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

It is a well known fact that the union of the Reverse H\"{o}lder 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 H\"{o}lder classes. We introduce the $RH_1$ condition as a limiting case of the $RH_p$ inequalities as $p$ tends to 1. Then we show sharp bound on $RH_1$ 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.