---
abstract: 'We extend the pointillist principles of Moon and Carrillo–de Guzmán to variational operators and jump functions.'
author:
- K Hughes
bibliography:
- 'references.bib'
title: The Pointillist Principle for Variation Operators and Jump functions
---

The pointillist principle
=========================

In [@Moon], Moon observed that for a sequence of sufficiently smooth convolution operators the weak (1,q) boundedness of their maximal operator is equivalent to restricted weak (1,q) boundedness of the maximal operator for any $q \geq 1$. For a sequence of operators $T_{m}$ for ${m}\in {\mathbb{N} }$, define their maximal function ${M}(T_{m}f(x) : {m}\in {\mathbb{N} }) := \sup_{{m}\in {\mathbb{N} }} |T_{m}f(x)|$ for $f:{\mathbb{R} }^{d}\to {\mathbb{C} }$ and $x \in {\mathbb{R} }^{d}$.

Suppose that $\{T_{m}\}_{{m}\in {\mathbb{N} }}$ is a sequence of convolution operators given by $T_{m}f := f {*}g_{m}$ for a sequence of functions $g_{m}\in L^1({\mathbb{R} }^{d})$. For any $q \geq 1$, ${M}(T_{m}f(x) : {m}\in {\mathbb{N} })$ is restricted weak-type $(1,q)$ with norm $C$ if and only if ${M}(T_{m}f(x) : {m}\in {\mathbb{N} })$ is weak-type $(1,q)$ with norm $C$.

Subsequently de Guzmán gave a version of Moon’s theorem where the class of characteristic functions is replaced by linear combinations of $\delta$-functions. To state his result, we introduce a little bit of terminology. Let $\delta_x$ denote the Dirac delta-function at the point $x \in {\mathbb{R} }^{d}$ and $g_{m}\in L^1({\mathbb{R} }^{d})$ be a sequence of functions. In analogy with restricted weak-type, let us say that ${M}(f*\mu_{m}(x) : {m}\in {\mathbb{N} })$ is *pointed weak-type $(p,p)$ with norm at most $C$* if for any finite subset of points $X \subset {\mathbb{R} }^{d}$, we have the inequality $$\label{eq:restricted_pointed_weak_type_inequality}
{\left \lvert {\left\{ x \in {\mathbb{R} }^{d}: \sup_{{m}\in {\mathbb{N} }} {\left \lvert \sum_{y \in X} g_{m}(x-y) \right \rvert} > \lambda \right\}} \right \rvert} 
\leq 
C \cdot \frac{\# X}{\lambda^{p}} 
.$$

Suppose that $\{T_{m}\}_{{m}\in {\mathbb{N} }}$ is a sequence of convolution operators given by $T_{m}f := f {*}g_{m}$ for a sequence of functions $g_{m}\in L^1({\mathbb{R} }^{d})$. For any $p \geq 1$, ${M}(T_{m}f(x) : {m}\in {\mathbb{N} })$ is weak-type $(p,p)$ with norm at most C if ${M}(T_{m}f(x) : {m}\in {\mathbb{N} })$ is pointed weak-type $(p,p)$ with norm at most C. Furthermore, the converse is true if $p=1$.

The converse can fail for $p>1$.

Grafakos–Mastylo extended Moon’s theorem to the multilinear setting in [@GM] while Carena extended Carrillo–de Guzmán’s theorem to more general metric measure spaces in [@Carena]. It should also be noted that Akcoglu–Baxter–Bellow–Jones showed that the analogue of Moon’s theorem over the discrete case ${\mathbb{Z} }$ may fail; see [@ABBJ:fail] and [@HJ:fail]. It is to these collections of theorem we refer to as the ‘pointillist principle’, taking its name from the Pointillism movement in art. The purpose of this short note is to extend Moon and Carrillo–de Guzmán’s instances of the pointillist principle to variational operators and jump functions. We recall these operators and discuss a few of their basic properties.

Suppose that $\{T_{m}\}_{{m}\in {\mathbb{N} }}$ is a sequence of operators acting on a Lebesgue measurable function $f:{\mathbb{R} }^{d}\to {\mathbb{C} }$. Let $r \in [1,\infty)$ and $\mathcal{R} \subset {\mathbb{N} }$. Define pointwise the $r$-variation operator over $T_{m}$ with respect to $\mathcal{R}$ as $$\label{def:variation}
{V_r}(T_{m}f(x) : {m}\in \mathcal{R})
:= 
\sup {\left( \sum_{i=1}^L |T_{{m}_i} f(x) - T_{{m}_{i+1}} f(x)|^r \right)}^{1/r}$$ where the supremum is over all finite, increasing subsequences $\{{m}_i\}$ of $\mathcal{R}$. One may make the usual modification using the essential supremum to extend to $r=\infty$. Note that ${V_r}(\cdot)$ is sublinear in its argument. Define the jump function ${N}_\lambda(T_{m}f(x) : {m}\in \mathcal{R})$ as given by the supremum over $M \in {\mathbb{N} }$ such that there exists a sequence $ s_0 < t_0 \leq s_1 < t_1 \leq \cdots  \leq s_M < t_M $ in $\mathcal{R}$ with $$|T_{s_i}f(x) - T_{t_i}f(x)| > \lambda$$ for each $0 \leq i \leq M$. Unlike the variation opeartors, the jump functions fail to be sublinear. However, we note the almost sub-additivity of the jump functions: $$\label{equation:jumpfunction_sublinearity}
{N}_{\lambda}(T_{m}[f+g](x) : {m}\in \mathcal{R}) 
\leq 
{N}_{\lambda/2}(T_{m}f(x) : {m}\in \mathcal{R}) + {N}_{\lambda/2}(T_{m}g(x) : {m}\in \mathcal{R})$$

