Revista Matemática Iberoamericana

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Volume 35, Issue 7, 2019, pp. 2151–2168
DOI: 10.4171/rmi/1115

Published online: 2019-08-26

Endpoint Sobolev and BV continuity for maximal operators, II

José Madrid[1]

(1) University of California Los Angeles, USA

In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both the continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto (\widetilde M_{\beta}f)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q={1}/{(1-\beta)}$. Here $\widetilde M_{\beta}$ denotes the non-centered fractional maximal operator on $\mathbb{R}$, with $\beta\in(0,1)$. The second one is related to the continuity of the discrete centered maximal operator in the space of functions of bounded variation ${\rm BV}(\mathbb{Z})$, complementing some recent boundedness results.

Keywords: Fractional maximal operator, discrete maximal operator, functions of bounded variation, Sobolev spaces

Madrid José: Endpoint Sobolev and BV continuity for maximal operators, II. Rev. Mat. Iberoam. 35 (2019), 2151-2168. doi: 10.4171/rmi/1115