Two-point inequalities,the Hermite semigroup and the Gauss-Weierstrass semigroup

Let e^?zH, Re z ? 0, be the Hermite semigroup on R with Gauss measure μ. Necessary and sufficient conditions for e^?zH to be a bounded map from L^p(μ) into L^q(μ), 1 ? p, q ? ∞, are found and in many cases it is proved that e^?zH: L^p(μ) → L^q(μ) is in fact a contraction. Furthermore, these results and a formula relating the Hermite semigroup with the Gauss-Weierstrass semigroup e^zΔ enable one to calculate the precise norm of e^zΔ:L^p(dx) → L^q(dx) in a large number of cases.