Abstract
The natural maximal and minimal functions commute pointwise with the logarithm on A_infty . We use this observation to characterize the spaces A_1 and RH_infty 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 A_1 and RH_infty , including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.
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