  The aim of this paper is to establish various functional inequalities for the
convolution of a compactly supported measure and a standard Gaussian
distribution on Rd. We especially focus on getting good dependence of the
constants on the dimension. We prove that the Poincar{\'e} inequality holds
with a dimension-free bound. For the logarithmic Sobolev inequality, we improve
the best known results (Zimmermann, JFA 2013) by getting a bound that grows
linearly with the dimension. We also establish transport-entropy inequalities
for various transport costs.

Probability;Functional Analysis