Interpolation of classical Lorentz spaces measuring oscillation

TL;DR

This paper provides an explicit characterization of the $K$-functional for pairs of weighted classical Lorentz spaces of type $S$, using a novel method involving fundamental functions, with applications to Lebesgue and Lorentz spaces.

Contribution

It introduces a new method to explicitly characterize the $K$-functional for Lorentz spaces of type $S$, simplifying the analysis through relations with $\\Lambda$-type spaces and fundamental functions.

Findings

Derived a formula for the $K$-functional of Lebesgue and Lorentz spaces with power weights.

Established a reverse Marchaud inequality using the new $K$-functional characterization.

Provided explicit tools for interpolation theory in weighted Lorentz spaces.

Abstract

We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the $K$-functional of a specific couple of spaces of type $\Lambda$, which are substantially more manageable than their companions of type $S$. The core of our techniques is a subtle manipulation with respective fundamental functions. We present several applications, in particular we nail down a formula for the $K$-functional of a Lebesgue space and a classical Lorentz space of type $S$ with a power weight, and using this formula we establish an inequality of a reverse Marchaud type.