On Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-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 $1\le \alpha \le 2$. 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.