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The centered maximal operator removes the non-concave Cantor part from the gradient

Panu Lahti, Julian Weigt

Abstract

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $M f(x)>f^*(x)$. 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 $M f$ can be upgraded to Sobolev regularity.

The centered maximal operator removes the non-concave Cantor part from the gradient

Abstract

We study regularity of the centered Hardy--Littlewood maximal function of a function of bounded variation in , . In particular, we show that at -a.e. point where has a non-concave blow-up, it holds that . We further deduce from this that if the variation measure of has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of can be upgraded to Sobolev regularity.

Paper Structure

This paper contains 11 sections, 29 theorems, 135 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Preliminaries
  4. Radon measures
  5. Functions of bounded variation
  6. One-dimensional sections of $\mathrm{BV}$ functions
  7. Blow-ups and nonconcavity
  8. Proof of Theorem 1.1
  9. Sobolev regularity
  10. Quasicontinuity
  11. Lusin property

Key Result

Theorem 1.1

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

Figures (1)

  • Figure 1: Neighborhood of $t'$ where ${\mathrm M}_{C(0,1)}w(t')-w^{*}(t')=\delta>0$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 1.6
  • proof
  • proof : Proof of corollary:gk24
  • Remark 1.7
  • Definition 2.1
  • ...and 46 more