A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type

Oleksiy Karlovych; Alina Shalukhina

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arXiv · 2403.10915

A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type

Oleksiy Karlovych·Alina Shalukhina

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Abstract

Let $(X,d,\mu)$ be a space of homogeneous type and $p(\cdot):X\to[1,\infty]$ be a variable exponent. We show that if the measure $\mu$ is Borel-semiregular and reverse doubling, then the condition ${\rm ess\,inf}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot)}(X,d,\mu)$.

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Oleksiy Karlovych, Alina Shalukhina. 2024-03-16. A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type. https://arxiv.org/abs/2403.10915

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