gtrsim 1\\}$
$\displaystyle\lesssim\lVert\sum_{l\geq
0}2^{-l(r-2)}(\mathcal{N}_{Q}F_{2^{-l}})^{2}\rVert_{1/2,\infty}^{1/2}$
$\displaystyle\lesssim\sum_{l\geq 0}\lVert
2^{-l(r-2)}(\mathcal{N}_{Q}F_{2^{-l}})^{2}\rVert_{1/2,\infty}^{1/2}$
$\displaystyle=\sum_{l\geq
0}2^{-l(r-2)/2}\lVert\mathcal{N}_{Q}F_{2^{-l}}\rVert_{1,\infty}$
$\displaystyle\lesssim\lVert f\rVert_{L^{1}(CQ)}\sum_{l\geq 0}2^{-l(r-2)/2}$
$\displaystyle\lesssim\frac{r}{r-2}\lVert f\rVert_{L^{1}(CQ)}.\qed$

##  5\. Singular integrals

In this section we prove Theorem 1.8. Let $\psi$ be a smooth function
supported on the interval $[1/\kappa,\kappa]$ with
$\sum_{k}\psi(\kappa^{-k}x)=1$ for all $x\in(0,\infty)$. Let

$K_{k}(x,y)=K(x,y)\psi(\kappa^{-k}\rho(x,y))$

be the smoothly truncated dyadic pieces of $K$ of scale $k$ and let $T_{k}$ be
the corresponding integral operators

$T_{k}f(x)=\int K_{k}(x,y)f(y)\mathrm{d}y.$

This coincides with the notation for truncated singular integrals (1.2), but
there should be no confusion between integer parameters $k$ and positive real
parameters $t$. The Hölder continuity hypothesis (1.9) implies that $K_{k}$
satisfies the same smoothness condition (1.11) as $K$ (up to a constant
factor). Standard calculations using the cancellation and smoothness
conditions show that

(5.1) $\lVert T_{k^{\prime}}^{*}T_{k}\rVert_{L^{2}(X)\to L^{2}(X)}+\lVert
T_{k^{\prime}}T_{k}^{*}\rVert_{L^{2}(X)\to L^{2}(X)}\lesssim
2^{-\epsilon\lvert k-k^{\prime}\rvert}.$

Hence by the Cotlar–Stein lemma [Ste93, p. 280] the operator
$T=\sum_{k\in\mathbb{Z}}T_{k}$ is bounded on $L^{2}(X)$.

###  5.1. Short variations

Fix $r>2$ and define

$F(Q^{\prime},Q):=\sup_{x\in
Q^{\prime}}\dot{V}^{r}(\int_{\rho(x,y)>t}K(x,y)f(y)\mathrm{d}y:\kappa^{k(Q^{\prime})}\leq
t\leq\kappa^{k(Q)}).$

The function $F$ is clearly monotonic and subadditive in the order interval
$[Q^{\prime},Q]$ and sublinear in $f$. Moreover, $F(Q^{\prime},Q)$ depends
only on the values of $f$ on $CQ$. Hence, in view of Lemma 2.17, Theorem 1.8
will follow if we show that the map $f\mapsto\mathcal{N}F$ has weak type
$(1,1)$ globally.

We split

(5.2) $\displaystyle\int_{r<\rho(x,y)}K(x,y)f(y)\mathrm{d}y$
$\displaystyle=\sum_{k=k_{0}(r)}^{\infty}\int K_{k}(x,y)f(y)\mathrm{d}y$ (5.3)
$\displaystyle-\int_{\kappa^{k_{0}(r)-1}<\rho(x,y)<\kappa^{k_{0}(r)}}K_{k_{0}(r)}(x,y)f(y)\mathrm{d}y$
(5.4)
$\displaystyle+\int_{r<\rho(x,y)<\kappa^{k_{0}(r)}}K(x,y)f(y)\mathrm{d}y,$

where $k_{0}(r)=\lceil\log_{\kappa}r\rceil$. The contributions of (5.3) and
(5.4) are controlled by the square function (3.2), so it remains to estimate
the contribution of (5.2).

###  5.2. Strong type $(2,2)$

Let

$F^{\prime}(Q^{\prime},Q):=\sup_{x\in Q^{\prime}}\dot{V}^{r}(\sum_{k\geq
k_{0}}T_{k}f(x):k(Q^{\prime})\leq k_{0}\leq k(Q)).$

First we show that the operator $f\mapsto\mathcal{N}F^{\prime}$ has strong
type $(2,2)$. Following [DR86, p. 548] we decompose

$\sum_{k\geq k_{0}}T_{k}=\sum_{l\in\mathbb{Z},k\geq
k_{0}}\mathbb{D}_{l}T_{k}=\sum_{k\in\mathbb{Z},l\geq
k_{0}}\mathbb{D}_{l}T_{k}-\sum_{l\geq
k_{0},k<k_{0}}\mathbb{D}_{l}T_{k}+\sum_{l<k_{0},k\geq
k_{0}}\mathbb{D}_{l}T_{k}=:I+II+III.$

The contribution of $I$ to $\mathcal{N}F^{\prime}(x)$ is bounded by

$\dot{V}^{r}(\sum_{l\geq k_{0}}\mathbb{D}_{l}Tf(x):k_{0}\in\mathbb{Z}).$

Notice that the supremum over $x\in Q^{\prime}$ disappeared because functions
in the image of $\mathbb{D}_{l}$ are constant on dyadic cubes of scale $\leq
l$. This operator is bounded on $L^{2}(X)$ with norm $\lesssim r/(r-2)$ since
$T$ is bounded on $L^{2}(X)$ and by Lépingle’s inequality for martingales in
the form [Bou89, Lemma 3.3].

The remaining two terms will be estimated by square functions.

#### Estimate for $II$

As in the estimate for $I$ we remove the supremum over $x\in Q^{\prime}$. Then
we estimate the $r$-variation by the $\ell^{2}$ norm. Using Minkowski’s
inequality for the sum over $m$ and Cauchy–Schwarz inequality for the sum over
$k$ we estimate

$\displaystyle\Big{(}\sum_{k_{0}\in\mathbb{Z}}\lvert\sum_{l\geq
k_{0},k<k_{0}}\mathbb{D}_{l}T_{k}f\rvert^{2}\Big{)}^{1/2}$
$\displaystyle=\Big{(}\sum_{k_{0}\in\mathbb{Z}}\lvert\sum_{m\geq
1}\sum_{k\in\mathbb{Z}}\mathbf{1}_{k<k_{0}\leq
k+m}\mathbb{D}_{k+m}T_{k}f\rvert^{2}\Big{)}^{1/2}$
$\displaystyle\leq\sum_{m\geq
1}\Big{(}\sum_{k_{0}\in\mathbb{Z}}m\sum_{k\in\mathbb{Z}}\lvert\mathbf{1}_{k<k_{0}\leq
k+m}\mathbb{D}_{k+m}T_{k}f\rvert^{2}\Big{)}^{1/2}$ $\displaystyle=\sum_{m\geq
1}m\Big{(}\sum_{k\in\mathbb{Z}}\lvert\mathbb{D}_{k+m}T_{k}f\rvert^{2}\Big{)}^{1/2}.$

Hence it suffices to find an estimate for the square function on the right-
hand side that decays sufficiently quickly with $m$. By orthogonality

$\lVert\Big{(}\sum_{k\in\mathbb{Z}}\lvert\mathbb{D}_{k+m}T_{k}f\rvert^{2}\Big{)}^{1/2}\rVert_{2}=\lVert\sum_{k\in\mathbb{Z}}\mathbb{D}_{k+m}T_{k}f\rVert_{2}.$

For each $k$ we have

$\lVert\mathbb{D}_{k+m}T_{k}\rVert_{L^{2}(X)\to L^{2}(X)}=\lVert
T_{k}^{*}\mathbb{D}_{k+m}\rVert_{L^{2}(X)\to L^{2}(X)}\lesssim 2^{-\epsilon
m}$

since $T_{k}^{*}\mathbf{1}_{Q}$ is bounded by a universal constant and
supported on $\partial_{C\kappa^{k}}Q$ for every $Q\in\mathcal{D}_{k+m}$ in
view of the cancellation condition.

Also, for all $k\neq k^{\prime}$ we have

$(\mathbb{D}_{k+m}T_{k})^{*}\mathbb{D}_{k^{\prime}+m}T_{k^{\prime}}=0$

and

$\lVert\mathbb{D}_{k^{\prime}+m}T_{k^{\prime}}(\mathbb{D}_{k+m}T_{k})^{*}\rVert_{L^{2}(X)\to
L^{2}(X)}\leq\lVert T_{k^{\prime}}T_{k}^{*}\rVert_{L^{2}(X)\to
L^{2}(X)}\lesssim 2^{-\epsilon\lvert k-k^{\prime}\rvert}$

by (5.1). By the Cotlar–Stein lemma [Ste93, p. 280] it follows that
$\lVert\sum_{k\in\mathbb{Z}}\mathbb{D}_{k+m}T_{k}\rVert_{L^{2}(X)\to
L^{2}(X)}\lesssim 2^{-\epsilon m}$.