The variation operators are connected to the jump functions by the inequality: $${N}_\lambda(T_{m}f(x) : {m}\in \mathcal{R}) 
\leq 
4 \lambda^{-r} [{V_r}(T_{m}f(x) : {m}\in \mathcal{R})]^r$$ for each $r \geq 1$. Surprisingly this can be reversed on average in $L^p({\mathbb{R} }^{d})$ for $1 \leq p < \infty$ when $r>2$. In practice the $L^p$ boundedness of $V_2$ often fails. However the jump function $ \lambda \sqrt{N_\lambda} $ may still be bounded in which case it acts as a surrogate ‘endpoint’ operator for $V_2$. See for instance [@JSW]. Moreover the variation operators are related to the maximal functions by $$V_\infty(T_{m}f(x) : {m}\in \mathcal{R})
= 
2{M}(T_{m}f(x) : {m}\in \mathcal{R})
\leq 
2\left[ V_\infty(T_{m}f(x) : {m}\in \mathcal{R}) + T_{{m}_0} f(x) \right]$$ for any ${m}_0 \in \mathcal{R}$. On the one hand, ${V_r}f(x)$ increases as $r$ decreases so that its $L^p$-boundedness becomes more difficult to prove. On the other hand, the jump inequalities and variational estimates give quantitative versions of pointwise ergodic theorems. For a more thorough discussion of variations and jump functions, see [@Bourgain_pointwise; @PX; @JSW; @MST].

Now we can precisely state our first theorem which is Moon’s theorem for variations and jump functions.

\[theorem:Moon\_variations\] Suppose that $T_{m}$ is a sequence of convolution operators given by $T_{m}f := f {*}g_{m}$ for a sequence of functions $g_{m}\in L^1({\mathbb{R} }^{d})$. Fix the variation exponent to be any $r \in [1,\infty]$. For any $q \geq 1$, ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is restricted weak-type $(1,q)$ with norm $C$ if and only if ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is weak-type $(1,q)$ with norm $C$. Moreover, $\lambda \sqrt[r]{{N}_{\lambda}}$is restricted weak-type $(1,q)$ if and only if $\lambda \sqrt[r]{{N}_{\lambda}}$is weak-type $(1,q)$.

We also prove the Carrillo–de Guzmán version of Theorem \[theorem:Moon\_variations\].

\[theorem:CdG\_variations\] Suppose that $T_{m}$ is a sequence of convolution operators given by $T_{m}f := f {*}g_{m}$ for a sequence of finite measures $g_{m}\in L^1({\mathbb{R} }^{d})$. Fix the variation exponent to be any $r \in [1,\infty]$. If $p \geq 1$ and ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is pointed weak-type $(p,p)$ with norm $C$, then ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is strong-type $(p,p)$ with norm at most $C$. Moreover the same is true for the jump functions $\lambda \sqrt[r]{{N}_{\lambda}}$.

