Sharp estimates involving $A_{\infty }$ and $LlogL$ constants, and their applications to PDE

Abstract

It is a well known fact that the union of the Reverse Hölder classes coincides with the union of the Muckenhoupt classes $A_p$, but the $A_{\infty }$ constant of the weight $w$, which is a limit of its $A_p$ constants, is not a natural characterization for the weight in Reverse Hölder classes. We introduce the $R{H}_1$ condition as a limiting case of the $R{H}_p$ inequalities as $p$ tends to 1. Then we show sharp bound on $R{H}_1$ constant of the weight $w$ in terms of its $A_{\infty }$ constant (from above and from below). We also prove the sharp version of the Gehring theorem for the case $p=1$, completing the answer to the famous question of Bojarski in dimension one. We illustrate our results by two straight-forward applications: to the Dirichlet problem for elliptic PDE's.