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

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 $L_{\mathrm{loc}}^p(w,\mathbb{R})$ 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 $L_w^p(\mathbb{R})$ 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.

34 pages, 3 pages of appendices. Clarified statement in abstract