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

AbstractA logarithmic Sobolev inequality with respect to the probability measure Aλ(1 − x2)λ − (12) dx on [−1, 1] is proved for all λ > −12. From this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n-sphere