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We prove new Bernstein and Markov type inequalities in $L^p$ spacesassociated with the normal and the tangential derivatives on the boundary of ageneral compact $C^\alpha$-domain with $1\leq \alpha\leq 2$. These estimatesare also applied to establish Marcinkiewicz type inequalities fordiscretization of $L^p$ norms of algebraic polynomials on $C^\alpha$-domainswith asymptotically optimal number of function samples used.

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