The centered maximal operator removes the non-concave Cantor part from the gradient
October 02, 2025
Abstract
We study regularity of the centered Hardy–Littlewood maximal function \({\mathrm M}f\) of a function \(f\) of bounded variation in \(\mathbb{R}^d\), \(d\in \mathbb{N}\). In particular, we show that at \(|{D}^{\mathrm{c}}{f}|\)-a.e.point \(x\) where \(f\) has a non-concave blow-up, it holds that \({\mathrm M}f(x)>f^*(x)\). We further deduce from this that if the variation measure of \(f\) has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of \({\mathrm M}f\) can be upgraded to Sobolev regularity.
1 Introduction↩︎
The so-called \(W^{1,1}\)-problem, posed in MR2041705?, asks whether the Hardy-Littlewood maximal function of a Sobolev function \(f\in W^{1,1}(\mathbb{R}^d)\), defined (for nonnegative \(f\)) by \[{\mathrm M}f(x) \mathrel{\vcenter{:}}= \sup_{r>0} \frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)}f(y)\mathop{}\!\mathrm{d}\mathcal{L}(y),\qquad x\in \mathbb{R}^d,\] is also locally in the class \(W^{1,1}(\mathbb{R}^d)\), with \[\Vert \nabla {\mathrm M}f\Vert_{L^1(\mathbb{R}^d)}\le C\Vert \nabla f\Vert_{L^1(\mathbb{R}^d)} .\] The problem arose as a natural extension of the case \(1<p<\infty\), where the analogous result is known to hold, as first shown by Kinnunen Kin?.
Many papers, for example APL?, AlPer09?, MR3695894?, MR2280193?, MR1898539?, weigt2024variation?, have considered this \(W^{1,1}\)-problem for the more general class of functions of bounded variation \(\mathrm{BV}(\mathbb{R}^d)\supset W^{1,1}(\mathbb{R}^d)\), and for different maximal operators, e.g. the uncentered maximal function, see zbMATH07215904? for a survey. In particular, Kurka Kur? gave a positive answer to the \(W^{1,1}\)-problem for BV functions in one dimension. Apart from BV or Sobolev regularity, one can also consider other regularity properties of the maximal function, such as approximate differentiability MR2550181? or quasicontinuity panubvsobolev?.
An addition to preserving regularity, APL? also addressed the question of whether maximal operators can even increase regularity. The authors proved that for \(f\in\mathrm{BV}(\mathbb{R})\), the gradient of its uncentered maximal function belongs to \(L^1(\mathbb{R})\), i.e.is not only a finite measure but an absolutely continuous one. For the centered maximal operator this does not hold: if a function has jumps, so can its centered maximal function. In general, the variation measure of a BV functions \(f\) can be divided into three parts: \[Df={D}^{\mathrm{a}}{f}+ {D}^{\mathrm{c}}{f}+ {D}^{\mathrm{j}}{f},\] see , . In Kur? Kurka also showed that if \(f\in W^{1,1}(\mathbb{R})\), i.e.\({D}^{\mathrm{c}}{f}={D}^{\mathrm{j}}{f}=0\), then also \(\nabla{\mathrm M}f\in L^1(\mathbb{R})\) i.e.\({D}^{\mathrm{c}}{{\mathrm M}f}={D}^{\mathrm{j}}{{\mathrm M}f}=0\). This leaves open the questions of whether the maximal function can have a nonzero Cantor part \({D}^{\mathrm{c}}{{\mathrm M}f}\). In addition, one can ask whether a nonzero jump part \({D}^{\mathrm{j}}{{\mathrm M}f}\) can be caused by anything else than a nonzero \({D}^{\mathrm{j}}{f}\). We address these two questions in the present manuscript.
In GoKo?, the authors consider the Cantor–Vitali function \(f\in \mathrm{BV}((0,1))\), which is the typical example of a function that does not have jumps but is not absolutely continuous either, in particular \(Df={D}^{\mathrm{c}}{f}\neq0\). More precisely, \(f\) is an increasing continuous function whose variation measure is \[Df=\mathcal{H}^{s}(C)^{-1}\mathcal{H}^{s}\,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!C,\] where \(s=\log 2/\log 3\) and \(C\) is the usual \(1/3\)-Cantor set. The authors show that \({\mathrm M}f(x)>f(x)\) for \(|Df|\)-a.e.\(x\in (0,1)\), and from this it follows that \({\mathrm M}f\) is absolutely continuous. More generally, in GoKo? the authors show absolute continuity of \({\mathrm M}f\) for any increasing continuous function \(f\in\mathrm{BV}_{\mathrm{loc}}(\mathbb{R})\) whose derivative is a finite sum of positive Radon measures \(\mu_i\) that are Ahlfors regular with dimension \(0<d_i<1\), i.e. \[\begin{align} \label{eq:gokoassumption} Df &= {D}^{\mathrm{c}}{f} = \sum_{j=1}^N \mu_i ,& \mu_i &\geq 0 ,& \mu_i(B(x,r)) &\sim r^{d_i} ,& 0<d_i &< 1 . \end{align}\tag{1}\]
In this manuscript we generalize this result substantially, see . We consider all dimensions \(d\ge 1\), and instead of Ahlfors regularity we consider the weaker pointwise condition that \(f\) has a non-concave blow-up \(|{D}^{\mathrm{c}}{f}|\)-almost everywhere, see . Then we show the following .
Theorem 1. Let \(f\in \mathrm{BV}(\mathbb{R}^d)\) and let \(R>0\). Let \(A\subset \mathbb{R}^d\) be the set of points where \(f\) has a non-concave blow-up. Then \[|{D}^{\mathrm{c}}{f}|(A\cap \{{\mathrm M}_{<R} f=f^*\})=0.\]
Here \({\mathrm M}_{<R} f\) denotes the maximal function where the radii in the supremum are limited to \(0<r< R\) and \(f^*\) the precise representative of \(f\), see . We always have \({\mathrm M}f\ge {\mathrm M}_{<R} f\ge f^*\). The expression \(\{{\mathrm M}_{<R} f=f^*\}\) is a shorthand for the set of all \(x\in\mathbb{R}^d\) with \({\mathrm M}_{<R}f(x)=f^*(x)\). We prove in .
Then in we consider Sobolev \(W^{1,1}_{\mathrm{loc}}\)-regularity of the maximal function. We let \(\Omega\) be an open subset of \(\mathbb{R}^d\), and then we understand \({\mathrm M}f\) to be the maximal function where the radii \(r\) are limited by the requirement \(B(x,r)\subset \Omega\).
Theorem 2. Suppose that \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\) and suppose that \({\mathrm M}f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\). Then \(|{D}^{\mathrm{j}}{{\mathrm M}f}|\le \tfrac 12 |{D}^{\mathrm{j}}{f}|\). If in addition \(f\) has a non-concave blow-up at \(|{D}^{\mathrm{c}}{f}|\)-a.e.point then \({D}^{\mathrm{c}}{{\mathrm M}f}=0\). If in addition \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\), then \({\mathrm M}f\) is ACL (absolutely continuous on lines) and in the class \(W_{\mathrm{loc}}^{1,1}(\Omega)\).
Here, we have to assume \(\mathrm{BV}_{\mathrm{loc}}\)-regularity of the maximal function, because it is not known except in one dimension due to Kur?.
By ACL we mean absolutely continuous on almost every line parallel to a coordinate axis; note, that \({\mathrm M}f\) being in \(W_{\mathrm{loc}}^{1,1}(\Omega)\) implies the ACL property for some pointwise representative of \({\mathrm M}f\), but by in fact \({\mathrm M}f\) itself already has this property.
Corollary 1. Let \(f\in L^1_\mathrm{loc}(\mathbb{R})\) with \(\mathop{\mathrm{var}}_\mathbb{R}f<\infty\). Then \(|{D}^{\mathrm{j}}{{\mathrm M}f}|\le \tfrac 12 |{D}^{\mathrm{j}}{f}|\). If in addition \(f\) has a non-concave blow-up at \(|{D}^{\mathrm{c}}{f}|\)-a.e.point then \({D}^{\mathrm{c}}{{\mathrm M}f}=0\). If in addition \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\), then \({\mathrm M}f\in W_{\mathrm{loc}}^{1,1}(\mathbb{R})\).
Remark 3. The bound \(|{D}^{\mathrm{j}}{{\mathrm M}f}|\le \tfrac 12 |{D}^{\mathrm{j}}{f}|\) is optimal: For example for \(f=1_{B(0,1)}\) we have \({D}^{\mathrm{j}}{{\mathrm M}f}=\tfrac 12{D}^{\mathrm{j}}{f}\).
In particular, generalizes the main result from GoKo? to more general functions and to all dimensions:
Corollary 2. Suppose \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\) and \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\) such that
at \(|{D}^{\mathrm{c}}{f}|\)-a.e.point, \(f\) has a blow-up with a singular part, or
\(f:\mathbb{R}\rightarrow\mathbb{R}\) and \(|{D}^{\mathrm{c}}{f}|\) is of the form of .
Moreover, suppose that \({\mathrm M}f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\). Then \({\mathrm M}f\) is ACL and in the class \(W_{\mathrm{loc}}^{1,1}(\Omega)\).
follows from the following .
Lemma 1. Suppose \(\gamma:(-1/2,1/2)\rightarrow\mathbb{R}\) is bounded and increasing such that \({D}^{\mathrm{s}}{\gamma}\) (the singular part of its derivative) is nonzero. Then \(\gamma\) is not concave.
Proof. For a contradiction suppose that \(\gamma\) is concave. Since \({D}^{\mathrm{s}}{\gamma}\) is nonzero, there exists a point \(x\in (-1/2,1/2)\) such that by , \[\lim_{r\to 0}\frac{\gamma(x+r)-\gamma(x-r)}{r} =\lim_{r\to 0}\frac{D\gamma((x-r,x+r))}{r}\\ =\infty.\] Consider \(s\in (-1/2,1/2)\) with \(s<x\), and \(r>0\) small enough \(s<x-r\). Then by concavity we have \[\gamma(s) \le \gamma(x-r)+(x-r-s)\frac{\gamma(x+r)-\gamma(x-r)}{2r} \to-\infty\] as \(r\searrow 0\). This contradicts the fact that \(\gamma\) is bounded. ◻
Proof of . Assume . Then implies the existence of a non-concave blow-up, and thus the conclusion follows from .
Assume . We can assume all the dimensions \(d_1,\ldots,d_N\) to be distinct. Then for \(|{D}^{\mathrm{s}}{f}|\)-a.e.\(x\in \mathbb{R}\), consider the smallest \(d_i\) such that \(x\) is in the support of \(\mu_i\). Then we have \[\lim_{r\to 0}\frac{Df(B(x,r))}{\mu_i(B(x,r))}=1.\] Then by a standard BV result, see , there exists a blow-up of \(f\) at \(x\), denoted by \(w\in \mathrm{BV}((-1/2,1/2))\), such that \(Dw\) is necessarily also Ahlfors \(d_i\)-regular. In particular, there exists a blow-up with a nonzero singular part, i.e. holds. ◻
Remark 4. Note that it is not necessarily the case that \(f\) has a non-concave blow-up at \(|{D}^{\mathrm{c}}{f}|\)-a.e.point. In MR890162?, Preiss gives an example of a measure on \(\mathbb{R}\) (which can always be taken to be \(Df\) for a BV function \(f\)) that is singular with respect to \(\mathcal{L}^1\), but at \(|Df|\)-a.e.point, all blow-ups of \(f\) are linear functions.
