  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.
