Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces

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 Ap 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 Ap 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.