It remains open whether the conclusions of continue to hold for \(A=\mathbb{R}^d\) and without any assumptions on \({D}^{\mathrm{c}}{f}\) respectively.
1.0.0.1 Acknowledgments.
The authors wish to thank Khanh Nguyen for pointing out some useful references.
2 Preliminaries↩︎
In this we will follow the monograph AFP?. The results discussed here are standard, and we only give references for some key results.
2.1 Radon measures↩︎
The Borel \(\sigma\)-algebra on a set \(H\subset \mathbb{R}^d\) is denoted by \(\mathcal{B}(H)\). We always work with the Euclidean norm \(|\cdot|\) for vectors \(v\in \mathbb{R}^d\) as well as for matrices \(A\in \mathbb{R}^{k\times d}\), \(k\ge 1\).
Let \(x\in\mathbb{R}^d\) and \(0<r<\infty\). We denote by \(B(x,r)\) the open ball in \(\mathbb{R}^d\) with center \(x\) and radius \(r\). We denote the open ball in \(\mathbb{R}^{d-1}\) with center \(z\in \mathbb{R}^{d-1}\) and radius \(r\) by \(B_{d-1}(z,r)\). We denote a cylinder centered in \(x\), parallel to \(e_d\) and with length and radius \(r\) by \[\label{eq:cylinder} C(x,r) \mathrel{\vcenter{:}}= B_{d-1}((x_1,\ldots,x_{d-1}),r)\times (x_d-r/2,x_d+r/2) .\tag{2}\] Given a unit vector \(v\in\mathbb{S}^{d-1}\), we denote by \(C_v(x,r)\) the corresponding cylinder oriented in the direction \(\nu\).
Let \(\Omega\subset\mathbb{R}^d\) be an open set and \(\ell \in\mathbb{N}\). We denote by \(\mathcal{M}(\Omega;\mathbb{R}^{\ell})\) the Banach space of \(\ell\)-vector-valued Radon measures. For \(\mu \in \mathcal{M}(\Omega;\mathbb{R}^{\ell})\), the total variation \(|\mu|(\Omega)\) is defined with respect to the Euclidean norm on \(\mathbb{R}^{\ell}\). We further denote the set of positive measures by \(\mathcal{M}^+(\Omega)\).
For a vector-valued Radon measure \(\nu\in\mathcal{M}(\Omega;\mathbb{R}^{\ell})\) and a positive Radon measure \(\mu\in\mathcal{M}^+(\Omega)\), we have the Lebesgue–Radon–Nikodym decomposition \[\nu={\nu}^{\mathrm{a}}+{\nu}^{\mathrm{s}}=\frac{d\nu}{d\mu}\mathop{}\!\mathrm{d}\mu+{\nu}^{\mathrm{s}}\] of \(\nu\) with respect to \(\mu\), where \(\frac{d\nu}{d\mu}\in L^1(\Omega,\mu;\mathbb{R}^{\ell})\).
For open sets \(E\subset \mathbb{R}^{d-1}\), \(F\subset \mathbb{R}\), a parametrized measure \((\nu_{z})_{y \in E}\) is a map from \(E\) to the set \(\mathcal{M}(F;\mathbb{R}^{\ell})\) of vector-valued Radon measures on \(F\). For \(\mu\in\mathcal{M}^+(E)\) it is said to be weakly* \(\mu\)-measurable if \(z\mapsto \nu_{z}(B)\) is \(\mu\)-measurable for all Borel sets \(B\in\mathcal{B}(F)\). Suppose, that we additionally have \(\int_{E}|\nu_{z}|(F)\mathop{}\!\mathrm{d}\mu(z)<\infty.\) For a set \(A \subset E\times F\) we denote its fibers by \[A_z\mathrel{\vcenter{:}}= \{t\in F \colon (z,t) \in A\},\qquad z\in E .\] Then we define the generalized product measure \(\mu\otimes(z\mapsto\nu_z)\) by \[\label{eq:product32measure} \mu\otimes(z\mapsto\nu_z)(A)\mathrel{\vcenter{:}}= \int_E \nu_z(A_z)\mathop{}\!\mathrm{d}\mu(z) ,\tag{3}\] for all \(A\subset E\times F\) that belong to the \(\sigma\)-algebra on \(E\times F\) generated by the set of rectangles \(\mathcal{B}_{\mu}(E) \times \mathcal{B}(F)\), where \(\mathcal{B}_{\mu}(E)\) denotes the \(\mu\)-completion of \(\mathcal{B}(E)\).
The Lebesgue measure in \(\mathbb{R}^d\) is denoted by \(\mathcal{L}\), and the Lebesgue measure in \(\mathbb{R}^{d-1}\) is denoted by \(\mathcal{L}^{d-1}\). The Hausdorff measure of dimension \(0\le s\le d\) is denoted by \(\mathcal{H}^s\). When we integrate with respect to Lebesgue measure we may omit it in the notation and write \[\int f \mathrel{\vcenter{:}}= \int f\mathop{}\!\mathrm{d}\mathcal{L} = \int f(y)\mathop{}\!\mathrm{d}\mathcal{L}(y) = \int f(y)\mathop{}\!\mathrm{d}y .\]
The Sobolev \(1\)-capacity of a set \(A\subset \mathbb{R}^d\) is defined by \[\mathop{\mathrm{Cap}}_1(A)\mathrel{\vcenter{:}}=\inf\Bigl\{ \int_{\mathbb{R}^n}|u|+|\nabla u|\mathop{}\!\mathrm{d}\mathcal{L}: u\in W^{1,1}(\mathbb{R}^d),\; u\ge 1\text{ in a nbhd of }A \Bigr\} .\] By e.g.HaKi? we know that for any \(A\subset \mathbb{R}^d\), \[\label{eq:null32sets32of32capa32and32Hausdorff} \mathop{\mathrm{Cap}}_1(A)=0\quad\textrm{if and only if}\quad\mathcal{H}^{d-1}(A)=0.\tag{4}\]
We have a function and a measure theoretic notion of restrictions. Given a map \(f:X\rightarrow Y\) and \(A\subset X\) we define define the function theoretic restriction \(f|_{A}:A\rightarrow Y\) by \(f|_{A}(x)=f(x)\). For a measure \(\mu\) on \(X\) we define the measure theoretic restriction \(\mu \,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!A\) by \(\mu \,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!A(B)=\mu(A\cap B)\).
Note, that while the measure theoretic restriction is in some sense a zero extension to the original domain, our function theoretic restriction is not. This is important for example in the following .
Definition 1. Let \(\Omega\subset \mathbb{R}^d\) be open. A function \(v:\Omega\rightarrow[-\infty,\infty]\) is \(1\)-quasicontinuous if for every \(\varepsilon>0\) there exists an open set \(G\subset \Omega\) such that \(\mathop{\mathrm{Cap}}_1(G)<\varepsilon\) and \(v|_{\Omega\setminus G}\) is finite and continuous.
2.2 Functions of bounded variation↩︎
We denote by \(\mathrm{BV}(\Omega)\) the set of all functions \(f\in L_\mathrm{loc}^1(\Omega)\) of bounded variation, that is, whose distributional derivative is an \(\mathbb{R}^{d}\)-valued Radon measure with finite total variation. This means that there exists a (necessarily unique) Radon measure \(Df\in \mathcal{M}(\Omega;\mathbb{R}^{d})\) such that for all \(\varphi\in C_{\mathrm c}^1(\Omega)\), the integration-by-parts formula \[\int_{\Omega}f\frac{\partial\varphi}{\partial y_i}\mathop{}\!\mathrm{d}\mathcal{L} =-\int_{\Omega}\varphi\mathop{}\!\mathrm{d}(Df)_i,\qquad i=1,\ldots,d\] holds. Similarly to \(L^1_\mathrm{loc}\) we denote by \(\mathrm{BV}_\mathrm{loc}(\Omega)\) the set of all \(f\in L^1_\mathrm{loc}(\Omega)\) such that for each open \(G\) which is compactly contained in \(\Omega\) we have \(f|_{G}\in\mathrm{BV}(G)\).
If we do not know a priori that a function \(f\in L^1_{\mathrm{loc}}(\Omega)\) is a BV function, we consider \[\label{eq:definition32of32total32variation} \mathop{\mathrm{var}}_\Omega f \mathrel{\vcenter{:}}=\sup\left\{\int_{\Omega}f\cdot \nabla \varphi\mathop{}\!\mathrm{d}\mathcal{L},\,\varphi\in C_c^{1}(\Omega), \,|\varphi|\le 1\right\}.\tag{5}\] If \(\mathop{\mathrm{var}}_\Omega f <\infty\), then the Radon measure \(Df\) exists and \(\mathop{\mathrm{var}}_\Omega f=|Df|(\Omega)\) by the Riesz representation theorem.
For a measurable \(A\subset\mathbb{R}^d\) with \(0<\mathcal{L}(A)<\infty\) and \(f\in L^1(A)\), we denote the integral average by \[\label{eq:integralaverage} f_A = \fint_A f = \frac{1}{\mathcal{L}(A)} \int_A f .\tag{6}\] The precise representative of \(f\) is defined by \[\label{eq:preciserepresentative} f^*(x)\mathrel{\vcenter{:}}=\limsup_{r\to 0}f_{B(x,r)},\quad x\in \Omega.\tag{7}\] This is easily seen to be a Borel function. We say that \(x\in\Omega\) is a Lebesgue point of \(f\) if \[\lim_{r\to 0}\,\fint_{B(x,r)}|f(y)-f^*(x)|\mathop{}\!\mathrm{d}y=0 .\] We denote by \(S_f\subset\Omega\) the set where this condition fails and call it the approximate discontinuity set.
We say that \(x\in \Omega\) is an approximate jump point of \(f\) if there exists a \(\nu\in\mathbb{S}^{d-1}\) and distinct numbers \(f^+(x), f^-(x)\in\mathbb{R}\) such that \[\label{eq95approximatejumpdefinition} \lim_{r\to 0}\,\fint_{\{y\in B(x,r)\colon \langle y-x,\pm \nu\rangle>0\}}|f(y)-f^\pm(x)|\mathop{}\!\mathrm{d}y=0 .\tag{8}\] The set of all approximate jump points is denoted by \(J_f\).
