𝐿log𝐿 results for the maximal operator in variable 𝐿^{𝑝} spaces

Cruz-Uribe, D., SFO; Fiorenza, A.

doi:10.1090/S0002-9947-08-04608-4

$L\log L$ results for the maximal operator in variable $L^p$ spaces
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by D. Cruz-Uribe SFO and A. Fiorenza PDF

Trans. Amer. Math. Soc. 361 (2009), 2631-2647 Request permission

Abstract:

We generalize the classical $L\log L$ inequalities of Wiener and Stein for the Hardy-Littlewood maximal operator to variable $L^p$ spaces where the exponent function $p(\cdot )$ approaches $1$ in value. We prove a modular inequality with no assumptions on the exponent function, and a strong norm inequality if we assume the exponent function is log-Hölder continuous. As an application of our approach we give another proof of a related endpoint result due to Hästö.

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