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 $\mathbb {R}^d$.References
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Additional Information
- Feng Dai
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 660750
- Email: fdai@ualberta.ca
- Han Feng
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 997105
- Email: hfeng3@ualberta.ca
- Sergey Tikhonov
- Affiliation: ICREA, Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C08193 Bellaterra (Barcelona), Spain – and – Universitat Autònoma de Barcelona
- MR Author ID: 706641
- Email: stikhonov@crm.cat
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1041-1051
- MSC (2010): Primary 33C55, 33C50, 42B15, 42C10
- DOI: https://doi.org/10.1090/proc/12986
- MathSciNet review: 3447658