The lower and upper approximate limits of a function \(f\in\mathrm{BV}_{\mathrm{loc}}(\Omega)\) are defined respectively by \[f^{\wedge}(x)\mathrel{\vcenter{:}}= \sup\left\{t\in\mathbb{R}\colon \lim_{r\to 0}\frac{\mathcal{L}(B(x,r))\cap\{f<t\})}{\mathcal{L}(B(x,r))}=0\right\}\] and \[f^{\vee}(x)\mathrel{\vcenter{:}}= \inf\left\{t\in\mathbb{R}\colon \lim_{r\to 0}\frac{\mathcal{L}(B(x,r)\cap\{f>t\})}{\mathcal{L}(B(x,r))}=0\right\},\] for all \(x\in\Omega\). We interpret the supremum and infimum of an empty set to be \(-\infty\) and \(\infty\), respectively. Note, that \[\label{eq:representatives32outside32jump32set} f^{*}(x)=f^{\wedge}(x)=f^{\vee}(x) \qquad\textrm{for }x\in \Omega\setminus S_f,\tag{9}\] and \[\label{eq:representatives32in32jump32set} f^{\wedge}(x)=\min\{f^{-}(x),f^+(x)\} , \qquad f^{\vee}(x)=\max\{f^{-}(x),f^+(x)\} \qquad\textrm{for } x\in J_f.\tag{10}\]
We write the Radon-Nikodym decomposition of the variation measure of \(f\) into the absolutely continuous and singular parts with respect to Lebesgue measure \(\mathcal{L}\) as \(Df={D}^{\mathrm{a}}{f}+{D}^{\mathrm{s}}{f}\). Furthermore, we define the Cantor and jump parts of \(Df\) by \[\begin{align} \label{eq:Dju32and32Dcu} {D}^{\mathrm{c}}{f} &\mathrel{\vcenter{:}}= {D}^{\mathrm{s}}{f}\,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!(\Omega\setminus S_f) ,& {D}^{\mathrm{j}}{f} &\mathrel{\vcenter{:}}= {D}^{\mathrm{s}}{f}\,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!J_f . \end{align}\tag{11}\] We have \[\label{eq:Sf32and32Jf} \mathcal{H}^{d-1}(S_f\setminus J_f)=0,\tag{12}\] and \(|Df|\) vanishes on \(\mathcal{H}^{d-1}\)-negligible sets, and so we have the decomposition (see AFP?) \[\label{eq:decomposition32of32variation32measure} Df={D}^{\mathrm{a}}{f}+ {D}^{\mathrm{c}}{f}+ {D}^{\mathrm{j}}{f}.\tag{13}\] From the fact that \(\mathcal{H}^{d-1}(S_f\setminus J_f)=0\), we also get \[\label{eq:f32star32and32vee} f^*(x)\le f^{\vee}(x)\quad\textrm{for }\mathcal{H}^{d-1}\textrm{-a.e.\;}x\in \Omega.\tag{14}\] For the jump part, we know \[\label{eq:jump32part32representation} |{D}^{\mathrm{j}}{f}|=|f^{+}-f^-|\mathcal{H}^{d-1}\,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!J_f.\tag{15}\]
2.3 One-dimensional sections of \(\mathrm{BV}\) functions↩︎
The following notation and results on one-dimensional sections of \(\mathrm{BV}\) functions are given in AFP?.
Let \(d=1\). Suppose \(f\in\mathrm{BV}_{\mathrm{loc}}(\mathbb{R})\). We have \(J_f=S_f\), \(J_f\) is at most countable, and \(Df (\{x\})=0\) for every \(x\in\mathbb{R}\setminus J_f\). For every \(x,y\in \mathbb{R}\setminus J_f\) with \(x<y\) we have \[\label{eq:fundamental32theorem32of32calculus32for32BV} f^*(y)-f^*(x)=Df((x,y)).\tag{16}\] For every \(x,y\in \mathbb{R}\) with \(x<y\) we have \[\label{eq:fundamental32theorem32of32calculus32for32BV322} |f^*(x)-f^*(y)|\le |Df|([x,y]),\tag{17}\] and the same with \(f^*\) replaced by \(f^{\wedge}\) or \(f^{\vee}\), or any pairing of these. We say that \[f^{\vee}(x)-f^{\wedge}(x)=|Df|(\{x\})\] is the jump size of \(f\) at point \(x\).
We denote by \(\pi:\mathbb{R}^d\rightarrow\mathbb{R}^{d-1}\) the projection given by \[\pi(x)\mathrel{\vcenter{:}}=(x_1,\ldots,,x_{d-1}) .\] We denote by \(e_1,\ldots,e_d\) the standard basis vectors in \(\mathbb{R}^d\). For a set \(A \subset \mathbb{R}^d\) we denote its fibers by \[A_z\mathrel{\vcenter{:}}= \{t\in\mathbb{R}\colon (z,t) \in A\},\qquad z\in \pi(A).\] For a map \(f\) defined on \(\Omega\) we denote \[f_z(t)\mathrel{\vcenter{:}}= f(z,t),\quad t\in \Omega_z,\;z\in \pi(\Omega).\] For \(f\in\mathrm{BV}(\Omega)\) we we have \(f_z\in \mathrm{BV}(\Omega_z)\) for \(\mathcal{L}^{d-1}\)-almost every \(z\in\pi(\Omega)\), see AFP?.
Recall . Denoting \(D_d f\mathrel{\vcenter{:}}= \langle Df,e_d\rangle\), we further have \[\begin{align} D_d f&=\mathcal{L}^{d-1}\otimes(z\mapsto D(f_z)) ,& {D}^{\mathrm{j}}_{d}f&=\mathcal{L}^{d-1}\otimes(z\mapsto {D}^{\mathrm{j}}{(f_z)}), \end{align}\] see AFP?. It follows that \[\begin{align} \label{eq:slice32representation32for32total32variation} |D_d f|&=\mathcal{L}^{d-1}\otimes(z\mapsto |D(f_z)|) ,& |{D}^{\mathrm{j}}_{d}f|&=\mathcal{L}^{d-1}\otimes(z\mapsto |{D}^{\mathrm{j}}{(f_z)}|), \end{align}\tag{18}\] see AFP?, and similarly \[\label{eq:sections32and32jump32sets323} |D^c_d f|=\mathcal{L}^{d-1}\otimes(z\mapsto |{D}^{\mathrm{c}}{(f_z)}|).\tag{19}\] Moreover, for \(\mathcal{L}^{d-1}\)-almost every \(z\in\pi(\Omega)\) we have \[\begin{align} \label{eq:sections32and32jump32sets} J_{f_{z}}&=(J_f)_{z} ,& (f^*)_{z}(t)&=(f_{z})^*(t)\;\;\textrm{for every }t\in \mathbb{R}\setminus J_{f_{z}}, \end{align}\tag{20}\] see AFP?. By AFP? we also know that for \(\mathcal{L}^{d-1}\)-almost every \(z\in\pi(\Omega)\), we have \[\label{eq:sections32and32jump32sets322} \{(f_z)^{-}(t),(f_z)^{+}(t)\}=\{(f^{-})_z(t),(f^{+})_z(t)\}\quad\textrm{for every }t\in (J_f)_z.\tag{21}\] Combining with , for \(\mathcal{L}^{d-1}\)-almost every \(z\in\pi(\Omega)\) and every \(t\in \Omega_z\) we have \[\begin{align} \label{eq:upper32and32lower32repr32sections} (f_z)^{\wedge}(t)&=(f^{\wedge})_z(t) ,& (f_z)^{\vee}(t)&=(f^{\vee})_z(t) ,& (f_z)^{*}(t)&=(f^{*})_z(t) . \end{align}\tag{22}\]
2.4 Blow-ups and nonconcavity↩︎
Let \(\Omega\subset \mathbb{R}^d\) be an open set and suppose \(f\in L^1_{\mathrm{loc}}(\Omega)\). Recall the definition of the integral average \(f_{B(x,r)}\) and for \(x\in\mathbb{R}^d\) and \(A\subset\mathbb{R}^d\) define \[\mathop{\mathrm{dist}}(x,A) \mathrel{\vcenter{:}}= \inf_{y\in A} |x-y| .\] We define the local maximal function \({\mathrm M}_\Omega f\) by \[\label{eq:HL32def} {\mathrm M}_\Omega f(x)\mathrel{\vcenter{:}}= \sup_{0<r<\mathop{\mathrm{dist}}(x,\mathbb{R}^d\setminus\Omega)} f_{B(x,r)} \qquad x\in \Omega.\tag{23}\] We usually drop \(\Omega\) from the notation, the exception being when instead of the (fixed) open set \(\Omega\) we consider the maximal function in a different set, e.g.a cylinder \(C(x,r)\). Note that in most of the literature one takes an absolute value of \(f\) in the integral; with our definition, this corresponds essentially to the special case of nonnegative functions. For \(R>0\) we define the auxiliary maximal functions \[\begin{align} \notag {\mathrm M}_{<R} f(x) &\mathrel{\vcenter{:}}= \sup_{0<r<\min\{R,\mathop{\mathrm{dist}}(x,\mathbb{R}^d\setminus\Omega)\}} f_{B(x,r)} ,\\ \label{eq:auxiliary32maximal32operator} {\mathrm M}_{\geq R} f(x) &\mathrel{\vcenter{:}}= \sup_{R\leq r<\mathop{\mathrm{dist}}(x,\mathbb{R}^d\setminus\Omega)} f_{B(x,r)} , \end{align}\tag{24}\] so that \({\mathrm M}f(x)=\max\{{\mathrm M}_{<R}f(x),{\mathrm M}_{\geq R}f(x)\}\). Note, that \({\mathrm M}_{\geq R}f(x)\) is only defined for those \(x\in\Omega\) with \(B(x,R)\subset\Omega\).
Definition 2. Let \(\Omega\subset \mathbb{R}^d\) be open and let \(f\in L^1_{\mathrm{loc}}(\Omega)\). We say that \(f\) is \(2\)-superharmonic if it is lower semicontinuous and \[f(x)\ge {\mathrm M}_{\Omega}f(x)\qquad \textrm{for all }x\in \Omega.\]
Definition 3. Let \(\Omega\subset \mathbb{R}^d\) be open and let \(f\in L^{1}_{\mathrm{loc}}(\Omega)\). We say that \(f\) is a \(2\)-supersolution if \[\int_{\Omega} f \Delta \varphi\le 0 \qquad \text{for all }\varphi\in C_c^{2}(\Omega).\]
For the following see e.g.Theorems 2.59 and 2.65 in the monograph MR1461542?.
Theorem 5. Let \(\Omega\subset \mathbb{R}^d\) be open and let \(f\in L^{1}_{\mathrm{loc}}(\Omega)\). If \(f\) is \(2\)-superharmonic, then it is a \(2\)-supersolution. Conversely, if \(f\) is a \(2\)-supersolution, then there exists a function \(\widetilde{f}=f\) a.e. which is \(2\)-superharmonic.
We have the following quasi-semicontinuity result from CDLP?. Alternatively, see LaSh? and L-SA? for a proof of in more general metric spaces.
Theorem 6. Let \(\Omega\subset \mathbb{R}^d\) be open, let \(f\in\mathrm{BV}_{\mathrm{loc}}(\Omega)\) and let \(\varepsilon>0\). Then there exists an open set \(G\subset\Omega\) such that \(\mathop{\mathrm{Cap}}_1(G)<\varepsilon\) and the map \(f^{\wedge}|_{\Omega\setminus G}\) is finite and lower semicontinuous, and \(f^{\vee}|_{\Omega\setminus G}\) is finite and upper semicontinuous.
