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arXiv · 1408.1877

Reverse H\"older's inequality for spherical harmonics

Feng Dai·Han Feng·Sergey Tikhonov
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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 $\mathbb{R}^d$.

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Feng Dai, Han Feng, Sergey Tikhonov. 2014-08-06. Reverse H\"older's inequality for spherical harmonics. https://arxiv.org/abs/1408.1877

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