Calculus of Variations and Geometric Measure Theory
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D. Cruz-Uribe - G. Di Fratta - A. Fiorenza

Modular inequalities for the maximal operator in variable Lebesgue spaces

created by difratta on 30 Apr 2021

[BibTeX]

preprint

Inserted: 30 apr 2021

Year: 2017

ArXiv: 1710.05217 PDF

Abstract:

A now classical result in the theory of variable Lebesgue spaces due to Lerner A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543 is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\cdot)}(\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \[ \int_\Omega Mf(x)^{p(x)}\,dx \ \leq c_1 \int_\Omega
f(x)
^{q(x)}\,dx + c_2, \] where $c_1,\,c_2$ are non-negative constants and $\Omega$ is any measurable subset of $\mathbb{R}^n$. As a corollary we get sufficient conditions for the modular inequality \[ \int_\Omega
Tf(x)
^{p(x)}\,dx \ \leq c_1 \int_\Omega
f(x)
^{q(x)}\,dx + c_2, \] where $T$ is any operator that is bounded on $L^p(\Omega)$, $1<p<\infty$.

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