Lemma 2. Let \(A\subset \mathbb{R}^d\). Then \[2\mathcal{H}^{d-1}(\pi(A))\le \mathop{\mathrm{Cap}}_1(A).\]
Proof. Consider \(u\in W^{1,1}(\mathbb{R}^d)\) with \(u\ge 1\) in a neighborhood of \(A\). Then for \(\mathcal{L}^{d-1}\)-a.e.\(z\in \pi(A)\), we have \[\int_{A_z}\left|\frac{\partial u}{\partial x_d}\right|\ge 2.\] Integrating over all \(z\in\mathbb{R}^{d-1}\), we get \[\int_{\mathbb{R}^d}|\nabla u|\ge 2\mathcal{H}^{d-1}(\pi(A)).\] Thus \(\Vert u\Vert_{W^{1,1}(\mathbb{R}^d)}\ge 2\mathcal{H}^{d-1}(\pi(A))\) and we get the result by taking infimum over all such \(u\). ◻
Recall that for a BV function \(f\) on the real line, we call \(f^{\vee}(x)-f^{\wedge}(x)=|Df|(\{x\})\) the jump size of \(f\) at point \(x\in\mathbb{R}\).
Lemma 3. Let \(\Omega\subset \mathbb{R}^d\) be open and let \(f\in \mathrm{BV}(\Omega)\). For every \(z\in \pi(\Omega)\) such that \(f_z\in \mathrm{BV}(\Omega_z)\) (which holds for \(\mathcal{L}^{d-1}\)-a.e.\(z\in \pi(\Omega)\)), let \(\alpha(z)\) be the maximal size of all jumps of \(f_z\). Then the map \(\alpha:\pi(\Omega)\rightarrow[0,\infty]\) is \(\mathcal{L}^{d-1}\)-measurable.
Proof. Note that \(|D(f_z)|\) is weakly* \(\mathcal{L}^{d-1}\)-measurable; recall . For a.e.\(z\in \pi(\Omega)\) we have \[\alpha(z)= \lim_{k\to\infty}\max_{i\in\mathbb{Z}} |D(f_z)|([i2^{-k},(i+1)2^{-k}]\cap \Omega_z) .\] ◻
Consider \(f\in \mathrm{BV}_{\mathrm{loc}}(\mathbb{R}^d)\). For \(|D f|\)-a.e.and in particular for \(|{D}^{\mathrm{c}}{f}|\)-a.e.\(x\in \mathbb{R}^d\) the Radon–Nikodym derivative exists: \[\label{eq:radon-nikodym} \frac{\mathop{}\!\mathrm{d}Df}{\mathop{}\!\mathrm{d}|Df|} (x) = \lim_{r\to 0}\frac{Df(B(x,r))}{|Df|(B(x,r))} \eqqcolon \xi(x)\tag{25}\] with \(|\xi(x)|=1\). Recall the integral average \(f_{C_{\xi(x)}(x,r)}\) over the cylinder \(C_{\xi(x)}(x,r)\), see . For \(r>0\) define the rescaling \[\label{eq:scalings32def} f_{r}(y)\mathrel{\vcenter{:}}=\frac{f(x+ry)-f_{C_{\xi(x)}(x,r)}}{|Df|(C(x,r))/r^{d-1}} , \qquad y\in C_{\xi(x)}(0,1) .\tag{26}\] The following blow-up behavior is known:
Theorem 7 (AFP?). Let \(f:\mathbb{R}^d\rightarrow\mathbb{R}\) for which \(|Df|\) is a Radon measure. Then for \(|Df|\)-a.e.\(x\in\mathbb{R}^d\) there exists a sequence \(r_1,r_2,\ldots>0\) with \(r_i\to 0\) and an increasing, nonconstant \(\gamma:(-1/2,1/2)\rightarrow\mathbb{R}\) such that for \(w:C_{\xi(x)}(0,1)\rightarrow\mathbb{R}\) given by \(w(y) = \gamma(\langle y, \xi(x)\rangle)\) we have \[f_{r_i} \to w \qquad\text{in }L^1(C_{\xi(x)}(0,1)) .\]
Definition 4. We call \(\gamma\) or \(w\) from a blow-up of \(f\) at \(x\).
There may exist several different blow-ups \(\gamma\) of \(f\) in \(x\), and if one of them is not concave then we say that \(f\) has a non-concave blow-up at \(x\).
For \(x\in\mathbb{R}^d\) and \(r>0\) denote by \(\mathcal{C}(x,r)\) the set of cylinders with length and radius \(r\) that are centered in \(x\). For Radon measures \(\mu,\nu\) denote \[\begin{align} \overline{D(\mathcal{C})}_\mu\nu(x) &= \limsup_{r\rightarrow0} \sup_{C\in\mathcal{C}(x,r)} \frac{ \nu(C) }{ \mu(C) } ,\\ \underline{D(\mathcal{C})}_\mu\nu(x) &= \liminf_{r\rightarrow0} \inf_{C\in\mathcal{C}(x,r)} \frac{ \nu(C) }{ \mu(C) } . \end{align}\]
Since (closed) cylinders also satisfy the covering theorem MR3409135?, due to Morse’s Covering Theorem (see e.g. FoLe?), we can prove the following the same way as MR3409135?.
Lemma 4. Let \(0<\alpha<\infty\). Then
\(A\subset\{\underline{D(\mathcal{C})}_\mu\nu\leq\alpha\}\) implies \(\nu(A)\leq\alpha\mu(A)\) ,
\(A\subset\{\overline{D(\mathcal{C})}_\mu\nu\geq\alpha\}\) implies \(\nu(A)\geq\alpha\mu(A)\) .
Lemma 5. Let \(\mu,\nu\) be Radon measures that are mutually singular. Then \[\begin{align} \text{For \nu-a.e. }&x & \underline{D(\mathcal{C})}_{\mu+\nu}\nu(x) = \overline{D(\mathcal{C})}_{\mu+\nu}\nu(x) &= 1 ,\\ \text{For \mu-a.e. }&x & \underline{D(\mathcal{C})}_{\mu+\nu}\nu(x) = \overline{D(\mathcal{C})}_{\mu+\nu}\nu(x) &= 0 . \end{align}\]
Proof. Since \(\mu\) and \(\nu\) are mutually singular there exists a set \(B\) with \(\nu=\nu \,\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!B\) and \(\mu(B)=0\). As a consequence of , the set of all points \(x\in B\) with \(\underline{D(\mathcal{C})}_\mu\nu(x)<\infty\) must have \(\nu(B)=0\). This means for \(\nu\)-almost every \(x\) we have \(\underline{D(\mathcal{C})}_\mu\nu(x)=\infty\) and thus \[1 \geq \overline{D(\mathcal{C})}_{\mu+\nu}\nu(x) \geq \underline{D(\mathcal{C})}_{\mu+\nu}\nu(x) = 1 .\] The second statement follows similarly. ◻
For \(|Df|\)-almost every \(x\in\Omega\) the Radon–Nikodym derivative exists. Since \({D}^{\mathrm{c}}{f}\) and \(Df-{D}^{\mathrm{c}}{f}\) are mutually singular, it follows from that for \(|{D}^{\mathrm{c}}{f}|\)-almost every \(x\in\Omega\) we have \[\label{eq:Cantor32density} \lim_{r\to 0}\frac{|{D}^{\mathrm{c}}{f}|(C_{\xi(x)}(x,r))}{|Df|(C_{\xi(x)}(x,r))}=1 .\tag{27}\]
3 Proof of↩︎
Fix a point \(x\in A\). Discarding a set of \(|Df|\)-measure zero, we can assume that the Radon–Nikodym derivative exists with \(|\xi(x)|=1\) and that the conclusion of and hold. Rotate, so that \(\xi(x)=e_d\).
Recall the definition and note that \({\mathrm M}_{C(0,1)}w(y) \ge w^{*}(y)\) for all \(y\in C(0,1)\). The main ingredient is the following simple .
Lemma 6. \({\mathrm M}_{C(0,1)}w(y) > w^{*}(y)\) for some \(y\in C(0,1)\).
Proof. For a contradiction suppose that \({\mathrm M}_{C(0,1)}w(y) \le w^{*}(y)\) for all \(y\in C(0,1)\), which means so that in fact equality holds. Then \(w^{*}\) is \(2\)-superharmonic on \(C(0,1)\), which means \(w\) is a \(2\)-supersolution by . But since \(w\) only depends on the \(d\)th coordinate, necessarily \(\gamma\) is a \(2\)-supersolution on \((-1/2,1/2)\). Thus by , there is a function \(\widetilde{\gamma}=\gamma\) a.e.in \((-1/2,1/2)\), which is \(2\)-superharmonic and thus necessarily a concave function (see e.g. MR1801253?) Since \(\gamma\) is increasing, necessarily \(\widetilde{\gamma}=\gamma\) everywhere in \((-1/2,1/2)\). But this contradicts the fact that \(\gamma\) is not concave, finishing the proof. ◻
By there exists a point \(y'=(z',t')\in B_{d-1}(0,1)\times(-1/2,1/2)\), for which \({\mathrm M}_{C(0,1)}w(y') > w^{*}(y')\). Thus there exists \(0<r\le 1/2\) such that \(B(y',r)\subset C(0,1)\) and \[w_{B(y',r)}> w^*(y').\] Recall that \(w\) is bounded. By continuity, that means there is an \(0<r'< 1/2\) such that \[\label{eq:delta32def} \delta \mathrel{\vcenter{:}}= w_{B(y',r')}- w^*(y') >0 .\tag{28}\] Since \(w\) only depends on the last coordinate, this inequality remains true for all \(y=(z,t')\) with \(z\in B_{d-1}(0,1-r')\).
To describe the idea of what we do next, assume for the moment that \(\gamma\) was continuous so that \(w^*=w\). Then for \[t'_1 = \inf\{t:\gamma(t)\geq\gamma(t')+\delta\} > t'\] we have \(w(z',t'_1) = w(y')+\delta\) and \(w(z',t)<w(y')+\delta\) for \(t<t'_1\). Since \(w\) is increasing in the last coordinate so is \(y\mapsto w_{B(y,r')}\), which means that for all \(y\in B_{d-1}(0,1-r')\times(t',t'_1)\) we would have (supposing that \(B(y,r')\subset C(0,1)\)) \[{\mathrm M}_{C(0,1)}w(y) \geq w_{B(y,r')} \geq w_{B(y',r')} = w(y')+\delta > w(y) .\]
Using \[|D\gamma'|((t',t'_1)) = \gamma(t'_1)-\gamma(t') = \delta,\] we could deduce that the maximal function is larger than the function itself in a \(|Dw|\)-large set: \[\begin{align} |Dw|(C(0,1)\cap \{{\mathrm M}_{C(0,1)}w>w^{*}\}) &\geq |Dw|(B_{d-1}(0,1-r')\times(t',t'_1)) \\ &= \delta \mathcal{L}^{d-1}(B_{d-1}(0,1-r')) , \end{align}\] see .
The following achives the corresponding estimate for the approximants \(f_{r_i}\) of \(w\).
Lemma 7. For \(\delta,r'>0\) from we have \[\limsup_{i\rightarrow\infty} |Df_{r_i}|(C(0,1)\cap \{{\mathrm M}_{C(0,1)} f_{r_i}> f_{r_i}^*\}) > \tfrac{\delta}{10} \mathcal{L}^{d-1}(B_{d-1}(0,1-r')) .\]
Before we prove we use it to conclude .
