Weighted fractional Bernstein’s inequalities and their applications
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- 作者：Feng Dai ; Sergey Tikhonov
- 刊名：Journal d'Analyse Math¨¦matique
- 出版年：2016
- 出版时间：July 2016
- 年：2016
- 卷：129
- 期：1
- 页码：33-68
- 全文大小：330 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis
Functional Analysis
Dynamical Systems and Ergodic Theory
Abstract Harmonic Analysis
Partial Differential Equations - 出版者：Hebrew University Magnes Press
- ISSN：1565-8538
- 卷排序：129
文摘
This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on Sd-1\(\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d\), 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.