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

It is a well-known fact that the union <!-- MATH $\bigcup _{p>1} RH_p$ --> 1} RH_p$"> of the Reverse Hölder classes coincides with the union <!-- MATH $\bigcup _{p>1} A_p = A_\infty$ --> 1} A_p = A_\infty $"> of the Muckenhoupt classes, but the constant of the weight , which is a limit of its constants, is not a natural characterization for the weight in Reverse Hölder classes. In the paper, the condition is introduced as a limiting case of the inequalities as tends to , and a sharp bound is found on the constant of the weight in terms of its constant. Also, the sharp version of the Gehring theorem is proved for the case of , completing the answer to the famous question of Bojarski in dimension one.