Weighted fractional Bernstein’s inequalities and their applications

Feng Dai, Sergey Tikhonov

    Research output: Contribution to journalArticleResearchpeer-review

    5 Citations (Scopus)


    © 2016, Hebrew University Magnes Press. This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on Sd-1 (0.1) ǁ(-Δ0)r/2 fǁp, ω ≤ Cωnr ǁfǁp, ω for all f ЄIInd, where Πnd denotes the space of all spherical polynomials of degree at most n on Sd-1 and (-Δ0)r/2 is the fractional Laplacian-Beltrami operator on Sd-1. A new class of doubling weights with conditions weaker than the Ap condition is introduced and used to characterize completely those doubling weights w on Sd-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 < p < 8 and r > 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ Lp(Sd-1) with 0 < p < 1 can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights.
    Original languageEnglish
    Pages (from-to)33-68
    JournalJournal d'Analyse Mathematique
    Issue number1
    Publication statusPublished - 1 Jul 2016


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