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

TL;DR

This work develops a comprehensive framework for Bernstein- and Markov-type inequalities in $L^p$ spaces for multivariate polynomials on general compact $C^\alpha$-domains with $1\le\alpha\le 2$, focusing on tangential derivatives on the boundary. By first proving weighted tangential Bernstein inequalities on $C^2$-graph domains of special type, then transitioning to $C^\alpha$-domains via Steklov smoothing and higher-dimensional extensions, the authors obtain explicit $n$-dependent bounds and a sharp $n^{2/\alpha}$ growth rate for tangential derivatives in $L^p$, along with corresponding Markov-type estimates. These results yield asymptotically optimal Marcinkiewicz-type discretization inequalities for discretizing $L^p$ norms of polynomials, in both two and higher dimensions, under partitions with cardinality scaling as $\mathcal{O}(n^d)$ for $d$-dimensional domains. The paper also proves sharpness of the tangential Markov inequalities and provides explicit constructions (via ellipsoids and Jacobi polynomials) to demonstrate lower bounds, with open questions about logarithmic factors and endpoint smoothness $\alpha=1$. Overall, the findings clarify how boundary smoothness $\alpha$ governs the behavior of tangential derivatives and enable near-optimal sampling schemes for polynomial approximation on complex 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 $C^α$-domain with $1\leq α\leq 2$. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^p$ norms of algebraic polynomials on $C^α$-domains with asymptotically optimal number of function samples used.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm-1-1:Bern']}
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm-2-2']}