Reverse H\"older's inequality for spherical harmonics

TL;DR

This paper establishes the precise asymptotic behavior of reverse H"older inequalities for spherical harmonics, revealing sharper bounds than classical polynomial estimates and improving results related to the restriction conjecture and Pitt inequalities.

Contribution

It provides the sharp asymptotic order of reverse H"older inequalities for spherical harmonics, surpassing previous polynomial bounds and enhancing related Fourier analysis results.

Findings

Sharp asymptotic order of reverse H"older inequalities for spherical harmonics.

Improved bounds over classical Nilkolskii inequalities for spherical polynomials.

Enhanced results on the restriction conjecture and Pitt inequalities.

Abstract

This paper determines the sharp asymptotic order of the following reverse H\"older inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$: \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C n^{\alpha(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nilkolskii 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 $\mathbb{R}^d$.