Continuity of weighted operators, Muckenhoupt $A_p$ weights, and Steklov problem for orthogonal polynomials

TL;DR

This paper studies the continuity of weighted operators with Muckenhoupt weights and applies these results to estimate norms of orthogonal polynomials on the unit circle, deriving asymptotics of polynomial entropy.

Contribution

It establishes the continuous dependence of weighted operators on Muckenhoupt weights and applies this to orthogonal polynomials and entropy asymptotics.

Findings

Weighted operators depend continuously on $A_p$ weights.

Estimated $L^p_w$ norms of orthogonal polynomials on the circle.

Derived asymptotics for polynomial entropy.

Abstract

We consider weighted operators acting on $L^p(\mathbb{R}^d)$ and show that they depend continuously on the weight $w\in A_p(\mathbb{R}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w(\mathbb{T})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2(\mathbb{T})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.