BV Continuity for the uncentered
Hardy–Littlewood maximal operator

The study of regularity properties for maximal operators has been an important topic of research over the last years. The most classical object in this context is the centered Hardy–Littlewood maximal operator, that for every $f\in L^{1}_{\text{loc}}(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$ is defined as

where $\mathchoice{\mathop{\vrule width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB}}{\mathop{\vrule width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B}}{\mathop{\vrule width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B}}{\mathop{\vrule width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B}}g:=\frac{\int_{B}g}{m(B)}$ and $m$ is the $d$–dimensional Lebesgue measure. The uncentered Hardy–Littlewood maximal operator, denoted by $\widetilde{M}$, is defined analogously by taking the supremum over balls that contain the point $x$ but that are not necessarily centered at $x$.

The program of studying the regularity of maximal functions was initiated by Kinnunen [15], who proved that the map

is bounded on $W^{1,p}(\mathbb{R}^{d})$ if $p>1.$ Since then, several authors have made many contributions in this direction, including connections with potential theory and partial differential equations (see, e.g., [4, 5, 7, 13, 16, 22, 23]).

An important direction has been the study of the continuity of the map (1.1) in several settings. Since it is not necessarily a sublinear map at the derivative level, its continuity does not follow from the boundedness. The first result answering a question of this kind was due to Luiro [18], who proved that the map (1.1) is in fact continuous from $W^{1,p}(\mathbb{R}^{d})$ to itself for each $p>1.$

This case is much more subtle. In fact, Kinnunen’s result certainly does not hold because for every non trivial $f\in L^{1}(\mathbb{R}^{d})$ we have $\widetilde{M}f\notin L^{1}(\mathbb{R}^{d}).$ Nevertheless, since we are interested at the derivative level of the maximal functions, the natural version of the problem at $p=1$ is to study the map

for $f\in W^{1,1}(\mathbb{R}^{d}).$ In fact, the boundedness of the map (1.2) from $W^{1,1}(\mathbb{R}^{d})$ to $L^{1}(\mathbb{R}^{d})$ is an important open problem for $d\geq 2.$ In dimension $d=1$ the answer is positive, as has been established for the uncentered version in [24] and for the centered version in [17]. In [19], Luiro proved that the map (1.2) is bounded when restricted to radial data in the uncentered setting. Other boundedness results in the radial setting have been achieved in [2, 3, 8, 14, 20].

An important work for our purposes is [1], where the boundedness

for every $f\in BV(\mathbb{R}),$ was proved. Moreover, the authors proved that $\widetilde{M}f$ is absolutely continuous for $f\in BV(\mathbb{R}),$ providing then an improvement over the original function.

In this article we are particularly interested in the continuity of the map (1.2) at the endpoint $p=1.$ We notice that the methods outlined by Luiro [18] cannot be adapted to our case, since they depend on the fact that the map

is bounded on $L^{p}(\mathbb{R}^{d})$ for $p>1$. In [12] the first continuity results at the endpoint were obtained. The authors proved, among other results, that the map

is continuous from $W^{1,1}(\mathbb{R})$ to $L^{1}(\mathbb{R})$ (cf. [12, Theorem 1]). This result was recently extended to the $W^{1,1}_{\rm rad}(\mathbb{R}^{d})$ setting (the subspace of $W^{1,1}(\mathbb{R}^{d})$ consisting of radial functions) in [9], where a more general approach, that can be applied in the context of other maximal functions, is proposed.

In [12] the authors also consider the space $BV(\mathbb{R})$ endowed with the norm $\|f\|_{BV}:=|f(-\infty)|+{\rm Var}(f).$ About this, they asked the following question:

This question, in case of being answered affirmatively, would provide a generalization of [12, Theorem 1] (since $W^{1,1}(\mathbb{R})$ embeds isometrically in $BV(\mathbb{R})$). It is important to notice that, in general, the continuity in the $BV(\mathbb{R})$ setting is more delicate that in the $W^{1,1}(\mathbb{R})$ setting. An example of this is that in the fractional setting the analogue of [12, Theorem 1] holds (see [21]) but the answer to the analogue of the previous question is negative (see [12, Theorem 3]).

The main goal of the present manuscript is to answer this question. We prove the following.