Proof of . First, note that \(\{{\mathrm M}_{<R} f>f^*\}\) is a Borel set and thus \(|Df|\)-measurable. Since \(r_i\rightarrow0\), in particular \(r_i<R\) for \(i\) large enough. Recall the definition of the rescalings . Then \[\begin{align} &\limsup_{r\to 0}\frac{|Df|(C(x,r)\cap \{{\mathrm M}_{<R} f>f^*\})}{|Df|(C(x,r))}\\ &\qquad \ge \limsup_{i\rightarrow\infty} \frac{|Df|(C(x,r_i)\cap \{{\mathrm M}_{C(x,r_i)} f>f^*\})}{|Df|(C(x,r_i))}\\ &\qquad = \limsup_{i\rightarrow\infty} |Df_{r_i}|(C(0,1)\cap \{{\mathrm M}_{C(0,1)} f_{r_i}> f_{r_i}^*\}) \\ &\qquad \ge \tfrac{\delta}{10} \mathcal{L}^{d-1}(B_{d-1}(0,1-r')) >0. \end{align}\] Due to this means \[\limsup_{r\to 0}\frac{|{D}^{\mathrm{c}}{f}|(C(x,r)\cap \{{\mathrm M}_{<R} f>f^*\})}{|{D}^{\mathrm{c}}{f}|(C(x,r))}>0.\] By this implies \[\lim_{r\to 0}\frac{|{D}^{\mathrm{c}}{f}|(C(x,r)\cap \{{\mathrm M}_{<R} f=f^*\})}{|{D}^{\mathrm{c}}{f}|(C(x,r))}=1\] for \(|{D}^{\mathrm{c}}{f}|\)-a.e.\(x\in \{{\mathrm M}_{<R} f=f^*\}\). Thus necessarily \(|{D}^{\mathrm{c}}{f}|(A\cap \{{\mathrm M}_{<R} f=f^*\})=0\). ◻
It remains to prove .
Proof of . Since \(|Df_r|(C(0,1))=1\), by lower semicontinuity we get \(|Dw|(C(0,1))\le1\), and then by , necessarily \(|D\gamma|((-1/2,1/2))\le1/\mathcal{L}^{d-1}(B_{d-1}(0,1))\), and so \(\gamma\) and \(w\) are bounded. The convergence \(f_{r_i}\to w\) in \(L^1(C(0,1))\) implies pointwise \(\mathcal{L}\)-a.e.convergence for a subsequence (not relabeled). By this means also \(f_{r_i}^{\vee}(y)\to w^{*}(y)=w^{\vee}(y)\) as \(i\to\infty\) for \(\mathcal{L}\)-a.e.\(y\in C(0,1)\).
From recall \(((f_{r_i})^{\wedge})_z(t)=((f_{r_i})_z)^{\wedge}(t)\) for \(\mathcal{L}^{d-1}\)-a.e.\(z\in B_{d-1}\) and every \(t\in (-1/2,1/2)\). Thus we can simply use the notation \((f_{r_i})^{\wedge}_z(t)\), and \((f_{r_i})^{\vee}_z(t)\). Similarly by , for \(\mathcal{L}^{d-1}\)-a.e.\(z\in B_{d-1}\) and every \(t\in (-1/2,1/2)\) we have \[\gamma^*(t)=w^*(z,t) ,\] which thus in fact holds for all \((z,t)\in C(0,1)\).
From and the continuity of the integral we find a \(-1/2<t<t'\), and \(0<r''\leq r'-(t'-t)\), such that for all \(z\in (1-r') B_{d-1}\) we still have \[w_{B((z,t),r'')}> w^{*}(z,t)+4\delta/5 ,\] and for \(\mathcal{L}^{d-1}\)-a.e.\(z\in B_{d-1}\) we have \(f_{r_i}^{\vee}(z,t)\to w^{\vee}(z,t)=w^*(z,t)\) as \(i\to\infty\). We also find \(t+r''<s<1/2\) such that \(f_{r_i}^{\vee}(z,s)\to w^{\vee}(z,s)\) as \(i\to\infty\) for \(\mathcal{L}^{d-1}\)-a.e. \(z\in B_{d-1}\). Then for all \(v\in [t,s]\), since \(\gamma\) is increasing, we have \[\label{eq:estimate32for32h} w_{B((z,v),\min\{r'',1/2-v\})}> w^{*}(z,t)+4\delta/5.\tag{29}\] By the convergence \(f_{r_i}\to w\) in \(L^1(C(0,1))\) as \(i\to\infty\), we have \[(f_{r_i}-w)_{B((z,v),\min\{r'',1/2-v\})} \to 0\] uniformly for all \(z\in (1-r') B_{d-1}\) and \(v\in [t,s]\). Thus for sufficiently large \(i\), we have \[\label{eq:uniform32closeness} | (f_{r_i})_{B((z,v),\min\{r'',1/2-v\})} - w_{B((z,v),\min\{r'',1/2-v\})} | <\delta/5\tag{30}\] Consider the set \(D_i\) of those \(z \in B_{d-1}\) for which \[\label{eq:only32small32jumps} (f_{r_i})_z\in\mathrm{BV}((-1/2,1/2))\textrm{ has jumps at most size } \;\delta/5.\tag{31}\] By , \(D_i\) is \(\mathcal{L}^{d-1}\)-measurable. Due to we have \(|{D}^{\mathrm{j}}{f_{r_i}}|(C(0,1))\rightarrow0\) and thus by we have \[\label{eq:size32of32Dj32sets} \lim_{i\to\infty}\mathcal{L}^{d-1}(B_{d-1}\setminus D_i)= 0.\tag{32}\] By Egorov’s theorem, we find an \(\mathcal{L}^{d-1}\)-measurable set \(H\subset (1-r') B_{d-1}\) such that \(\mathcal{L}^{d-1}(H)> \frac{1}{2} \mathcal{L}^{d-1}((1-r') B_{d-1})\), and \(f_{r_i}^{\vee}(z,t)\to w^{*}(z,t)\) and \(f_{r_i}^{\vee}(z,s)\to w^{\vee}(z,s)\) as \(i\to\infty\) uniformly for all \(z\in H\). From we then get for sufficiently large \(i\), for all \(z\in H\) and for all \(v\in [t,s]\), that \[\label{eq:max32func32much32bigger} (f_{r_i})_{B((z,v),\min\{r'',1/2-v\})}\ge f_{r_i}^{\vee}((z,t))+3\delta/5.\tag{33}\] For all sufficiently large \(i\) and for all \(z\in H\), we also have \[\begin{align} f_{r_i}^{\vee}(z,s) &\ge w^{\vee}(z,s)-\delta/5\\ &\ge w_{B((z,t),r'')}-\delta/5&& \text{since \gamma is increasing}\\ &>(f_{r_i})_{B((z,t),r'')} -2\delta/5&&\text{by \zcref{eq:uniform32closeness}}\\ &>f_{r_i}^{\vee}((z,t))+\delta/5&&\text{by \zcref{eq:max32func32much32bigger}}\\ &\ge f_{r_i}^{\wedge}((z,t))+\delta/5. \end{align}\] By and we can assume that \((z,t)\mapsto(f_{r_i})_z^{\vee}(t)\) restricted to \(H\times (-1/2,1/2)\) is upper semicontinuous, and that \((z,t)\mapsto(f_{r_i})_z^{\wedge}(t)\) restricted to \(H\times (-1/2,1/2)\) is lower semicontinuous, for every \(i\in\mathbb{N}\). For all \(z\in H\) and for all sufficiently large \(i\), we find the smallest \(t\le t_{z}\le s\) such that \[f_{r_i}^{\vee}((z,t_{z})) \ge f_{r_i}^{\wedge}((z,t))+\delta/5.\] Note that now \[\label{eq:lsc} H\ni z\mapsto t_z\quad\textrm{is lower semicontinuous.}\tag{34}\] By for every \(z\in H\cap D_i\) and \(v\in [t,t_{z}]\) we have \[\label{eq:choice32of32tjz} f_{r_i}^{\vee}((z,v))\\ \le f_{r_i}^{\wedge}((z,t))+2\delta/5.\tag{35}\] By , we have \[\begin{align} |D[(f_{r_i})_z]|([t,t_{z}]) &\ge (f_{r_i}^{\vee})_z(t_{z})-(f_{r_i}^{\wedge})_z(t) \ge \delta/5. \end{align}\] By , we have for all \(v\in [t,t_{z}]\), \[(f_{r_i})_{B((z,v),\min\{r'',1/2-v\})} \ge f_{r_i}^{\vee}((z,v))+\delta/5.\] Recalling also , for every \(z\in H\cap D_i\) we get \[|D[(f_{r_i})_z]|([t,t_{z}] \cap \{{\mathrm M}_{C(0,1)} f_{r_i}> f_{r_i}^*\}) \ge \delta/5.\] Integrating over \(H\cap D_i\), where the required measurability is guaranteed by , by we get \[|Df_{r_i}|(C(0,1)\cap \{{\mathrm M}_{C(0,1)} f_{r_i}> f_{r_i}^*\}) \ge \tfrac{\delta}{5}\mathcal{L}^{d-1}(H\cap D_i) > \tfrac{\delta}{10}\mathcal{L}^{d-1}((1-r') B_{d-1})\] for sufficiently large \(i\), due to . ◻
4 Sobolev regularity↩︎
In this , we will use the results proved in to prove Sobolev regularity of the maximal function of certain functions of bounded variation.
4.1 Quasicontinuity↩︎
First, we prove that \({\mathrm M}f\) is 1-quasicontinuous outside of \(J_f\), see . The proof of is analogous to the one given in panubvsobolev? for the uncentered maximal function; we note that a few statements in panubvsobolev? were incorrect since only \(f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\) was assumed, when in fact \(\mathop{\mathrm{var}}f<\infty\) is needed.
As before, \(\Omega\subset \mathbb{R}^d\) is always an open set. We start with some preliminary results.
Lemma 8 (lecturenoteshajlasz?). Let \(\Omega\) be a John domain and let \(f\in W^{1,1}(\Omega)\). Then \(f\in L^{d/(d-1)}(\Omega)\) and \[\Bigl( \int_\Omega |f-f_\Omega|^{\frac{d}{d-1}} \Bigr)^{\frac{d-1}{d}} \lesssim \int_\Omega |\nabla f| .\]
We require the following variant.
Lemma 9. Let \(\Omega\) be a John domain and let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\). Then \(f\in L^1(\Omega)\) and \[\frac{1}{\mathcal{L}(\Omega)^{\frac{1}{d}}} \int_\Omega |f-f_\Omega| \lesssim \mathop{\mathrm{var}}_\Omega f .\]
Proof. Observe, that for \(f\in L^1(\Omega)\) we have \[\label{eq:fomegavsinf} \inf_{c\in\mathbb{R}} \int_\Omega |f-c| \leq \int_\Omega |f-f_\Omega| \leq \inf_{c\in\mathbb{R}} \int_\Omega |f-c| + |f_\Omega-c| \leq 2 \inf_{c\in\mathbb{R}} \int_\Omega |f-c| .\tag{36}\] As a consequence of Cavalieri’s principle, \[\int_\Omega |f-c| = \int_{-\infty}^c \mathcal{L}(\Omega\cap\{f\leq\lambda\}) \mathop{}\!\mathrm{d}\lambda + \int_c^\infty \mathcal{L}(\Omega\cap\{f>\lambda\}) \mathop{}\!\mathrm{d}\lambda ,\] the infimum is attained for the median, \[c = \mathrm m(\Omega ,f) \mathrel{\vcenter{:}}= \inf\{\lambda\in\mathbb{R}:\mathcal{L}(\Omega\cap\{f>\lambda\})\leq\mathcal{L}(\Omega)/2\} ,\] which is finite for any \(f\) that is finite almost everywhere.
