Functional Inequalities for Convolution Probability Measures

Let $\mu$ and $\nu$ be two probability measures on ${\mathbb{R}}^d$, where $\mu(\d x)= \e^{-V(x)}\d x$forsome$V\in C^1(\R^d)$\mathrm{.}Explicitsufficientconditionson$V$and$\nu$arepresentedsuchthat$\mu*\nu$ satisfies the log-Sobolev, Poincar'e and super Poincar'e inequalities. In particular, the recent results on the log-Sobolev inequality derived in \cite{Z} for convolutions of the Gaussian measure and compactly supported probability measures are improved and extended.