On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^\alpha$-domains

TL;DR

This paper establishes new Bernstein, Markov, and Marcinkiewicz inequalities for polynomial approximation on general smooth domains, providing optimal discretization methods in $L^p$ spaces.

Contribution

It introduces novel Bernstein and Markov inequalities on $C^eta$-domains and applies them to derive optimal Marcinkiewicz inequalities for polynomial norm discretization.

Findings

Derived Bernstein and Markov inequalities for $C^eta$-domains.

Established Marcinkiewicz inequalities with asymptotically optimal sampling.

Extended classical inequalities to more general multivariate domains.

Abstract

We prove new Bernstein and Markov type inequalities in $L^p$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^\alpha$-domain with $1\leq \alpha\leq 2$. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^p$ norms of algebraic polynomials on $C^\alpha$-domains with asymptotically optimal number of function samples used.