# Complex interpolation of smoothness Triebel-Lizorkin-Morrey spaces

## TL;DR

This paper generalizes the complex interpolation results for Triebel-Lizorkin-Morrey spaces by including additional parameters and relaxing previous restrictions, using a new approach that avoids sequence space methods.

## Contribution

It extends prior work by incorporating the smoothness parameter s and the second smoothness parameter r, broadening the applicability of interpolation results for these spaces.

## Key findings

- Extended interpolation results to include parameters s and r.
- Relaxed conditions on parameters s and r from previous work.
- Applied Bergh's formula to avoid sequence space techniques.

## Abstract

This paper extends the result in \cite{HNS15} to Triebel-Lizorkin-Morrey spaces which contains $4$ parameters $p,q,r,s$. This paper reinforces our earlier paper \cite{HNS15} by Nakamura, the first and the third authors in two different directions. First, we include the smoothness parameter $s$ and the second smoothness parameter $r$. In \cite{HNS15} we assumed $s=0$ and $r=2$. Here we relax the conditions on $s$ and $r$ to $s \in {\mathbb R}$ and $1 < r \le \infty$. Second, we apply a formula obtained by Bergh in 1978 to prove our main theorem without using the underlying sequence spaces.