We can extend Theorem \[theorem:Moon\_variations\] to a slightly more general set-up. In addition to working with convolutions of $L^1$ functions, we will work with convolutions of smoothing, possibly singular, measures. This extension first appeared for the maximal function of lacunary dilates of a smoothing measure in unpublished work of Seeger–Tao–Wright connected with [@STW_Radon; @STW_pointwise_lacunary]. Inspired by the set-up of [@SW:lacunary_problems], we use a weak version of condition (2) of Seeger–Wright’s Theorem 1.1 in [@SW:lacunary_problems]. Let $\mu_{m}$ be a sequence of finite measures of bounded variation and $T_{m}$ denote convolution with $\mu_{m}$. Assume that for some fixed $p \geq 1$ we have $$\label{eq:smoothing_measures}
\sup_{m}\|T_{m}\circ P_{>k}\|_{p \to p} 
= 
o(1) 
\text{ as } k \to \infty
.$$ Here, and throughout, $P_k$ denotes a smooth Littlewood–Paley ‘projection’ operator adapted to frequency band of frequency size $2^k$. To be precise, let ${{\bf 1}_{[-1,1]}} \leq \phi \leq {{\bf 1}_{[2,2]}}$ be a smooth function on ${\mathbb{R} }$. Define by the multiplier $\hat{P_k}(\xi) = \phi(|\xi|) - \phi(2|\xi|)$. Then for a function $f : {\mathbb{R} }^{d}\to {\mathbb{C} }$, $\widehat{P_k f} := \widehat{P_k} \cdot \widehat{f}$ the Fourier transform of $P_k f$ has support in $\{ |\xi| \in [2^{k-1},2^{k+1}] \}$ while $\sum_{k \in {\mathbb{Z} }} \phi(|\xi|) - \phi(2|\xi|) \equiv 1$ for $\xi \in {\mathbb{R} }^{d}$ so that $\sum_{k \in {\mathbb{Z} }} P_k f = f$ in many senses. We write $P_{\leq k} f = \sum_{j \leq k} P_j f$ and $P_{> k} f = \sum_{j > k} P_j f$. As a motivating example one may consider the lacunary spherical averages given by the measures $\mu_{m}:= \sigma_{2^{m}}$ for ${m}\in {\mathbb{N} }$ where $\sigma_{r}$ is the spherical measure on a sphere of radius $r>0$ normalized to have mass 1. It is known that $\|P_k \mu_r\|_{L^2({\mathbb{R} }^{d})} \lesssim (1+r2^{-k})^{\frac{1-{d}}{2}}$ for ${d}\geq 2$. We have the following ‘smoothing’ version of Moon’s theorem and Theorem \[theorem:Moon\_variations\].

\[theorem:Moon\_smoothing\] Suppose that $T_{m}$ is a sequence of convolution operators given by $T_{m}f := f {*}\mu_{m}$ for a sequence of finite measures $\mu_{m}$ each of bounded total variation satisfying the smoothing property . Fix the variation exponent to be any $r \in [1,\infty]$. For any $q \geq 1$, ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is restricted weak-type $(1,q)$ with norm $C$ if and only if ${V_r}(T_{m}f : {m}\in {\mathbb{N} })$ is weak-type $(1,q)$ with norm at most $C$. Moreover, $\lambda \sqrt[r]{{N}_{\lambda}}$is restricted weak-type $(1,q)$ if and only if $\lambda \sqrt[r]{{N}_{\lambda}}$is weak-type $(1,q)$.

We close the introduction with a little bit of notation that will be useful in the proof of our theorems. First, $f(x) \lesssim g(x)$ if there exists a constant $f(x) \leq C g(x)$ for some implicit constant $C>0$. Second, for a subset $F \subset {\mathbb{R} }^{d}$, let ${{\bf 1}_{F}}$ denote the indicator or characteristic function of $F$.

Acknowledgements {#acknowledgements .unnumbered}
================

I thank Jim Wright for informing me of the extension of Moon’s theorem to smoothing mesaures, as well as for insightful conversations. I also thank Ben Krause, Mariusz Mirek and Bartosz Trojan for discussions concerning variational operators and jump function inequalities.

Moon’s theorem for variations
=============================

The proof of Moon’s theorem hinges on how to approximate simple functions. The following proposition is implicit in [@Moon]. It says that the set $I_\epsilon$ approximates $f$ very well, in the sense that it has the same size as $f$ and it is close to the convolution of $f$ with a prescribed finite sequence of smooth functions. Since we will use it in Theorem \[theorem:Moon\_variations\], we include its proof for completeness.

\[proposition:Moon\_approximation\] For a sequence ${\left\{ h_{m}\right\}}_{1 \leq {m}\leq {M}}$ of $C^1({\mathbb{R} }^{d})$ functions, if $f$ is a simple function on ${\mathbb{R} }^{d}$, then for any $\epsilon>0$, there exists a set $I_\epsilon \subseteq supp(f)$ such that

1.  ${\left\| f \right\|_{L^{\infty}}} {\left \lvert I_\epsilon \right \rvert} = {\left\| f \right\|_{L^{1}}}$,

2.  ${\left \lvert f {*}h_{m}(x) - ({\left\| f \right\|_{L^{\infty}}} {{\bf 1}_{I_\epsilon}}) {*}h_{m}(x) \right \rvert} < {\left\| f \right\|_{L^{1}}} \epsilon$ for each $1 \leq {m}\leq {M}$ and all $x \in {\mathbb{R} }^{d}$.

