Logarithmic Sobolev inequalities for the heat-diffusion semigroup

An explicit formula relating the Hermite semigroup e~'H on R with Gauss measure and the heat-diffusion semigroup e'A on R with Lebesgue measure is proved. From this formula it follows that Nelson's hypercontractive estimates for e~'H are equivalent to the best norm estimates for e'A as a map Lq(R) into LP(R), 1 < q <p < oo. Furthermore, the inequality |iogw;<¿iog q2 Re<-A<J.,.7»<¡>> 2-nnei.q — 1) + M*!!«. where 1 < q < oo, /*<(> = (sgn$)|$|*-1, and the norms and sesquilinear form <, } are taken with respect to Lebesgue measure on R", is shown to be equivalent to the best norm estimates for e'4 as a map from Z, *(/?") into Lp(R"). This inequality is analogous to Gross' logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.