Riesz transforms and fractional integration for orthogonal expansions on spheres, balls and simplices

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)549-614
Journal / PublicationAdvances in Mathematics
Online published5 Jul 2016
Publication statusPublished - 1 Oct 2016
Externally publishedYes


This paper studies the Hardy–Littlewood–Sobolev (HLS) inequality and the Riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres, balls and simplexes with weights being invariant under a general finite reflection group on ℝd. The sharp index for the validity of the HLS inequality is determined and the Lp-boundedness of the Riesz transforms is established. In particular, our results extend a classical inequality of Muckenhoupt and Stein on conjugate ultraspherical polynomial expansions. Our idea is based on a new decomposition of the Dunkl–Laplace–Beltrami operator on the sphere and some sharp asymptotic estimates of the weighted Christoffel functions.

Research Area(s)

  • Christoffel functions, Dunkl operators, Fractional integration, Hardy–Littlewood–Sobolev inequality, Riesz transforms, Spherical h-harmonics