Endpoint regularity of the discrete multisublinear fractional maximal operators
type: Item
instance of: scholarly article
title: Endpoint regularity of the discrete multisublinear fractional maximal operators
zbMATH Open document ID: 1472.42035
author: Xiao Zhang
published in: Results in Mathematics
publication date: 2021-07-07T00:00:00Z
review text: The main results of this article are about the discrete centered and uncentered \(m\)-sublinear fractional maximal operators, \( \mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \) (respectively). First, a theorem on endpoint regularity is obtained for \( \widetilde{ \mathfrak{M} }_\alpha \) from the \(m\)-fold Cartesian product of \( \text{BV}(\mathbb{Z}) \) (\(0 \leq \alpha < 1 \)) or \( \ell^1 (\mathbb{Z}) \) (\( m-1 \leq \alpha < m \)) into the space of functions with bounded \(q\)-variation, where \(q\) depends on the subscript \(\alpha\). Here \( \text{BV}(\mathbb{Z}) \) denotes the space of functions of bounded variation defined on \(\mathbb{Z}\). After that, a theorem showing the boundedness of \(\mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \), \( 0 \leq \alpha < m \), from the \(m\)-fold Cartesian product of \( \ell^1 (\mathbb{Z}) \) into \( \text{BV}(\mathbb{Z}) \) is obtained. Both theorems can be considered as the discrete version of the corresponding results contained in [\textit{F. Liu} and \textit{H. Wu}, Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and also extend the corresponding already known results of \textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] and \textit{F. Liu} [Bull. Aust. Math. Soc. 95, No. 1, 108--120 (2017; Zbl 1364.42020)].
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 7369089
zbMATH Keywords: discrete multisublinear fractional maximal operator
zbMATH Keywords: Sobolev space
zbMATH Keywords: bounded variation
zbMATH Keywords: boundedness and continuity
reviewed by: Guillermo J. Flores
MaRDI profile type: Publication
full work available at URL: s00025-021-01387-5
OpenAlex ID: W3140999012
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DOI: 10.1007/S00025-021-01387-5
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description: scientific article; zbMATH DE number 7369089