#### Estimate for $III$

This estimate is similar to $II$, but this time we have to keep the supremum
over $x^{\prime}\in Q^{\prime}$:

$\displaystyle\Big{(}\sum_{k_{0}\in\mathbb{Z}}\sum_{Q\in\mathcal{D}_{k_{0}}}\mathbf{1}_{Q}(x)\sup_{x^{\prime}\in
Q}\lvert\sum_{l<k_{0}\leq
k}\mathbb{D}_{l}T_{k}f(x^{\prime})\rvert^{2}\Big{)}^{1/2}$
$\displaystyle\leq\Big{(}\sum_{k_{0}\in\mathbb{Z}}\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k_{0}}}\lvert\sum_{m\geq
1}\sum_{k\in\mathbb{Z}}\mathbf{1}_{k-m<k_{0}\leq
k}\mathbb{D}_{k-m}T_{k}f(x^{\prime})\rvert^{2}\Big{)}^{1/2}$
$\displaystyle\leq\Big{(}\sum_{k_{0}\in\mathbb{Z}}\Big{(}\sum_{m\geq
1}\sum_{k\in\mathbb{Z}}\mathbf{1}_{k-m<k_{0}\leq
k}\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k}}\lvert\mathbb{D}_{k-m}T_{k}f(x^{\prime})\rvert\Big{)}^{2}\Big{)}^{1/2}$
$\displaystyle\leq\sum_{m\geq
1}\Big{(}\sum_{k_{0}\in\mathbb{Z}}m\sum_{k\in\mathbb{Z}}\Big{(}\mathbf{1}_{k-m<k_{0}\leq
k}\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k}}\lvert\mathbb{D}_{k-m}T_{k}f(x^{\prime})\rvert\Big{)}^{2}\Big{)}^{1/2}$
$\displaystyle=\sum_{m\geq
1}m\Big{(}\sum_{k\in\mathbb{Z}}\sup_{\rho(x,x^{\prime})\leq
C\kappa^{k}}\lvert\mathbb{D}_{k-m}T_{k}f(x^{\prime})\rvert^{2}\Big{)}^{1/2}$

We view the square sum on the right-hand side as an intrinsic square function.
Indeed, the condition (1.9) implies that $K_{k}$ also has Hölder type
regularity. Notice that $\mathbb{D}_{k-m}g(x^{\prime})=\int
gh_{k-m,x^{\prime}}$, where $h_{k-m,x^{\prime}}$ is a function with mean $0$
and bounded $L^{1}$ norm supported in $B(x,C\kappa^{k-m})$. It follows that
for each $x^{\prime}$ the function
$y\mapsto\mathbb{D}_{k-m}K_{k}(x^{\prime},y)$, where $\mathbb{D}_{k-m}$ acts
in the first variable, has absolute value $O(\kappa^{-kD}\kappa^{-\epsilon
m})$. Moreover, it is Hölder continuous since it is an average of Hölder
continuous functions. The required decay in $m$ now comes from the estimate
for the intrinsic square function, [Zor17, Theorem 1.3], with
$\Phi=\kappa^{-\epsilon m}$.

###  5.3. Weak type $(1,1)$

By homogeneity it suffices to show

$\lvert\\{\mathcal{N}F^{\prime}>1\\}\rvert\lesssim\lVert f\rVert_{1}.$

We make a CZ decomposition $f=g+b$. The good part $g$ is controlled by the
$L^{2}$ estimate. Let $Q\in\mathcal{D}$ be a bad cube, $b_{Q}$ the
corresponding bad function, and $\tilde{Q}\supset Q$ a ball containing $Q$
with a much larger, but still comparable, radius. We estimate

$\displaystyle\int_{X\setminus\tilde{Q}}\mathcal{N}F^{\prime}(b_{Q})(x)\mathrm{d}x$
$\displaystyle\lesssim\int_{X\setminus\tilde{Q}}\sum_{k>k(Q)}\sup_{\rho(x,x^{\prime})\lesssim\kappa^{k}}\lvert
T_{k}b_{Q}(x^{\prime})\rvert\mathrm{d}x$
$\displaystyle\lesssim\sum_{k>k(Q)}\kappa^{kD}\lVert
T_{k}b_{Q}\rVert_{\infty}$
$\displaystyle\lesssim\sum_{k>k(Q)}\kappa^{-\epsilon(k-k(Q))}\lVert
b_{Q}\rVert_{1}$ $\displaystyle\lesssim\lVert b_{Q}\rVert_{1},$

where we have used that $T_{k}b_{Q}$ is supported in a ball or radius
$O(\kappa^{k})$, the mean zero property of $b_{Q}$, and Hölder continuity of
$K_{k}$. Summing over all bad cubes we obtain the claim.

######  Remark 5.5.

The above proof also yields $L^{p}$ and weak type estimates for the jump
counting function $\mathcal{J}_{\lambda}(T_{t}f(x):t>0)$, and even its
localized non-tangentional maximal version. However, the jump counting
function is not subadditive, and unlike in the case of averages we have been
unable to construct a subadditive majorant that still satisfies the localized
non-tangentional weak type $(1,1)$ estimate.

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arxiv:1409.7120MR3742545 arxiv:1508.03872MR3649476 arXiv:1512.07523MR3681393
arxiv:1510.05789MR3625128 arXiv:1501.05818MR3625108