Complex interpolation of function spaces with general weights

TL;DR

This paper studies the complex interpolation of Besov and Triebel-Lizorkin spaces with general weights, providing new results especially for weighted spaces and their interpolation properties.

Contribution

It introduces new interpolation results for weighted Besov and Triebel-Lizorkin spaces with generalized smoothness, extending existing theories.

Findings

Derived interpolation formulas for weighted Triebel-Lizorkin spaces.

Extended the theory to include spaces with generalized smoothness.

Provided conditions on parameters and weights for interpolation.

Abstract

In this paper, we present the complex interpolation of Besov and Triebel-Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel-Lizorkin spaces. An application, we obtain the complex interpolation between the weighted Triebel-Lizorkin spaces $\dot{F}_{p_{0},q_{0}}^{s_{0}}(\omega_{0})$ and $\dot{F}_{\infty ,q_{1}}^{s_{1}}(\omega_{1})\ $with suitable assumptions on the parameters $\ s_{0},s_{1},p_{0},$ $q_{0}$ and$\ q_{1}$, and the pair of weights $(\omega_{0},\omega_{1})$.