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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
DOI: https://doi.org/10.1090/proc/12986
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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  • [1] Feng Dai and Yuan Xu, Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3060033
  • [2] F. Dai and Z. Ditzian, Combinations of multivariate averages, J. Approx. Theory 131 (2004), no. 2, 268-283. MR 2106541 (2006h:41012), https://doi.org/10.1016/j.jat.2004.10.003
  • [3] L. De Carli, D. Gorbachev, and S. Tikhonov, Pitt and Boas inequalities for Fourier and Hankel transforms, J. Math. Anal. Appl. 408 (2013), no. 2, 762-774. MR 3085070, https://doi.org/10.1016/j.jmaa.2013.06.045
  • [4] Laura de Carli and Loukas Grafakos, On the restriction conjecture, Michigan Math. J. 52 (2004), no. 1, 163-180. MR 2043403 (2004k:42013), https://doi.org/10.1307/mmj/1080837741
  • [5] Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635 (95f:41001)
  • [6] Javier Duoandikoetxea, Reverse Hölder inequalities for spherical harmonics, Proc. Amer. Math. Soc. 101 (1987), no. 3, 487-491. MR 908654 (88m:42040), https://doi.org/10.2307/2046394
  • [7] A. I. Kamzolov, Approximation of functions on the sphere $ S^n$, Serdica 10 (1984), no. 1, 3-10 (Russian). MR 764160 (85j:41049)
  • [8] Christopher D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43-65. MR 835795 (87g:42026), https://doi.org/10.1215/S0012-7094-86-05303-2
  • [9] Robert J. Stanton and Alan Weinstein, On the $ L^{4}$ norm of spherical harmonics, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 343-358. MR 600249 (82d:58015), https://doi.org/10.1017/S0305004100058229
  • [10] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals., Princeton Mathematical Series, vol. 43, With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [11] Gábor Szegő, Orthogonal polynomials. American Mathematical Society Colloquium Publications, Vol. 23, 3rd ed., American Mathematical Society, Providence, R.I., 1967. MR 0310533 (46 #9631)

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Additional Information

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: fdai@ualberta.ca

Han Feng
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
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
Email: stikhonov@crm.cat

DOI: https://doi.org/10.1090/proc/12986
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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