By the scaling homogeneity of the problem we may normalize ${\left\| f \right\|_{L^{\infty}}}=1$. Let $f = \sum_{k=1}^K a_k {{\bf 1}_{F_k}}$ be a simple function with coefficients $a_k \in {\mathbb{R} }$ and $F_k \subset {\mathbb{R} }^{d}$ of finite Lebesgue measure. We may assume that the $F_k$ are open balls with diameter at most $\delta>0$ a small parameter that we will optimize later. Let $I_k$ be *any* open ball in $F_k$ such that ${\left \lvert I_k \right \rvert} = a_k {\left \lvert F_k \right \rvert}$. Now set $I = \cup_{k} I_k$ so that ${\left\| f \right\|_{L^{\infty}}} |I| = |I|$. We want to show that the difference between $f$ and ${{\bf 1}_{I}} = {\left\| f \right\|_{L^{\infty}}} {{\bf 1}_{I}}$ is small. First note that $$\begin{aligned}
f {*}h_{m}(x) 
& = 
\int_{{\mathbb{R} }^{d}} \sum_k a_k {{\bf 1}_{F_k}}(y) h_{m}(x-y) \; dy 
= \sum_k a_k \int_{F_k} h_{m}(x-y) \; dy 
\\ & = 
\sum_k a_k {\left \lvert F_k \right \rvert} h_{m}(x-y_k) 
= \sum_k {\left\| f \right\|_{L^{\infty}}} {\left \lvert I_k \right \rvert} h_{m}(x-y_k) 
\\ & = 
\sum_k {\left \lvert I_k \right \rvert} h_{m}(x-y_k) \end{aligned}$$ for some $y_k \in F_k$ since the $h_{m}$ are smooth by the Mean Value Theorem. Similarly since $I_k \subset F_k$, we can write $$\begin{aligned}
{{\bf 1}_{I}} {*}h_{m}(x) 
& = 
\int_{{\mathbb{R} }^{d}} \sum_k {{\bf 1}_{I_k}}(y) h_{m}(x-y) \; dy 
\\ 
& = 
\sum_k \int_{I_k} h_{m}(x-y) \; dy 
\\ 
& = 
\sum_k {\left \lvert I_k \right \rvert} h_{m}(x-y_k') \end{aligned}$$ for some $y_k' \in I_k$. Therefore we have the pointwise estimate $$\begin{aligned}
{\left \lvert f {*}h_{m}(x) - {{\bf 1}_{I}} {*}h_{m}(x) \right \rvert} 
& = 
{\left \lvert \sum_k {\left \lvert I_k \right \rvert} h_{m}(x-y_k) - \sum_k {\left \lvert I_k \right \rvert} h_{m}(x-y_k') \; dy \right \rvert} 
\\ & \leq 
\sum_k {\left \lvert I_k \right \rvert} \cdot {\left \lvert h_{m}(x-y_k) - h_{m}(x-y_k') \right \rvert} 
.\end{aligned}$$ Since the functions $h_{m}$ are smooth and ${M}$ is finite, we can choose $\delta$ small enough so that ${\left \lvert h_{m}(x-y_k) - h_{m}(x-y_k') \right \rvert} < \epsilon$ for each $1 \leq {m}\leq {M}$. Take $I_\epsilon$ to be $I$ to conclude the proof.

With Proposition \[proposition:Moon\_approximation\] in hand, it is an easy matter to prove Theorem \[theorem:Moon\_variations\].

Weak-type obviously implies restricted weak-type so we only prove that restricted weak-type implies weak-type. Fix $q \geq 1$. We make two quick reductions. By Monotone Convergence, reduce to the truncated variation operator ${V_r}(f*g_{m}(x) : {m}\in [{M}])$ where the supremum is over all finite, increasing subsequences of $[{M}] := {\left\{ 1, \dots, {M}\right\}}$ as long as our bounds at the end are independent of ${M}$. Normally one would also reduce to simple functions, however we cannot do this since we do not yet know that the variation operator is continuous. Assume for now that $f$ is a simple function. (We will remove this restriction at the end of the argument.) By dilational symmetry of $L^1({\mathbb{R} }^{d})$, normalize our simple function so that ${\left\| f \right\|_{L^{\infty}}}=1$.

Let $\epsilon>0$. Our first step is to approximate $g_{m}\in L^1({\mathbb{R} }^{d})$ by smooth $h_{m}\in L^1({\mathbb{R} }^{d})$. We can do this so that ${\left\| g_{m}-h_{m}\right\|_{L^{1}}} < \epsilon$. Note that for each $x \in {\mathbb{R} }^{d}$ $$\label{eq:measure_approximation}
{\left \lvert f {*}(g_{m}-h_{m})(x) \right \rvert} 
\leq 
{\left\| f \right\|_{L^{\infty}}} {\left\| g_{m}-h_{m}\right\|_{L^{1}}}
< 
\epsilon 
.$$ Sublinearity of the variation operators implies $$\begin{aligned}
\label{equation:variation_sublinearity}
{V_r}(f*g_{m}(x) : {m}\in [{M}]) 
& \leq 
{V_r}(f*[g_{m}-h_{m}](x) : {m}\in [{M}])  + {V_r}(f*h_{m}(x) : {m}\in [{M}]) 
.\end{aligned}$$ The union bound then implies the pointwise bound $$\begin{aligned}
{V_r}(f*g_{m}(x) : {m}\in [{M}]) 
& < 
{M}^{1/r} \cdot \epsilon + {V_r}(f*h_{m}(x) : {m}\in [{M}]) 
.\end{aligned}$$

