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

Abstract

AbstractThe natural maximal and minimal functions commute pointwise with the logarithm on $$A_\infty $$ A ∞ . We use this observation to characterize the spaces $$A_1$$ A 1 and $$RH_\infty $$ R H ∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $$A_p$$ 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 $$A_p$$ A p and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $$A_1$$ A 1 and $$RH_\infty $$ R H ∞ , including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.