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摘要
Let tot and nu be two probability measures on R-d, where mu(dx) = e(-v(x))dx/integral R-d e(-v(x))dx for some V epsilon C-1 (R-d). Explicit sufficient conditions on V and nu are presented such that mu*nu satisfies the log-Sobolev, Poincare and super Poincare inequalities. In particular, if V(x) = lambda vertical bar x vertical bar(2) for some A, > 0 and nu(e(lambda theta vertical bar center dot vertical bar 2)) < infinity for some theta > 1, then mu*nu satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in (J. Funct. Anal. 265 (2013) 1064-1083) for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for mu*nu implies nu(e(epsilon vertical bar center dot vertical bar 2)) < infinity for some epsilon > 0.
