Real interpolation with weighted rearrangement invariant Banach function spaces

TL;DR

This paper investigates real interpolation spaces based on weighted rearrangement invariant Banach function spaces, establishing key equivalences and extending important theorems relevant to maximal regularity and functional calculus.

Contribution

It introduces a unified approach to real interpolation in weighted rearrangement invariant Banach spaces, proving equivalence of methods and extending Dore's theorem.

Findings

Equivalence of trace and K-method in this setting

Identification of interpolation spaces between a Banach space and a sectorial operator domain

Extension of Dore's theorem on boundedness of $H^$-functional calculus

Abstract

Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the $K$-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of $H^\infty$-functional calculus to this general setting.