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arXiv · 2406.03382

Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type

Alina Shalukhina
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Abstract

We prove the self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hyt\"onen--Kairema dyadic cubes and studying the maximal operator alongside its dyadic version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.

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Alina Shalukhina. 2024-06-05. Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type. https://arxiv.org/abs/2406.03382

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