Limiting conditions of Muckenhoupt and reverse Hölder classes on metric measure spaces

2022/04/04 by Emma-Karoliina Kurki, Kurki, Emma-Karoliina
Mathematics · #Advanced Harmonic Analysis Research #Advanced Banach Space Theory #Fixed Point Theorems Analysis

paper · pdf · doi:10.48550/arxiv.2204.01441

Abstract

The natural maximal and minimal functions commute pointwise with the logarithm on A_∞. We use this observation to characterize the spaces A₁ and RH_∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt A_p and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between A_p and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A₁ and RH_∞, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.

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