Curvature dimension bounds on the deltoid model

TL;DR

This paper investigates curvature-dimension inequalities for diffusion operators on the deltoid domain, deriving bounds on orthogonal polynomials and Sobolev inequalities, extending classical Jacobi operator results to a two-dimensional setting.

Contribution

It introduces curvature-dimension bounds for the deltoid model, a novel two-dimensional extension of Jacobi operators, and derives related polynomial bounds and Sobolev inequalities.

Findings

Derived curvature-dimension inequalities for the deltoid model

Established bounds on associated orthogonal polynomials

Proved Sobolev inequalities for the Dirichlet forms

Abstract

The deltoid curve in R 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvec-tors of diffusion operators. As such, they may be considered as a two dimensional extension of the classical Jacobi operators. They belong to one of the 11 families of such bounded domains in R 2. We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet forms