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Reverse Hölder's inequality for spherical harmonics

Authors: Feng Dai, Han Feng and Sergey Tikhonov
Journal: Proc. Amer. Math. Soc. 144 (2016), 1041-1051
MSC (2010): Primary 33C55, 33C50, 42B15, 42C10
Published electronically: November 20, 2015
MathSciNet review: 3447658
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Abstract | References | Similar Articles | Additional Information

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 $:

$\displaystyle \Vert Y_n\Vert _{L^q(\mathbb{S}^{d-1})}\leq C n^{\alpha (p,q)}\Vert Y_n\Vert _{L^p(\mathbb{S}^{d-1})},\ 0<p<q\leq \infty .$

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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Additional Information

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Han Feng
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Sergey Tikhonov
Affiliation: ICREA, Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C08193 Bellaterra (Barcelona), Spain – and – Universitat Autònoma de Barcelona

Keywords: Spherical harmonics, polynomial inequalities, restriction theorems
Received by editor(s): August 7, 2014
Received by editor(s) in revised form: December 7, 2014
Published electronically: November 20, 2015
Additional Notes: The first and the second authors were partially supported by the NSERC Canada under grant RGPIN 04702 Dai
The third author was partially supported by MTM 2014-59174-P, 2014 SGR 289, and by the Alexander von Humboldt Foundation
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society

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