Apply Proposition \[proposition:Moon\_approximation\] to find a subset $I_\epsilon \subset supp(f)$ such that ${\left \lvert I_\epsilon \right \rvert} = {\left\| f \right\|_{L^{1}}}$ and satisfying the inequality ${\left \lvert f {*}g_{m}(x) - {{\bf 1}_{I_\epsilon}} {*}g_{m}(x) \right \rvert} < \epsilon$ simultaneously for each ${m}\in [{M}]$ and every $x \in {\mathbb{R} }^{d}$. This latter condition implies that for any ${m}_1, {m}_2 \in [{M}]$ and $x \in {\mathbb{R} }^{d}$, $${\left \lvert (f-{{\bf 1}_{I_\epsilon}}) {*}h_{{m}_1}(x) - (f-{{\bf 1}_{I_\epsilon}}) {*}h_{{m}_2}(x) \right \rvert} 
< 
2 \epsilon 
.$$ Therefore, $${\left\| {V_r}([f-{{\bf 1}_{I_\epsilon}}]*h_{m}: {m}\in [{M}])  \right\|_{L^{\infty}}} 
< 
{M}^{1/r} \cdot 2 \epsilon 
.$$

Since ${V_r}$ is sublinear, we have $${V_r}(f*h_{m}(x) : {m}\in [{M}])
\leq 
{V_r}([f-{{\bf 1}_{I_\epsilon}}]*h_{m}(x) : {m}\in [{M}]) 
    + {V_r}({{\bf 1}_{I_\epsilon}}*h_{m}(x) : {m}\in [{M}]) 
.$$ Choosing $\epsilon = \lambda/100 {M}^{1/r}$, we see that $$\begin{aligned}
{\left \lvert {\left\{ {V_r}(f*g_{m}(x) : {m}\in [{M}]) > \lambda \right\}} \right \rvert} 
& 
\leq 
{\left \lvert {\left\{ {V_r}(f*[g_{m}-h_{m}](x) : {m}\in [{M}]) > \lambda/3 \right\}} \right \rvert} 
\\ & \hspace{10mm} + {\left \lvert {\left\{ {V_r}([f-{{\bf 1}_{I_\epsilon}}]*h_{m}(x) : {m}\in [{M}]) > \lambda/3 \right\}} \right \rvert} 
\\ & \hspace{10mm} + {\left \lvert {\left\{ {V_r}({{\bf 1}_{I_\epsilon}}*h_{m}(x) : {m}\in [{M}]) > \lambda/3 \right\}} \right \rvert} 
\\ & 
= 
{\left \lvert {\left\{ {V_r}({{\bf 1}_{I_\epsilon}} : \{T_{m}\}_{{m}\in [{M}]}) > \lambda/3 \right\}} \right \rvert} 
.\end{aligned}$$ Applying our hypothesis that the variation is restricted weak-type $(1,q)$, we have $${\left \lvert {\left\{ {V_r}(f*g_{m}(x) : {m}\in [{M}]) > \lambda \right\}} \right \rvert} 
\leq 
C (\lambda/3)^{-q} {\left \lvert I_\epsilon \right \rvert} 
= 
3^q C \lambda^{-q} {\left\| f \right\|_{L^{1}}}
.$$ Choosing $\epsilon$ smaller, we see that we can replace $3^qC$ in the above bound by $C$.

We extend our estimates to $f$ in $L^1({\mathbb{R} }^{d})$. Find a simple function $ \sum_{k=1}^K a_k {{\bf 1}_{F_k}} $ for subsets $F_k$ of ${\mathbb{R} }^{d}$ of finite Lebesgue measure such that $ \|f - \sum_{k=1}^K a_k {{\bf 1}_{F_k}} \|_{L^1({\mathbb{R} }^{d})} < \epsilon $ for $\epsilon = \lambda/10 {M}^{1/r}>0$. Then $${\left\| {V_r}(f-\sum_{k=1}^K a_k {{\bf 1}_{F_k}}) \right\|_{L^{\infty}}} 
< 
{M}^{1/r} \cdot 2 \epsilon 
<
\lambda/3$$ and $$\begin{aligned}
{\left \lvert {\left\{ {V_r}(f : \{T_{m}\}_{{m}\in [{M}]}) > \lambda \right\}} \right \rvert} 
& 
\leq 
{\left \lvert {\left\{ {V_r}(f-\sum_{k=1}^K a_k {{\bf 1}_{F_k}} : \{T_{m}\}_{{m}\in [{M}]}) > \lambda/2 \right\}} \right \rvert} 
\\ 
& \hspace{10mm}
+ {\left \lvert {\left\{ {V_r}(\sum_{k=1}^K a_k {{\bf 1}_{F_k}} : \{T_{m}\}_{{m}\in [{M}]}) > \lambda/2 \right\}} \right \rvert} 
\\ & 
= 
{\left \lvert {\left\{ {V_r}(\sum_{k=1}^K a_k {{\bf 1}_{F_k}} : \{T_{m}\}_{{m}\in [{M}]}) > \lambda/2 \right\}} \right \rvert} 
\\ & 
\leq 
C (\lambda/2)^{-q} \sum_{k=1}^K |a_k| {\left \lvert F_k \right \rvert} 
\\ & 
\leq  
2^qC \lambda^{-q} (1+\epsilon) {\left\| f \right\|_{L^{1}}}
.\end{aligned}$$ Once again taking $\epsilon \to 0^+$ in our approximation, we may drop the factor of $2^q$.

