Abstract
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 Π d n 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 A p 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 ∈ L p (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.
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The first author was partially supported by the NSERC Canada under grant RGPIN 04702 Dai.
The second author was partially supported by by MTM2014-59174-P and 014-SGR-289.
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Dai, F., Tikhonov, S. Weighted fractional Bernstein’s inequalities and their applications. JAMA 129, 33–68 (2016). https://doi.org/10.1007/s11854-016-0014-z
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DOI: https://doi.org/10.1007/s11854-016-0014-z