Reverse Hölder’s inequality for spherical harmonics

Dai, Feng; Feng, Han; Tikhonov, Sergey

doi:10.1090/proc/12986

Reverse Hölder’s inequality for spherical harmonics
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by Feng Dai, Han Feng and Sergey Tikhonov PDF

Proc. Amer. Math. Soc. 144 (2016), 1041-1051 Request permission

Abstract:

This paper determines the sharp asymptotic order of the following reverse Hölder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb {S}^{d-1}$ of $\mathbb {R}^d$ as $n\to \infty$: \[ \|Y_n\|_{L^q(\mathbb {S}^{d-1})}\leq C n^{\alpha (p,q)}\|Y_n\|_{L^p(\mathbb {S}^{d-1})},\ 0<p<q\leq \infty .\] 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 $\mathbb {R}^d$.

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