# Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces

## TL;DR

This paper advances the understanding of variable exponent Morreyfied Triebel-Lizorkin spaces by providing atomic and molecular characterizations, reducing moment conditions, and establishing a Sobolev-type theorem for related sequence spaces.

## Contribution

It introduces new atomic and molecular decompositions for these spaces and shows that fewer zero moments are needed for convergence, improving existing theoretical frameworks.

## Key findings

- Atomic and molecular characterizations of the spaces.
- Reduced zero moment requirements for molecules.
- A Sobolev-type theorem for related sequence spaces.

## Abstract

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest.