A generalized version of Holmstedt's formula for the K-functional

TL;DR

This paper extends Holmstedt's formula to a broader class of quasi-normed spaces, providing a generalized method to compute the K-functional for interpolated couples using two parameters.

Contribution

It introduces a generalized version of Holmstedt's formula applicable to quasi-normed spaces with two interpolation parameters, expanding the theoretical framework of K-interpolation.

Findings

Derived a new formula for the K-functional in quasi-normed spaces

Unified the computation of K-functionals for complex interpolation couples

Extended classical results to more general quasi-normed settings

Abstract

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces, and let $\Phi_0$ and $\Phi_1$ be two general parameters of $K$-interpolation method. We compute $K$-functional for the couple $((A_0,A_1)_{\Phi_0}, (A_0, A_1)_{\Phi_1})$ in terms of $K$-functional for the couple $(A_0, A_1)$.