We prove a new Bernstein type inequality in $L^p$ spaces associated with the tangential derivatives on the boundary of a general compact $C^2$-domain. We give two applications: Marcinkiewicz type inequality for discretization of $L^p$ norm and positive cubature formula. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.

翻译：我们证明伯恩斯泰因式的不平等是新的,与一般契约$C$2$-域边界上的相近衍生物相关,以美元为单位。我们给出了两种应用:马克辛基维茨式的不平等(Marcinkiewicz型),以美元为单位分解标准值和正的幼稚公式。 这两种结果都是最佳的,因为所使用的功能样本数量与代数多元体相应空间的大小相近。