Sharp Bernstein inequalities on simplex

TL;DR

This paper establishes new Bernstein inequalities on the simplex, including sharp $L^2$ inequalities with Jacobi weights and $L^p$ inequalities for doubling weights, expanding the theoretical understanding of polynomial bounds.

Contribution

It introduces novel Bernstein inequalities on the simplex, utilizing spectral operators and covering both $L^2$ and $L^p$ norms with different weight conditions.

Findings

Derived sharp $L^2$ Bernstein inequalities with Jacobi weights.

Established $L^p$ Bernstein inequalities for doubling weights.

Demonstrated that the two types of inequalities are not necessarily special cases of each other.

Abstract

We prove several new families of Bernstein inequalities of two types on the simplex. The first type consists of inequalities in $L^2$ norm for the Jacobi weight, some of which are sharp, and they are established via the spectral operator that has orthogonal polynomials as eigenfunctions. The second type consists of inequalities in $L^p$ norm for doubling weight on the simplex. The first type is not necessarily a special case of the second type when $d \ge 3$.