Several of the arguments in [12] (also the arguments in [9]) rely on the regularity of the original function, therefore they are not enough to conclude Theorem 1. In fact, the authors (also the authors of [9]) used in their work a reduction of the problem to the analogous question stated for “lateral” maximal operators, but in our case that reduction causes several problems. For instance, the one-sided maximal function of a function in $BV(\mathbb{R})$ is not necessarily continuous, which brings additional difficulties. Therefore, a different idea is required in order to achieve the result. Our methods, although inspired by [12], provide a new approach to this kind of problems.

Having taken into account the main result obtained here, we summarize the situation of the endpoint continuity program (originally proposed in [12, Table 1]) in the table below. The word YES in a box means that the continuity of the corresponding map has been proved, whereas the word NO means that it has been shown that it fails. The remaining boxes are marked as OPEN problems. We notice that after this work the only open problems in this program are the ones related to the centered Hardy–Littlewood maximal operator in a continuous setting.

In this subsection we develop the main tools required in our work. We start by stating the following result which describes the behavior of the maximal function at infinity.

The next goal is to use the $BV(\mathbb{R})$ norm to control the difference between two $BV(\mathbb{R})$ functions or between their maximal functions at a given point $x$. The following estimates, although very basic, will be extremely useful later on.

Contrasting with the $W^{1,1}(\mathbb{R})$ setting (see [12, Lemma 14]), in our context to make the reduction to the case $f\geq 0$ is much more problematic. In order to deal with this issue we require several results describing the relations between $f$ and $|f|$.

In the following, for given $g\in BV(\mathbb{R})$ we define $\underset{y\uparrow x}{\lim}\ g(y)=:g(x^{-})$ and $\underset{y\downarrow x}{\lim}\ g(y)=:g(x^{+})$. Also, for each $-\infty\leq a<b\leq\infty$ we introduce the quantity

where the supremum is taken over all $K\in\mathbb{N}$ and all sequences $a<a_{0}<\dots<a_{K}<b$ (notice that if $g$ is continuous at $a$ and $b$, then the sequences satisfying $a=a_{0}<\dots<a_{K}=b$ can be considered instead and the supremum will not change). For a given partition $\mathcal{P}=\{a_{0}<a_{1}<\dots<a_{K}\}$ we denote ${\rm Var}(g,\mathcal{P}):=\displaystyle\sum_{i=1}^{K}|g(a_{i})-g(a_{i-1})|$. Finally, we write $D_{l}(g):=\{x\in\mathbb{R};g(x)\neq g(x^{-})\}$ and $D_{r}(g):=\{x\in\mathbb{R};g(x)\neq g(x^{+})\}.$

Now, we use Lemma 4 to show that the map $f\mapsto{\rm Var}_{(a,b)}(|f|)$ is continuous from $BV(\mathbb{R})$ to $[0,\infty)$.

Let us now take a closer look at the properties of the maximal operator. Recall that the total variation of $\widetilde{M}f$ can be controlled by the total variation of $f$. There is also a local version of this principle, where we focus on an interval $(a,b)$. However in this case some boundary terms must be included. Thus, to avoid the possibility that $f$ behaves badly at $a$ or $b$, we use its adjusted version $\overline{|f|}$ defined by

It is known that $\overline{|f|}$ is upper semicontinuous and that $\overline{|f|}\leq\widetilde{M}f$ (see [1, Lemma 3.3]).

The next result gives us the uniform control (with respect to $j$) on the behavior of $\big{(}\widetilde{M}f_{j}\big{)}^{\prime}$ near infinity, provided that $\{f_{j}\}_{j\in\mathbb{N}}$ is a converging sequence in $BV(\mathbb{R})$. This, in turn, allows one to restrict the attention to a bounded interval, while dealing with the total variations of the maximal functions $\widetilde{M}f_{j}$. We point out that it is also possible to proceed without this reduction, but then for all considered functions the extended domain $[-\infty,\infty]$ should be used instead of $\mathbb{R}$.

Before we prove our key result regarding the variation of maximal functions, we need the following definition. A given partition $\mathcal{P}=\{a_{0}<a_{1}<\dots<a_{n}\},$ with $n\geq 2,$ has property (V) with respect to $f$ if for each $i\in\{0,1,\dots,n-2\},$ we have $\operatorname{\mathrm{sgn}}(f(a_{i+2})-f(a_{i+1}))\cdot\operatorname{\mathrm{sgn}}(f(a_{i+1})-f(a_{i}))<0$.