Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere

Abstract

A logarithmic Sobolev inequality with respect to the probability measure $\text{A}{}_{\mathrm{\lambda }}\text{(1 − x}{}^2\text{)}{}^{\mathrm{<mtext>\lambda \; -\; (</mtext><mtext>1</mtext><mtext>2</mtext><mtext>)</mtext>}}\text{dx}$ on [−1, 1] is proved for all $\text{λ > −}\text{1}\text{2}$. From this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n-sphere.