$L^p$-Bernstein inequalities on $C^2$-domains and applications to discretization

TL;DR

This work develops $L^p$-Bernstein inequalities for polynomials on general compact $C^2$-domains in ${\mathbb R^d}$, focusing on tangential boundary derivatives. The authors introduce a decomposition of any such domain into domains of special type attached to the boundary and a tractable boundary metric, enabling reduction to 2D-type estimates and then extension to higher dimensions and general domains. The resulting Bernstein bounds underpin Marcinkiewicz-type discretization inequalities and the construction of positive cubature formulas with sampling counts that match the polynomial space dimension, achieving optimal order. These tools advance discretization of $L^p$ norms and stable numerical integration for multivariate polynomial approximation on general geometric domains. Key techniques include parabolic curve representations near the boundary, a special-type domain framework, and meticulous multi-scale partitioning arguments, all tied together by a robust boundary metric equivalence.

Abstract

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.

Theorems & Definitions (61)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10