Reverse hölders inequality for spherical harmonics

Abstract

This paper determines the sharp asymptotic order of the following reverse Holder inequality for spherical harmonics Yn of degree n on the unit sphere Sd-1 of Rd as n→∞: ║Yn║Lq(sd-1)≤Cnα(p,q)║Yn║Lp(Sd-1), 0 <p<q ≤∞. In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nikolskii inequality for spherical polynomials. Furthermore, they allow us to improve two recent results on the restriction conjecture and the sharp Pitt inequalities for the Fourier transform on Rd. © 2015 American Mathematical Society.