On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions

In the line of investigation of the works by D. Bakry and M. Emery (Lecture Notes in Math., Vol. 1123, pp. 175–206, Springer-Verlag, New York/Berlin 1985) and O. S. Rothaus ( J. Funct. Anal. 42 (1981) 102–109; J. Funct. Anal. 65 (1986) 358–367), we study an integral inequality behind the “Γ2 criterion” of D. Bakry and M. Emery (see previous reference) and its applications to hypercontractivity of diffusion semigroups. With, in particular, a short proof of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup, our exposition unifies in a simple way several previous results, interpolating smoothly from the spectral gap inequalities to logarithmic Sobolev inequalities and even true Sobolev inequalities. We examine simultaneously the extremal functions for hypercontractivity and logarithmic Sobolev inequalities of the Ornstein-Uhlenbeck semigroup and heat semigroup on spheres.