Bernstein inequality on parabolic domains

TL;DR

This paper establishes sharp Bernstein inequalities on weighted L^2 spaces over various parabolic domains using a second-order differential operator linked to orthogonal polynomials.

Contribution

It introduces new Bernstein inequalities on parabolic domains with a novel approach involving differential operators and orthogonal polynomials.

Findings

Sharp Bernstein inequalities are proved for weighted L^2 spaces on parabolic domains.

The method employs a second-order differential operator related to orthogonal polynomial bases.

Results include inequalities on both bounded and unbounded parabolic surfaces.

Abstract

Several families of sharp Bernstein inequalities are established on the weighted $L^2$ space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted $L^2$ space.