The proof for jump inequalities is essentially similar but replaces with . Unfortunately we lose a small power of 2 in this inequality and thus cannot conclude that the norm bounds are the same.

Our strategy for the proof of Theorem \[theorem:Moon\_smoothing\] is to take $h_{m}$ to be $P_{\leq k} \mu_{m}$ for some large k as an approximation to $\mu_{m}$ and bound the rest as error. We assumed that $\mu_{m}$ is a finite measure of bounded total variation so that $P_k \mu_{m}$, which is the convolution of $\mu_{m}$ with a Schwartz function, is well-defined, and $\| P_{\leq k} \mu_{m}\|_{p} \lesssim_k \|\mu_{m}\|_{TV}$ where $\|\mu_{m}\|_{TV}$ denotes the total variation of $\mu_{m}$. We remark that the implicit bound is not uniform in $k$; this presents a minor technicality.

Reduce to the truncated variation opeartor ${V_r}(f*\mu_{m}: {m}\in [{M}])$ for large ${M}\in {\mathbb{N} }$ as before. For the moment choose $f$ to be a simple function normalized so that $\|f\|_\infty = 1$. Let $\lambda > 0$ and choose $$\epsilon 
= 
\min \{ \lambda^{p-q} \|f\|_1 /{M}^2 \|f\|_p , \lambda/10M, \lambda/4\|f\|_1 \}.$$ Choose $k$ sufficiently large so that our assumption , implies that $$\| f * P_{>k}\mu_{m}\|_{L^p} 
< 
\epsilon \| f \|_{L^p}$$ uniformly in ${m}$. Apply Proposition \[proposition:Moon\_approximation\] with $g_{m}:= P_{\leq k}\mu_{m}$ to find a subset $I_\epsilon$ satisfying the conclusions of Proposition \[proposition:Moon\_approximation\].

The decomposition, $$\begin{aligned}
f*\mu_{{m}} 
= 
f*P_{>k}\mu_{{m}} 
+ [f-{{\bf 1}_{I_\epsilon}}]*P_{\leq k}\mu_{{m}} 
+ {{\bf 1}_{I_\epsilon}}*\mu_{{m}} - {{\bf 1}_{I_\epsilon}}*P_{>k}\mu_{{m}} 
,\end{aligned}$$ implies $$\begin{aligned}
|\{ {V_r}(f*\mu_{{m}} : {{m}\in {M}}) > \lambda \}| 
& \leq 
|\{ {V_r}(f*P_{>k}\mu_{{m}} : {{m}\in {M}}) > \lambda/4 \}| 
\\ & \hspace{10mm} + |\{ {V_r}([f-{{\bf 1}_{I_\epsilon}}]*P_{\leq k}\mu_{{m}} : {{m}\in {M}}) > \lambda/4 \}| 
\\ & \hspace{10mm} + |\{ {V_r}({{\bf 1}_{I_\epsilon}}*P_{>k}\mu_{{m}}: {{m}\in {M}}) > \lambda/4 \}| 
\\ & \hspace{10mm} + |\{ {V_r}({{\bf 1}_{I_\epsilon}}*\mu_{{m}}: {{m}\in {M}}) > \lambda/4 \}| 
.\end{aligned}$$ The union bound implies that $$\begin{aligned}
|\{ {V_r}(f*P_{>k}\mu_{m}: {{m}\in {M}}) > \lambda \}| 
& \lesssim 
{M}^{1/r} \sup_{{m}\leq {M}} |\{ |f {*}P_{>k} \mu_{m}| > \lambda/{M}\}| 
\\
& \lesssim 
{M}^{1+1/r} \lambda^{-p} \epsilon \|f\|_{L^p}
\\
& < 
\lambda^{-q} \|f\|_1
.\end{aligned}$$ The same holds true for ${{\bf 1}_{I_\epsilon}}$ so that $$\begin{aligned}
|\{ {V_r}({{\bf 1}_{I_\epsilon}}*P_{>k}\mu_{m}: {{m}\in {M}}) > \lambda \}| 
\lesssim 
\lambda^{-q} \|{{\bf 1}_{I_\epsilon}}\|_1 
= 
\lambda^{-q} \|f\|_1 
.\end{aligned}$$ From our assumption on $I_\epsilon$ in the conclusion of Proposition \[proposition:Moon\_approximation\], we also have that $$\begin{aligned}
\{ {V_r}([f-{{\bf 1}_{I_\epsilon}}]*P_{\leq k}\mu_{{m}} : {{m}\in {M}}) > \lambda/4 \} 
= \emptyset
.\end{aligned}$$ Combining these estimates we have $$\begin{aligned}
|\{ {V_r}(f*\mu_{{m}} : {{m}\in {M}}) > \lambda \}| 
& \lesssim 
|\{ {V_r}({{\bf 1}_{I_\epsilon}}*\mu_{{m}}: {{m}\in {M}}) > \lambda/4 \}| 
+ \lambda^{-q} \|f\|_1
.\end{aligned}$$ Apply our restricted weak-type hypothesis to conclude the theorem for simple functions.

To extend to all $f$ in $L^1$ (rather than $f$ in $L^1 \cap L^p$), repeat the argument at the end of the proof of Theorem \[theorem:Moon\_variations\].

Carrillo–de Guzmán’s Theorem for variational operators
======================================================

The following proposition is the Carrillo–de Guzmán analogue of Proposition \[proposition:Moon\_approximation\].

\[proposition:CdG\_approximation\] Let ${\left\{ g_{m}\right\}}_{1 \leq {m}\leq {M}}$ be a sequence of uniformly continuous functions, and $f = \sum_{k=1}^K a_k {{\bf 1}_{F_k}}$ be a simple function on ${\mathbb{R} }^{d}$ with $F_k$ dyadic cubes from the standard dyadic mesh on ${\mathbb{R} }^{d}$. If $\epsilon>0$, then $f$ can be refined into a sum of dyadic cubes $f = \sum b_j {{\bf 1}_{Q_j}}$ where $Q_j$ is in some $F_k$, and for any points $y_j$ in the interior of $Q_j$, we have $$\label{CdG:pointwise_comparison}
{\left \lvert f {*}g_{m}(x) - \sum_j b_j |Q_j| g_{m}(x-y_j) \right \rvert} 
< 
2{\left\| f \right\|_{L^{1}}} \epsilon
,$$ for each $1 \leq {m}\leq {M}$ and all $x \in {\mathbb{R} }^{d}$.

This is very similar to Proposition \[proposition:Moon\_approximation\], but has the technical advantage of choosing $F_k$ dyadic to partition ${\mathbb{R} }^{d}$. We could instead use open balls with diameter $<\delta$ that partition ${\mathbb{R} }^{d}$. The main point of Proposition \[proposition:CdG\_approximation\] is the pointwise comparison .

Since each of the $g_{m}$ are uniformly continuous and there are finitely many of them, they are altogether uniformly continuous. This means that for any $\epsilon>0$, which we pick and fix now, if $|x-y|<\delta$, then $|g_{m}(x)-g_{m}(y)| < \epsilon$ simultaneously for all ${m}$. With this in mind, use the dyadic structure in ${\mathbb{R} }^{d}$ to approximate each $F_k$ by a disjoint union $\cup_{j} Q_{k,j}$ of dyadic cubes, each with length at most $\delta/\sqrt{{d}}$, so that $|F_k \setminus \cup_{j} Q_{k,j}| < \epsilon$. Then $${\left\| f - \sum_k a_k {{\bf 1}_{\cup_{j} Q_{k,j}}}  \right\|_{L^{1}}} 
< 
(\sum_{k=1}^K |a_k|) \epsilon 
= 
{\left\| f \right\|_{L^{1}}} \epsilon 
.$$ Partitioning and reordering the cubes and coefficients, we define $h = \sum_{j} b_j {{\bf 1}_{Q_j}} = \sum_k a_k {{\bf 1}_{\cup_{j} Q_{k,j}}}$.

Let $y_j$ be a point in the interior of $Q_j$. For each cube $Q_j$ and $x \in {\mathbb{R} }^{d}$, we have $$|{{\bf 1}_{Q_j}} {*}g_{m}(x) - |Q_j| g_{m}(x-y_k)| 
< 
|Q_j| \epsilon$$ by our imposed uniform continuity of $\{g_{m}\}_{1 \leq {m}\leq {M}}$. This implies for each $x \in {\mathbb{R} }^{d}$ $${\left \lvert h {*}g_{m}(x) - \sum_j b_j |Q_j| g_{m}(x-y_k) \right \rvert}
< 
\sum_j |b_j| |Q_j| \epsilon 
= 
{\left\| h \right\|_{L^{1}}} \epsilon 
< 
2{\left\| f \right\|_{L^{1}}} \epsilon 
.$$

