Weighted Fractional Bernstein's inequalities and their applications

TL;DR

This paper establishes weighted fractional Bernstein inequalities for spherical polynomials, characterizes weights for which these inequalities hold, and applies results to approximation theory and Sobolev embeddings on the sphere.

Contribution

It introduces a new class of doubling weights with weaker conditions than $A_p$, fully characterizes weights for weighted Bernstein inequalities, and applies these to approximation and embedding theorems.

Findings

Characterization of weights satisfying weighted Bernstein inequalities.

Extension of inequalities to fractional Laplacian operators on the sphere.

Applications to approximation of functions in $L_p$ spaces with $0<p<1$.

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 Vall\'ee 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.