Let \(f\) be bounded. Since a John domain is by definition bounded, we have \(f\in L^1(\Omega)\), and by e.g.AFP? we find a sequence of functions \(f_i\in W^{1,1}(\Omega)\) with \(f_i\to f\) in \(L^1(\Omega)\) and \(\mathop{\mathrm{var}}_\Omega f_i\to \mathop{\mathrm{var}}_\Omega f\). By , Hölder’s inequality and we can conclude \[\frac{1}{\mathcal{L}(\Omega)^{\frac{1}{d}}} \int_\Omega |f-\mathrm m(\Omega ,f)| \leq \frac{1}{\mathcal{L}(\Omega)^{\frac{1}{d}}} \int_\Omega |f-f_\Omega| \leq \Bigl( \int_\Omega |f-f_\Omega|^{\frac{d}{d-1}} \Bigr)^{\frac{d-1}{d}} \lesssim \mathop{\mathrm{var}}_\Omega f.\]
For a general \(f\in L^1_\mathrm{loc}(\Omega)\) and \(N\in\mathbb{N}\) truncate \(f_N=\max\{\min\{f_N,N\},-N\}\). Then for \(N>|\mathrm m(\Omega ,f)|\) we have \(\mathrm m(\Omega,f_N)=\mathrm m(\Omega ,f)\). Thus, we can conclude from monotone convergence and the previous bounded case \[\begin{align} \frac{1}{\mathcal{L}(\Omega)^{\frac{1}{d}}} \int_\Omega |f-\mathrm m(\Omega ,f)| &= \lim_{N\rightarrow\infty} \frac{1}{\mathcal{L}(\Omega)^{\frac{1}{d}}} \int_\Omega |f_N-\mathrm m(\Omega,f_N)| \\ &\lesssim \lim_{N\rightarrow\infty} \mathop{\mathrm{var}}_\Omega(f_N) = \mathop{\mathrm{var}}_\Omega f . \end{align}\] In particular \(f\in L^1(\Omega)\), and we can conclude the proof from . ◻
Lemma 10. Let \(f\in\mathrm{BV}_{\mathrm{loc}}(\Omega)\). Then \({\mathrm M}f(x)\ge f^{\vee}(x)\) for every \(x\in \Omega\setminus S_f\).
Proof. Recall that always \({\mathrm M}f\ge f^*\), and \(f^*(x)=f^{\vee}(x)\) for every \(x\in \Omega\setminus S_f\) by . ◻
Proposition 8. Let \(f\in L^1_{\mathrm{loc}}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\). Then \[\mathop{\mathrm{Cap}}_1(\{x\in \Omega\colon {\mathrm M}|f|(x)=\infty\})=0.\]
Proof. For \(x\in\Omega\) we call a sequence \(r_1,r_2,\ldots\in\mathbb{R}\) a maximizing sequences if \(f_{B(x,r_i)}\rightarrow{\mathrm M}|f|(x)\), We group the set of all \(x\in\Omega\) with \({\mathrm M}|f|(x)=\infty\) into three parts, based on their maximizing sequences.
First consider those \(x\) for which all maximizing sequences satisfy \(r_i\rightarrow0\). Then \({\mathrm M}|f|(x)=|f|^*(x)\) and we know that \(|f|^*(x)<\infty\) for \(\mathop{\mathrm{Cap}}_1\)-a.e.\(x\in \Omega\), due to .
Next we consider those \(x\) for which there exists an \(0<r<\infty\) and a maximizing sequence with \(r_i\rightarrow r\). Then \(B(x,r)\subset\Omega\) and \({\mathrm M}|f|(x)=|f|_{B(x,r)}\) and by \(\mathop{\mathrm{var}}_{B(x,r)} f\le\mathop{\mathrm{var}}_\Omega f<\infty\) and we have \(f\in L^1(B(x,r))\).
For all remaining \(x\) there exists a maximizing sequence with \(r_i\rightarrow\infty\). Now necessarily \(\Omega=\mathbb{R}^d\). Then by e.g.AFP? there exists a \(c\in\mathbb{R}\) such that \(f-c\in L^{d/(d-1)}(\mathbb{R}^d)\). That means \[\fint_{B(x,r)}|f-c| \le \left(\fint_{B(x,r)}|f-c|^{d/(d-1)}\right)^{(d-1)/d}\to 0\] as \(r\to \infty\) and therefore \({\mathrm M}|f|(x)=c<\infty\). ◻
Lemma 11. panubvsobolev? For any dimension \(d\in\mathbb{N}\) there exists a \(C_1\geq0\) such that the following holds: Let \(f\in\mathrm{BV}_{\mathrm{loc}}(\Omega)\), let \(G\subset \Omega\), and let \(\varepsilon>0\). Then there exists an open set \(U\supset G\) such that \(\mathop{\mathrm{Cap}}_1(U)\le C_1\mathop{\mathrm{Cap}}_1(G)+\varepsilon\) and \[\frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)\cap G}|f| \to 0\quad\textrm{as }r\to 0\] locally uniformly for \(x\in \Omega\setminus U\).
Next, we find an exceptional set outside of which several regularity properties hold, which will be used repeatedly.
Lemma 12. Let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\). Then for every \(\varepsilon>0\) exists a set \(G\supset(S_f\setminus J_f)\) such that \({\mathrm M}f|_{\Omega\setminus G}\) is finite and \(f^{\vee}|_{\Omega\setminus G}\) is finite and upper semicontinuous, and an open set \(U\supset G\) with \(\mathop{\mathrm{Cap}}_1(U)<\varepsilon\) such that \[\label{eq:uniform32convergence32G} \frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)\cap G}|f| \to 0\quad\textrm{as }r\to 0\tag{37}\] locally uniformly for \(x\in \Omega\setminus U\).
Proof. By there is a set \(G\) with \(\mathop{\mathrm{Cap}}_1(G)<\varepsilon/(2C_1)\) such that \(f^{\vee}|_{\Omega\setminus G}\) is finite and upper semicontinuous. By , the set where \({\mathrm M}|f|\) is infinite has zero capacity, and by we have \(\mathop{\mathrm{Cap}}_1(S_f\setminus J_f)=\mathcal{H}^{d-1}(S_f\setminus J_f)=0\). That means we can include both previous sets in \(G\) without increasing its capacity. Finally, we find \(U\supset G\) by as desired. ◻
Next, we show that \(f^*\) is 1-quasicontinuous outside of \(J_f\).
Proposition 9. Let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\). Then \(f^*\) is \(1\)-quasicontinuous outside of \(J_f\), that is, for every \(\varepsilon>0\) there exists an open set \(G\subset \Omega\) such that \(\mathop{\mathrm{Cap}}_1(G)<\varepsilon\) and \(f^*|_{\Omega\setminus G}\) is finite and continuous at every \(x\in \Omega\setminus (J_f\cup G)\).
Proof. We apply , and we note that by we have \(\mathop{\mathrm{Cap}}_1(S_f\setminus J_f)=\mathcal{H}^{d-1}(S_f\setminus J_f)=0\), so that \(S_f\setminus J_f\) can be included in the exceptional set \(G\). Recalling also , the result follows. ◻
Finally, we can show the same for the maximal function.
Theorem 10. Let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\). Then \({\mathrm M}f\) is \(1\)-quasicontinuous outside of \(J_f\). If \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\), then \({\mathrm M}f\) is \(1\)-quasicontinuous.
Note that we have to exclude \(J_f\) because it may be a set with nonzero capacity in which \({\mathrm M}f\) is discontinuous, as can be seen for example for \(f=1_{B(0,1)}\).
Proof. For \(\varepsilon>0\) take \(G,U\) from . Since \({\mathrm M}f\) is finite and lower semicontinuous in \(\Omega\setminus U\), it is sufficient to prove the upper semicontinuity of \({\mathrm M}f|_{\Omega\setminus U}\) on \(\Omega\setminus(U\cup J_f)\). Fix \(x\in\Omega\setminus(U\cup J_f)\). Take a sequence \(x_i\to x\), \(x_i\in \Omega\setminus U\), such that \[\lim_{i\to\infty}{\mathrm M}f(x_i)=\limsup_{\Omega\setminus U\ni y\to x}{\mathrm M}f(y).\] (At this stage we cannot exclude the possibility that the \(\limsup\) is \(\infty\).) Now we only need to show \({\mathrm M}f(x)\ge \lim_{i\to\infty}{\mathrm M}f(x_i)\).
We find radii \(r_i>0\) such that \[\label{eq:choice32of32almost32optimal32balls} \lim_{i\to\infty}{\mathrm
M}f(x_i)=\lim_{i\to\infty}\,f_{B(x_i,r_i)}\tag{38}\] Now we consider three cases.
Case 1. Suppose that by passing to a subsequence (not relabeled), we have \(r_i\to 0\). Fix \(\delta>0\). By the map \(f^{\vee}|_{\Omega\setminus G}\) is upper semicontinuous. That means for some \(r>0\) we have \(B(x,r)\subset\Omega\) and \[f^{\vee}(x)\ge \sup_{B(x,r)\setminus G}f^{\vee}-\delta.\] Note that for sufficiently large \(k\in\mathbb{N}\), we have \(B(x_i,r_i)\subset B(x,r)\). Note also
that \(f\in L^1(B)\) for every open ball \(B\subset \Omega\), due to \(\mathop{\mathrm{var}}_\Omega f<\infty\) and the Poincaré inequality (see ) which
justifies subtracting integrals over (subsets of) balls below. Using (recall that \(x\notin U\cup J_f\supset S_f\)), for large \(i\in\mathbb{N}\) we get \[\begin{align} {\mathrm M}f(x) &\ge f^{\vee}(x) \ge \sup_{B(x,r)\setminus G}f^{\vee}-\delta\\ &\ge \frac{1}{\mathcal{L}(B(x_i,r_i))}\int_{B(x_i,r_i)\setminus G}f-\delta\\ &=
\frac{1}{\mathcal{L}(B(x_i,r_i))}\int_{B(x_i,r_i)}f- \frac{1}{\mathcal{L}(B(x_i,r_i))}\int_{B(x_i,r_i)\cap G}f-\delta \\ &\rightarrow \lim_{i\to\infty}{\mathrm M}f(x_i)-\delta \qquad\text{ by \zcref{eq:uniform convergence G,eq:choice of almost optimal
balls}}.
\end{align}\] Letting \(\delta\to 0\), we obtain the desired inequality.
Case 2. The second alternative is that by passing to a subsequence (not relabeled), we have \(r_i\to r\in (0,\infty)\). Take \(k\in \mathbb{N}\) so large that \(|x_i-x|\le \tfrac 1{10} \inf_{m\ge k}r_m\) for all \(i\ge k\) and denote \[\Omega_x = B(x,r) \cup \bigcup_{i\ge k}B(x_i,r_i) .\] Then \(\Omega_x \subset \Omega\) is a John-domain: The curve \(\gamma\) witnessing this for \(y\in\Omega_x\) can be taken to be a straight line from the center point \(x\) to \(x_k\) and then a straight line from \(x_k\) to \(y\). By we can conclude \(f\in L^1(\Omega_x)\). Since \(\mathcal{L}( B(x_i,r_i) \Delta B(x,r) )\) tends to \(0\) as \(i\rightarrow0\) this also means \[\lim_{r\rightarrow\infty} \int_{ B(x_i,r_i) \Delta B(x,r) } f =0 .\] Since \(\mathcal{L}(B(x_i,r_i))\rightarrow\mathcal{L}(B(x,r))>0\) we can conclude \[{\mathrm M}f(x)\ge \fint_{B(x,r)}f =\lim_{i\to\infty}\,\fint_{B(x_i,r_i)}f =\lim_{i\to\infty}{\mathrm M}f(x_i) .\]
Case 3. Finally, we have the possibility that passing to a subsequence (not relabeled), we have \(r_i\to \infty\). Note that now necessarily \(\Omega=\mathbb{R}^d\). For \(i\) sufficiently large that \(|x_i-x|<r_i\), for \(d\ge 2\) by we have \[\begin{align} |f_{B(x,r_i)}-f_{B(x_i,r_i)}| &\le |f_{B(x,r_i)}-f_{B(x,2r_i)}| + |f_{B(x_i,r_i)}-f_{B(x,2r_i)}| \\ &\le \frac{ 2^{d+1}r_i }{ \mathcal{L}(B(x,2r_i)) }|Df|(B(x,2r_i)) \to 0 \end{align}\] as \(i\to \infty\) since \(|Df|(\mathbb{R}^d)<\infty\). For \(d=1\), we can estimate \[\begin{align} |f_{B(x,r_i)}-f_{B(x_i,r_i)}| &\le \frac{ \mathcal{L}(B(x,r_i)\Delta B(x_i,r_i)) }{2r_i} \|f\|_{L^{\infty}(\mathbb{R})} \to 0. \end{align}\] Then \[{\mathrm M}f(x)\ge \limsup_{i\to\infty}f_{B(x,r_i)} = \limsup_{i\to\infty} f_{B(x_i,r_i)} = \lim_{i\to\infty}{\mathrm M}f(x_i).\] This completes the proof.
Finally, if \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\), then also the set \(J_f\) can be included in \(G\), giving the result. ◻
4.2 Lusin property↩︎
Recall the definition .
Proposition 11. Let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\), and let \(R>0\). Then \({\mathrm M}_{\geq R} f\) is \(2\mathop{\mathrm{var}}_\Omega f/\mathcal{L}(B(0,R))\)-Lipschitz on \(\{x:\mathop{\mathrm{dist}}(x,\mathbb{R}^d\setminus\Omega)\geq R\}\) with respect to shortest path distance.
Proof. For any \(r>0\) and \(x\in\Omega\) define \(r(x)=\min\{r,\mathop{\mathrm{dist}}(x,\Omega^\complement)\}\) and \[g_r(x) = \fint_{B(x,r(x))} |f| = \int_{B(0,1)} |f|(x+r(x)y) \mathop{}\!\mathrm{d}y .\] Note that the map \(r(\cdot)\) is 1-Lipschitz. Let \(\nu\in\mathbb{S}^{d-1}\). Then \[\begin{align} |\nabla g_r(x)| &= \Bigl| \int_{B(0,1)} \nabla(\cdot+r(\cdot)y)(x) \cdot \nabla|f|(x+r(x)y) \mathop{}\!\mathrm{d}y \Bigr| \\ &= \Bigl| \int_{B(0,1)} (\mathrm{Id}+\nabla r(x)\otimes y) \cdot \nabla|f|(x+r(x)y) \mathop{}\!\mathrm{d}y \Bigr| \\ &\leq2 \int_{B(0,1)} |\nabla f|(x+r(x)y) \mathop{}\!\mathrm{d}y \\ &\leq 2\frac{\mathop{\mathrm{var}}_\Omega f}{\mathcal{L}(B(x,r(x)))} \end{align}\] That means \({\mathrm M}_{\geq R}f\) is a supremum of maps that are \(2\mathop{\mathrm{var}}_\Omega f/\mathcal{L}(B(0,R))\)-Lipschitz, and thus so is itself. ◻
Lemma 13. panubvsobolev? Let \(V\subset\mathbb{R}\) be open and let \(f\in\mathrm{BV}_{\mathrm{loc}}(V)\). If \(N\subset V\setminus S_f\) with \(|Df|(N)=0\), then \[\mathcal{L}^1(f^*(N))=0.\]
The Lusin property for a function \(v\) defined on \(V\subset\mathbb{R}\) states that \[\textrm{if }N\subset V\textrm{ with }\mathcal{L}^1(N)=0,\;\textrm{ then } \mathcal{L}^1(v(N))=0.\]
The measure-theoretic boundary \(\mathop{\partial_*}{E}\) of a set \(E\subset \mathbb{R}^d\) is defined as the set of points where \[\limsup_{r\to 0} \frac{ \mathcal{L}(B(x,r)\cap E) }{ \mathcal{L}(B(x,r)) }>0 \quad\textrm{and}\quad \limsup_{r\to 0} \frac{ \mathcal{L}(B(x,r)\setminus E) }{ \mathcal{L}(B(x,r)) }>0.\] By the coarea formula for BV functions, see e.g.AFP?, for any \(f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\) we have \[|Df|(\Omega)=\int_{-\infty}^\infty\mathcal{H}^{d-1}(\Omega\cap\mathop{\partial_*}{\{x\in\Omega:f(x)>t\}})\mathop{}\!\mathrm{d}t .\]
Lemma 14. Suppose \(W\subset \mathbb{R}\) is open and \(u,v\in \mathrm{BV}_{\mathrm{loc}}(W)\). Suppose \(A\subset W\) is such that \(u\) is continuous and zero in all points in \(A\). Then \[|Du|(A)=0.\]
Proof. Note that \(A\cap \partial^*\{u>t\}=\emptyset\) for all \(t\neq 0\). Now the result follows from the coarea formula. ◻
Lemma 15. Let \(G,U\) be as in . Then for every \(x_0\in\Omega\setminus U\) with \({\mathrm M}f(x_0)>f^\vee(x_0)\) there exists an \(R>0\) such that for all \(x\in B(x_0,R)\setminus U\) we have \({\mathrm M}f(x)={\mathrm M}_{\geq R}f(x)\).
Proof. Abbreviate \(\alpha\mathrel{\vcenter{:}}={\mathrm M}f(x_0)\in (-\infty,\infty)\) and \(\delta\mathrel{\vcenter{:}}=\alpha-f^{\vee}(x_0)>0\). By upper semicontinuity of \(f^{\vee}\) on \(\Omega\setminus G\), there exists an \(R_1>0\) such that for all \(x\in B(x_0,R_1)\setminus G\) we have \[f^{\vee}(x)<\alpha-\frac{3\delta}{4} .\] By lower semicontinuity of the maximal function, there exists an \(R_2>0\) such that \({\mathrm M}f(x)>\alpha-\delta/4\) for all \(x\in B(x_0,R_2)\). By the choice of \(U\) there exists an \(R_3>0\) such that for all \(x\in B(x_0,R_3)\setminus U\) we have \[\frac{1}{\mathcal{L}(B(x,s))}\int_{G\cap B(x,s)}|f| \le\frac{\delta}{2} \qquad\textrm{for all }0<s<R_3.\] Set \(R=\min\{R_1,R_2,R_3\}/2\) and let \(x\in B(x_0,R)\setminus U\) and \(r\in (0,R)\). Then we can conclude \[\begin{align} \frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)}f &= \frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)\cap G}f +\frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)\setminus G}f\\ &\le \frac{1}{\mathcal{L}(B(x,r))}\int_{B(x,r)\cap G}f +\alpha-\frac{3\delta}{4}\\ &\le\frac{\delta}{2}+\alpha-\frac{3\delta}{4} =\alpha-\frac{\delta}{4}. \end{align}\] On the other hand, we had \({\mathrm M}f(x)>\alpha-\delta/4\) for all \(x\in B(x_0,R)\). That means \({\mathrm M}f(x)={\mathrm M}_{\geq R}f(x) .\) ◻
Theorem 12. Let \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\) such that at \(|{D}^{\mathrm{c}}{f}|\)-a.e.\(x\in \Omega\), we have \({\mathrm M}f(x)>f^*(x)\). Then \({\mathrm M}f\) has the Lusin property on almost every line parallel to a coordinate axis.
Proof. Recall the notation and results from and that \(\pi\) denotes the orthogonal projection onto \(\mathbb{R}^{d-1}\), and recall which says \(2\mathcal{H}^{d-1}\circ\pi\le \mathop{\mathrm{Cap}}_1 .\) That means that for \(\mathcal{H}^{d-1}\)-almost every \(z\in\pi(\Omega)\) there exist sets \(G,U\) from with \(z\in \pi(\Omega)\setminus \pi(U)\). In other words, for almost every line parallel to the \(d\)th coordinate axis exist \(G,U\) which do not intersect said line. In addition, the following hold for almost every \(z\in \pi(\Omega)\), i.e.for almost every line parallel to the \(d\)th coordinate axis: We have \(f_z\in\mathrm{BV}_{\mathrm{loc}}(\Omega_z)\), the set \((J_f)_z=J_{f_z}\) is at most countable, and holds; recall . Denote \[H_f\mathrel{\vcenter{:}}=\{{\mathrm M}f>f^{\vee}\}.\] By , for \(|{D}^{\mathrm{c}}{f}|\)-a.e.\(x\in\Omega\) we have \(f^{\vee}(x)=f^*(x)\). Then by assumption \(|{D}^{\mathrm{c}}{f}|(\Omega\setminus H_f)=0\), and so due to we have \[\label{eq:cantorzzero} |{D}^{\mathrm{c}}{(f_z)}|(\Omega_z\setminus (H_f)_z)=0\tag{39}\] for \(\mathcal{L}^{d-1}\)-a.e.\(z\in\pi(\Omega)\). By symmetry all of the above also holds for almost every line parallel to any other coordinate axis.
