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On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded

Research output: Contribution to journalArticlepeer-review

Eugene Shargorodsky, Lars Diening, Alexei Karlovich

Original languageEnglish
Pages (from-to)347-352
Number of pages6
JournalGeorgian Mathematical Journal
Volume29
Issue number3
Early online date26 Mar 2022
DOIs
Accepted/In press10 Jan 2022
E-pub ahead of print26 Mar 2022
Published11 Jun 2022

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Publisher Copyright: © 2022 Walter de Gruyter GmbH, Berlin/Boston.

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Abstract

We show that if the Hardy-Littewood maximal operator M is bounded on a reflexive variable exponent space Lp(·) (ℝd), then for every q ϵ (1, ∞), the exponent p(·) admits, for all sufficiently small θ > 0, the representation 1/p(x) = θ/q + 1 - θ/ r(x), x ϵ ℝd, such that the operator M is bounded on the variable Lebesgue space Lr(·) (ℝd). This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.

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