Weighted Fractional Bernstein'\''s inequalities and their applications

Abstract

This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on $ \sph$ : \begin{equation}\label{4-1-TD-ab} \|(-\Delta_0)^{r/2} f\|_{p,w}\leq C_w n^{r} \|f\|_{p,w}, \ \ \forall f\in \Pi_n^d, \end{equation} where $ \Pi_n^d$ denotes the space of all spherical polynomials of degree at most $ n$ on $ \sph$ , and $ (-\Delta_0)^{r/2}$ is the fractional Laplacian-Beltrami operator on $ \sph$ . A new class of doubling weights with conditions weaker than the $ A_p$ is introduced, and used to fully characterize those doubling weights $ w$ on $ \sph$ for which the weighted Bernstein inequality \eqref{4-1-TD-ab} holds for some $ 1\leq p\leq \infty$ and all $ r>\tau$ . In the unweighted case, it is shown that if $ 0<p<\infty$ and $ r>0$ is not an even integer, then \eqref{4-1-TD-ab} with $ w\equiv 1$ holds if and only if $ r>(d-1)(\f 1p-1)$ . As applications, we show that any function $ f\in L_p(\sph)$ 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.