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

It is a well-known fact that the union $\bigcup_{p>1}RH_p$ of the Reverse Hölder classes coincides with the union $\bigcup_{p>1}A_p=A_\infty$ of the Muckenhoupt classes, 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. In the paper, the $RH_1$ condition is introduced as a limiting case of the $RH_p$ inequalities as $p$ tends to 1, and a sharp bound is found on the $RH_1$ constant of the weight $w$ in terms of its $A_\infty$ constant. Also, the sharp version of the Gehring theorem is proved for the case of $p=1$, completing the answer to the famous question of Bojarski in dimension one.
The results are illustrated by two straightforward applications to the Dirichlet problem for elliptic PDE's.
Despite the fact that the Bellman technique, which is employed to prove the main theorems, is not new, the authors believe that their results are useful and prove them in full detail.