On Bernstein inequalities on the unit ball

TL;DR

This work advances weighted Bernstein inequalities on the unit ball by proving strong $L^p$ bounds for doubling weights through the novel $\Phi_i$-type refinements, and by establishing sharp $L^2$ bounds for Jacobi weights via a new self-adjoint spectral formulation. The authors leverage a ball–simplex correspondence built from the map $x \mapsto (x_1^2,\dots,x_d^2)$ and the associated extremal function theory to derive improved inequalities that unify and extend previous simplex results. They provide detailed proofs, including a parity-aware $L^2$ theory and a second, independent decomposition of the spectral operator that yields further sharp Bernstein-type estimates. Overall, the paper deepens our understanding of polynomial approximation on the ball under general doubling and Jacobi weights, with connections to plurisubharmonic potential theory and orthogonal polynomials in several variables.

Abstract

Two types of Bernstein inequalities are established on the unit ball in $\mathbb{R}^d$, which are stronger than those known in the literature. The first type consists of inequalities in $L^p$ norm for a fully symmetric doubling weight on the unit ball. The second type consists of sharp inequalities in $L^2$ norm for the Jacobi weight, which are established via a new self-adjoint form of the spectral operator that has orthogonal polynomials as eigenfunctions.

Theorems & Definitions (17)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma 3.2
• Theorem 4.1