The Steklov problem and Remainder Estimates for Krein Systems generated by a Muckenhoupt weight

TL;DR

This paper investigates solutions to Krein systems generated by Muckenhoupt weights, proving their boundedness in certain function spaces and introducing a novel remainder estimate that highlights unique features of Krein systems compared to orthogonal polynomials.

Contribution

It establishes boundedness of Krein system solutions for weights close to 1 in L^p spaces and introduces a new remainder estimate specific to Krein systems.

Findings

Solutions are uniformly bounded in L^p_loc(w, R) for weights close to 1.

A new remainder measure between Krein system solutions and polynomial-like approximants is defined.

Remainder estimates are provided in weighted L^p spaces for certain Muckenhoupt weights.

Abstract

We show that solutions to Krein systems, the continuous frequency analogue of orthogonal polynomials on the unit circle, generated by an $A_2 (\mathbb{R})$ weight $w$ satisfying $w-1 \in L^1 (\mathbb{R}) + L^2 (\mathbb{R})$, are uniformly bounded in $L^p_{\mathrm{loc}} (w, \mathbb{R})$ for $p$ sufficiently close to $2$. This provides a positive answer to the Steklov problem for Krein systems. Furthermore, we define a "remainder" which measures the difference between the solution to a Krein system and a polynomial-like approximant, and we estimate these remainders in $L^p_w (\mathbb{R})$ for $w \in A_2 (\mathbb{R})$ satisfying some additional conditions. Such polynomial-like approximants, and hence remainder estimates, seem unique to Krein systems, with no analogue for orthogonal polynomials on the unit circle.