Best constants in Sobolev inequalities for ultraspherical polynomials

Abstract

A family of sharp Sobolev-type inequalities for functions on the classical measure spaces associated with the ultraspherical or Gegenbauer polynomials is obtained. These estimates generalize the Sobolev inequalities for the n-sphere S n given by Beckner, and are derived from a sharp Sobolev inequality for functions on the real line. Spectral considerations allow these estimates to be expressed as multiplier inequalities for functions which have expansions in terms of Gegenbauer polynomials.