JournalsrmiVol. 35, No. 7pp. 2151–2168

Endpoint Sobolev and BV continuity for maximal operators, II

  • José Madrid

    University of California Los Angeles, USA
Endpoint Sobolev and BV continuity for maximal operators, II cover
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Abstract

In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space W1,1W^{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(M~βf)f \mapsto (\widetilde M_{\beta}f)' from W1,1(R)W^{1,1}(\mathbb{R}) to Lq(R)L^q(\mathbb{R}), for q=1/(1β)q={1}/{(1-\beta)}. Here M~β\widetilde M_{\beta} denotes the non-centered fractional maximal operator on R\mathbb{R}, with β(0,1)\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 BV(Z){\rm BV}(\mathbb{Z}), complementing some recent boundedness results.

Cite this article

José Madrid, Endpoint Sobolev and BV continuity for maximal operators, II. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2151–2168

DOI 10.4171/RMI/1115