For simplicity we only consider a line \(\{(z,t):t\in \mathbb{R}\}\) parallel to the \(d\)th coordinate axis. For a set \(N\subset \mathbb{R}\) with zero \(1\)-dimensional Lebesgue measure, we split \[\begin{align} {\mathrm M}f(\{z\}\times N) &= {\mathrm M}f((\{z\}\times N)\cap H_f) \cup {\mathrm M}f((\{z\}\times N)\cap J_f) \\ &\qquad\cup {\mathrm M}f((\{z\}\times N)\setminus(H_f\cup J_f). \end{align}\] The first set has zero one-dimensional Lebesgue measure since \({\mathrm M}f\) is locally Lipschitz on \(H_f\setminus U\) due to . So does the second set since we assumed \((\{z\}\times \Omega_z)\cap J_f\) is at most countable. For the third set, since \(U\supset S_f\setminus J_f\) and since we assumed to hold, we have \[\begin{align} \notag {\mathrm M}f((\{z\}\times N)\setminus (H_f\cup J_f)) &= {\mathrm M}f((\{z\}\times N)\setminus (H_f\cup S_f)) \\ \notag &= f^*((\{z\}\times N)\setminus (H_f\cup S_f)) \\ \label{eq:Mf32union} &\subset f_z^*(N\setminus (S_{f_z}\cup (H_f)_z) . \end{align}\tag{40}\] By our assumptions \[\begin{align} |{D}^{\mathrm{c}}{(f_z)}| (\Omega_z\setminus (H_f)_z)) &=0 ,& |{D}^{\mathrm{a}}{(f_z)}| (N) &= 0 ,& |{D}^{\mathrm{j}}{(f_z)}| (\Omega_z\setminus S_{f_z}) &= 0 \end{align}\] so that by the third set in has zero \(\mathcal{L}^1\)-measure. In conclusion, \(\mathcal{L}^1({\mathrm M}f(z,N))=0\), and so \({\mathrm M}f\) has the Lusin property on almost every line parallel to a coordinate axis. ◻
Theorem 13. Suppose \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\), such that at \(|{D}^{\mathrm{c}}{f}|\)-a.e.\(x\in \Omega\), we have \({\mathrm M}f(x)>f^*(x)\). Moreover, suppose that \({\mathrm M}f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\). Then \(|D^c {\mathrm M}f|(\Omega)=0\).
Proof. As in the beginning of the proof of , for \(\mathcal{L}^{d-1}\)-a.e.line parallel to a coordinate axis, indexed by \(z\in\pi(\Omega)\), there exist \(G,U\) from which do not intersect that line. Moreover, \(f_z\in\mathrm{BV}_{\mathrm{loc}}(\Omega_z)\) and holds. In addition, \(f^*\) and \({\mathrm M}f\) are continuous outside of \(J_f\) on almost every line by .
For simplicity we only consider a line \(\{(z,t):t\in \mathbb{R}\}\) parallel to the \(d\)th coordinate axis. We consider a set \(N\subset \Omega_z\) with zero \(1\)-dimensional Lebesgue measure. By we have \[\begin{align} |{D}^{\mathrm{c}}{[({\mathrm M}f)_z]}|(\Omega_z\setminus (H_f)_z) &=|{D}^{\mathrm{c}}{[({\mathrm M}f)_z]}|(\Omega_z\setminus ((H_f)_z\cup (J_f)_z))\\ &=|{D}^{\mathrm{c}}{[f_z]}|(\Omega_z\setminus ((H_f)_z\cup (J_f)_z)) =0 . \end{align}\] By the map \({\mathrm M}f|_{\Omega\setminus U}\) is locally Lipschitz and thus \[|D[({\mathrm M}f)_z]|(N\cap (H_f)_z)=0 .\] That means \(|{D}^{\mathrm{c}}{[({\mathrm M}f)_z]}|(N)=0 .\) Since \({D}^{\mathrm{c}}{[({\mathrm M}f)_z]}\) is carried by a null set, in fact \(|{D}^{\mathrm{c}}{[({\mathrm M}f)_z]}|(\Omega_z)=0\). Using again , we can conclude \(|{D}^{\mathrm{c}}{{\mathrm M}f}|(\Omega)=0\). ◻
Proposition 14. Suppose \(f\in L^1_\mathrm{loc}(\Omega)\) with \(\mathop{\mathrm{var}}_\Omega f<\infty\), and suppose that \({\mathrm M}f\in \mathrm{BV}_{\mathrm{loc}}(\Omega)\). Then \(|{D}^{\mathrm{j}}{{\mathrm M}}f|\le \tfrac 12 |{D}^{\mathrm{j}}{f}|\).
Proof. Take sets \(G,U\) from . Given \(G\), by applying to a constant function, we obtain \[\label{eq95Gfillsball} \frac{\mathcal{L}(G\cap B(x,r))}{\mathcal{L}(B(x,r))} \to 0\quad\textrm{as }r\to 0\tag{41}\] locally uniformly for \(x\in \Omega\setminus U\) after slightly enlarging \(U\) while still respecting \(\mathop{\mathrm{Cap}}_1(U)<\varepsilon\). By , we can also assume that \({\mathrm M}f|_{\Omega\setminus G}\) is continuous at every point \(x\in \Omega\setminus(G\cup J_f)\).
In particular, for any \(x\in \Omega\setminus (U\cup J_f)\) there exists a \(c\in\mathbb{R}\) for which \(({\mathrm M}f-c)|_{B(x,r)\cap\Omega\setminus G}\rightarrow0\) uniformly as \(r\rightarrow0\). By the set \(G\) where this convergence happens eventually fills all of \(B(x,r)\). This is not compatible with \(x\in J_{({\mathrm M}f)_z}\), which, due to , would require the blow-up of \({\mathrm M}f\) to converge to two distinct values on two halves of the ball. Since \(\mathop{\mathrm{Cap}}_1(U)\) can be made arbitrarily small, we can conclude \(\mathop{\mathrm{Cap}}_1(J_{{\mathrm M}f}\setminus J_f)=0\) which by implies \(\mathcal{H}^{d-1}(J_{{\mathrm M}f}\setminus J_f)=0\).
Suppose \(x\in J_{{\mathrm M}f}\cap J_f\). We have \[{\mathrm M}f(x)\ge f^*(x)=\frac{f^{\wedge}(x)+f^{\vee}(x)}{2},\] and by the lower semicontinuity of \({\mathrm M}f\) also \[({\mathrm M}f)^{\wedge}(x)\ge \frac{f^{\wedge}(x)+f^{\vee}(x)}{2}.\] If \(({\mathrm M}f)^\vee(x)\le f^{\vee}(x)\), then we get \[\label{eq:less32than32half} ({\mathrm M}f)^\vee(x)-({\mathrm M}f)^\wedge(x)\le \tfrac 12 (f^\vee(x)-f^\wedge(x)).\tag{42}\] Otherwise, for \(\kappa\mathrel{\vcenter{:}}=({\mathrm M}f)^\vee(x)-f^{\vee}(x)>0\), for some half-space \(H\) with \(x\) on its boundary and for all \(0<\delta<1/2\) we have \[\label{eq:half32space32H} \lim_{r\to 0}\frac{\mathcal{L}(B(x,r)\cap H\cap \{{\mathrm M}f>f^{\vee}(x)+(1-\delta)\kappa\})}{\mathcal{L}(B(x,r)\cap H)} = 1 .\tag{43}\] Fix such \(\delta\). We want to show that there is an \(R>0\) and sequence \(H\ni y_i\to x\) with \[\begin{align} \label{eq:Rsequence} {\mathrm M}f(y_i) &> f^{\vee}(x)+(1-\delta)\kappa ,& {\mathrm M}f(y_i) &= {\mathrm M}_{\geq R} f(y_i) \end{align}\tag{44}\] Take \(R>0\) sufficiently small so that for all \(0<r\le R\), \[\label{eq:4532d} \frac{1}{\delta\kappa}\fint_{B(x,3r)}(f-f^{\vee}(x)-(1-2\delta)\kappa)_+<\frac{1}{45^d}.\tag{45}\] Fixed such \(r\), let \(\mathcal{B}\) be the set of balls \(B\) with center in \(B(x,r)\) with radius at most \(R\) and \(f_B>f^\vee(x)+(1-\delta)\kappa\). Then \[B(x,r)\cap\{{\mathrm M}_{<R}f >f^\vee(x)+(1-\delta)\kappa\}\subset\bigcup\mathcal{B}.\] By the \(5\)-covering lemma we find a disjoint subcover of \(\mathcal{B}\), denoted by \(\{B_j\}_{j}\), such that \(\bigcup_{j}5B_j\) still contains \(B(x,r)\cap\{{\mathrm M}f>f^\vee(x)+(1-\delta)\kappa\}\). If there were a ball \(B_j\) with radius \(r_j\ge 2r\), we would have \[\begin{align} & \frac{1}{\delta\kappa}\fint_{B(x,3r_j)}(f-f^{\vee}(x)-(1-2\delta)\kappa)_+ \\ &\qquad\ge \frac{1}{2^d\delta\kappa}\fint_{B_j}(f-f^{\vee}(x)-(1-2\delta)\kappa)_+ \ge \frac{1}{2^d}, \end{align}\] contradicting . Thus all balls \(B_j\) have radius at most \(2r\), and so \[\begin{align} & \mathcal{L}(B(x,r)\cap\{{\mathrm M}_{<R}f>f^\vee(x)+(1-\delta)\kappa\}) \le 5^d\sum_j\mathcal{L}(B_j) \\ &\qquad\le 5^d\frac{1}{\delta\kappa}\sum_j\int_{B_j}(f-f_{B_j}-(1-2\delta)\kappa)_+ \\ &\qquad\le 5^d\frac{1}{\delta\kappa}\int_{B(x,3r)}(f-f_{B_j}-(1-2\delta)\kappa)_+ <\frac{1}{3}\mathcal{L}(B(x,r)). \end{align}\] Due to we obtain . By the continuity of \({\mathrm M}_{\geq R}f\) we can conclude \[\liminf_{y\rightarrow x} {\mathrm M}f(y) \geq \liminf_{y\rightarrow x} {\mathrm M}_{\geq R}f(y) = \lim_{i\to\infty}{\mathrm M}_{\geq R} f(y_i) \ge f^{\vee}(x)+(1-\delta)\kappa.\] Letting \(\delta\to0\) we obtain \[({\mathrm M}f)^\wedge(x) \ge \liminf_{y\rightarrow x} {\mathrm M}f(y) \ge f^{\vee}(x)+\kappa = ({\mathrm M}f)^\vee(x)\] which means holds also in this case as the left hand side is zero. Finally, we can conclude the proof from and . ◻
Proof of . The estimate \(|{D}^{\mathrm{j}}{{\mathrm M}f}|\leq\frac{1}{2}|{D}^{\mathrm{j}}{f}|\) is proved in .
The second claim follows by combining and .
For the third claim, suppose that \(|{D}^{\mathrm{j}}{f}|(\Omega)=0\). Now by the first claim, the singular part \(|{D}^{\mathrm{j}}{{\mathrm M}f}|\) vanishes and so \({\mathrm M}f\in W^{1,1}_{\mathrm{loc}}(\Omega)\). Consider the \(d\):th coordinate direction; the other directions are analogous. Since \({\mathrm M}f\in W^{1,1}_{\mathrm{loc}}(\Omega)\), the function \({\mathrm M}f(z,\cdot)\) is in the class \(W^{1,1}_{\mathrm{loc}}(\Omega_z)\) for almost every \(z\in \pi(\Omega)\). By , and , \({\mathrm M}f(z,\cdot)\) is continuous for almost every \(z\in \pi(\Omega)\). Thus \({\mathrm M}f(z,\cdot)\) is absolutely continuous for almost every \(z\in \pi(\Omega)\), and in conclusion \({\mathrm M}f\) is ACL. ◻