A class of subspaces of Morrey spaces and norm inequalities on Riesz potential operators
type: Item
instance of: scholarly article
title: A class of subspaces of Morrey spaces and norm inequalities on Riesz potential operators
zbMATH Open document ID: 1329.42021
author: Ibrahim Fofana
author: Falé Roland Faléa
author: Bérenger Akon Kpata
published in: Afrika Matematika
publication date: 2015-09-23T00:00:00Z
review text: The Riesz potential \(I_{\gamma}\) defined by \[ I_{\gamma}f(x) = \int \limits _{\mathbb{R}^{d}} |x-y|^{d(\gamma-1)}f(y)dy, \] for \(f \in C_{c}(\mathbb{R}^{d})\), \(x \in \mathbb{R}^{d}\), and \(\gamma \in (0,1)\), is known to map \(L^{p}\) to \(L^{p^{*}}\) when \(p \in [1,\frac{1}{\gamma})\) and \(\frac{1}{p^{*}}=\frac{1}{p}-\gamma\). The paper under review deals with refinements of this result, aimed at describing the space \(B=\{f \in L^{0}(\mathbb{R}^{d}))\;;\; I_{\gamma}(|f|) \in L^{p^{*}}\}\). This space contains \(L^{p}\), but only as a proper subspace (a counterexample is constructed in Proposition 1.14). To describe \(B\), the authors introduce a space \(F\) of measurable functions \(f\) such that \[ \sum \limits _{Q \in S}(|Q|^{-\frac{1}{p'}}\|f.1_{Q}\|_{1})^{p^{*}} \] is finite for all countable partitions \(S\) of \(\mathbb{R}^{d}\) by disjoint cubes. Their main result gives a description of \(B\) that is close to optimal: \(B\) is contained in the closure of \(L^{p^{*}}\) in \(F\), and, for \(f\) in that closure, \(I_{\gamma}(f) \in L^{p^{*},\infty}\). To shed some light on the nature of \(F\), the authors also compare their space to more classical function spaces used in harmonic analysis. They show, for instance, that \(F\) is a subspace of the Herz space of measurable functions such that \[ \sum \limits _{n \in \mathbb{Z}} 2^{-\frac{dnp^{*}}{p'}}\|f.1_{C_{n}}\|_{1}^{p^{*}} \] is finite, where \(C_{n} = \{x \in \mathbb{R}^{d} \;;\; 2^{n}<|x|\leq 2^{n+1}\}\). They also compare \(F\) to certain amalgam spaces introduced by the first author in an earlier work. In the second half of the paper, they generalise their results to analogous spaces of measures \(\mu\), and to the corresponding extension of the Riesz potential defined by \( I_{\gamma}\mu(x) = \int \limits _{\mathbb{R}^{d}} |x-y|^{d(\gamma-1)}d\mu(y) \).
reviewed by: Pierre Portal
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 6484524
zbMATH Keywords: Morrey spaces
zbMATH Keywords: Riesz potential operators
zbMATH Keywords: Herz space
zbMATH Keywords: amalgams spaces
zbMATH Keywords: fractional maximal operator
zbMATH Keywords: Radon measure
MaRDI profile type: Publication
full work available at URL: s13370-014-0241-3
OpenAlex ID: W2083836962
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DOI: 10.1007/S13370-014-0241-3
description: scientific article; zbMATH DE number 6484524