中文摘要：

研究了一维Cauchy分布的加权Poincare不等式和加权log-Sobolev不等式．我们给出并证明了所给权函数的最优性，同时对不等式中的常数进行了阶的估计．

英文摘要：

We study the weighted Poincare inequality and log-Sobolev inequality for one dimensional Cauchy distribution. We offer the optimal weighted functions, prove the sharpness, and estimate the orders of the constants in the inequalities.