The proof will be similar to that of Carrillo–de Guzmán’s theorem and Theorem \[theorem:Moon\_variations\]. We begin with several standard reductions which we outline. The first step is to reduce to the truncated variation function ${V_r}(f*\mu_{m}: {{m}\in [{M}]})$ for arbitrarily large but finite ${M}$. Our results will be independent of ${M}$ so this is fine. The second step is to boost to the same inequality with arbitrary positive coefficients $a_k > 0$: $$\label{eq:pointed_weak_type_inequality}
{\left \lvert {\left\{ x: {V_r}(\sum_{k} a_k g_{m}(x-x_k) : {m}\in [{M}]) > \lambda \right\}} \right \rvert} 
\lesssim {\left( \sum_{k} a_k^p \right)} \lambda^{-p} 
.$$ This step follows a standard method: First prove it for $a_k \in {\mathbb{Z} }$. Then extend to rational coefficients. Finish by taking limits to conclude it for real coefficients. The next step is to reduce to smooth $g_{m}\in L^1$ using as in the proof of Theorem \[theorem:Moon\_variations\]. At this point, we may now assume that for all $\epsilon>0$, there exists a $\delta>0$ depending on $\epsilon$ such that $ |g_{{m}_1}(x)-g_{{m}_2}(y)| < \epsilon $ for all $1 \leq {m}_1, {m}_2 \leq {M}$ and all $|x-y|<\delta$.

Proceed as in Moon’s theorem where the delta functions replace the sets $I_k$ by choosing the delta function at a point inside $F_k$. Let $\epsilon>0$ and choose $\delta$ so that the above holds. Suppose that $f := \sum_{k=1}^K a_k {{\bf 1}_{Q_k}}$ is a simple function where the $Q_k$ are dyadic cubes. Suppose further that all the dyadic cubes $Q_k$ have the same sidelength $\delta \leq 1$, then $$\|f\|_p 
= 
\left( \sum_{k=1}^K |a_k|^p \delta^{d}\right)^{1/p} 
\geq 
\left( \sum_{k=1}^K |a_k|^p \right)^{1/p} \delta^{d}=
\left( \sum_{k=1}^K |a_k Q_k|^p \right)^{1/p} 
.$$ For each $1 \leq k \leq K$, let $x_k$ be any point in the interior of $Q_k$. Note that $$\begin{aligned}
f*g_{m}(x)-\sum_{k=1}^K a_k |Q_k| g_{m}(x-x_k) 
& = 
\int \sum_{k=1}^K a_k {{\bf 1}_{Q_k}}(y) g_{m}(x-y)-\sum_{k=1}^K a_k |Q_k| g_{m}(x-x_k) 
\\
& = 
\sum_{k=1}^K a_k \int {{\bf 1}_{Q_k}}(y) g_{m}(x-y) - {{\bf 1}_{Q_k}}(y) g_{m}(x-x_k) \; dy 
\\
& \leq 
\sum_{k=1}^K a_k \int_{Q_k} |g_{m}(x-y) - g_{m}(x-x_k)| \; dy 
\\
& \leq 
\left( \sum_{k=1}^K a_k |Q_k| \right) \epsilon 
= 
\|f\|_1 \epsilon
.\end{aligned}$$ Then $$\begin{aligned}
{V_r}(f*g_{m}(x) : {m}\in [{M}]) 
& \leq 
{V_r}(f*g_{m}(x)-\sum_{k} a_k |Q_k| g_{m}(x-x_k) : {m}\in [{M}]) 
\\
& \hspace{10mm}
+ {V_r}(\sum_{k} a_k |Q_k| g_{m}(x-x_k) : {m}\in [{M}])
.\end{aligned}$$ Choosing $\epsilon < \frac{\lambda}{8 {M}^{1/r} \|f\|_1}$ we have that $$\begin{aligned}
|{\left\{ {V_r}(f*g_{m}(x) : {m}\in [{M}]) > \lambda \right\}}| 
& \leq 
|{\left\{ {V_r}(\sum_{k} a_k |Q_k| g_{m}(x-x_k) : {m}\in [{M}]) > \lambda/2 \right\}}|
\\
& \leq 
2^pC \cdot \left( \sum_{k=1}^K |a_k|^p |Q_k|^p \right)^{1/p} 
\\
& \leq 
2^pC \cdot \|f\|_p^p \lambda^{-p}
.\end{aligned}$$ Once again choosing $\epsilon>0$ smaller we may remove the extraneous power of $2$.

The final step is to extend from simple functions formed by the standard dyadic mesh on ${\mathbb{R} }^{d}$ to general functions in $L^p({\mathbb{R} }^{d})$ by adapting the argument at the end of the proof of Theorem \[theorem:Moon\_variations\].

The modifications for jump inequalities are like Theorems \[theorem:Moon\_variations\] and \[theorem:Moon\_smoothing\]. We leave the details to the reader.

<span style="font-variant:small-caps;">Heilbronn Institute for Mathematical Research School of Mathematics, University of Bristol, Howard House, Queens Avenue, Bristol BS8 1SN, United Kingdom</span>

*Email address:* `khughes.math@gmail.com`
