Mutual estimates for the dyadic Reverse H\"{o}lder and Muckenhoupt constants for the dyadically doubling weights

Abstract

Muckenhoupt and Reverse H\"{o}lder classes of weights play an important role in harmonic analysis, PDE's and quasiconformal mappings. In 1974 Coifman and Fefferman showed that a weight belongs to a Muckenhoupt class $A_p$ for some $1<p<\infty$ if and only if it belongs to a Reverse H\"{o}lder class $RH_q$ for some $1<q<\infty$. In 2009 Vasyunin found the exact dependence between $p$, $q$ and the corresponding characteristic of the weight using the Bellman function method. The result of Coifman and Fefferman works for the dyadic classes of weights under an additional assumption that the weights are dyadically doubling. We extend the Vasyunin's result to the dyadic Reverse H\"{o}lder and Muckenhoupt classes and obtain the dependence between $p$, $q$, the doubling constant and the corresponding characteristic of the weight. More precisely, given a dyadically doubling weight in $RH_p^d$ on a given dyadic interval $I$, we find an upper estimate on the average of the function $w^{q}$ over the interval $I$. From the bound on this average we can conclude, for example, that $w$ belongs to the corresponding $A_{s_1}^d$ class or that $w^p$ is in $A_{s_2}^d$ for some values of $s_i$. We obtain our results using the method of Bellman functions.