Endpoint Sobolev and BV continuity for maximal operators

Abstract

In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\tilde{M}$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f\mapsto (\tilde{M}f{)}^{\prime }$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\tilde{M}:BV(\mathbb{Z})\to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.

24 pages. To appear in J. Funct. Anal

References in corpus (1)

• Derivative bounds for fractional maximal functions

Cited by in corpus (5)

• Double phase image restoration
• Weak differentiability for fractional maximal functions of general $L^p$ functions on domains
• Regularity of maximal operators: recent progress and some open problems
• Sobolev contractivity of gradient flow maximal functions
• Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces