包含$A_\infty$和$LlogL$常数的精确估计及其对PDE的应用
Sharp estimates involving $A_\infty$ and $LlogL$ constants, and their applications to PDE
摘要：
众所周知，反向Hölder类的并集与Muckenhoupt类$A_p$的并集重合，但权重$w$的$A_\infty$常数是其$A_p$常数的限制，不是反向Hölder类中权重的自然表征。我们引入了$RH_1$条件作为$p$趋于1时的$RH_p$不等式的一个极限情形。然后，我们显示了尖锐的限制$RH_1$常数的重量$w$在其$A_\infty$常数（从上到下）。我们还证明了尖锐版本的格林定理的情况下$p=1$，完成了回答著名的问题Bojarski在一维。我们说明我们的结果由两个直接的应用程序：椭圆型偏微分方程的狄利克雷问题。
英文摘要：
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 $RH_1$ 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.
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