Holmstedt's formula for the $K$-functional: the limit case $\theta_0=\theta_1$

TL;DR

This paper investigates Holmstedt's formula for K-interpolation spaces with slowly varying functions, establishing conditions for the limit case where the parameters are equal at the endpoints, with applications to various function spaces.

Contribution

It provides necessary and sufficient conditions for Holmstedt's formula in the limiting case of equal parameters, extending the understanding of K-interpolation spaces involving slowly varying functions.

Findings

Derived conditions for Holmstedt's formula in limit cases

Extended results to Lorentz-Karamata and Besov spaces

Analyzed the case for parameters in (0,1)

Abstract

We consider $K$-interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt-type formula to be held in the limiting case $\theta_0=\theta_1\in\{0,1\}.$ We also study the case $\theta_0=\theta_1\in (0,1).$ Applications are given to Lorentz-Karamata spaces, generalized gamma spaces and Besov spaces.