  Let $\mu$ and $\nu$ be two probability measures on $\R^d$, where $\mu(\d x)=
\e^{-V(x)}\d x$ for some $V\in C^1(\R^d)$. Explicit sufficient conditions on
$V$ and $\nu$ are presented such that $\mu*\nu$ 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.
