Muckenhoupt inequality with three measures and applications to Sobolev orthogonal polynomials

TL;DR

This paper extends the Muckenhoupt inequality to three measures, providing new insights into the boundedness and asymptotic behavior of Sobolev orthogonal polynomials for a broad class of measures.

Contribution

It introduces a generalized Muckenhoupt inequality with three measures and characterizes the boundedness of the multiplication operator in this context.

Findings

Characterization of the boundedness of the multiplication operator

Conditions for the zeros and asymptotic behavior of Sobolev orthogonal polynomials

Applicability to common measure examples in the literature

Abstract

We generalize the classical Muckenhoupt inequality with two measures to three under appropriate conditions. As a consequence, we prove a simple characterization of the undedness of the multiplication operator and thus of the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes the most usual examples in the literature.