# Embeddings of shearlet coorbit spaces into Sobolev spaces

## TL;DR

This paper studies how shearlet coorbit spaces relate to Sobolev spaces in three dimensions, revealing how group features affect smoothness properties and embedding behavior.

## Contribution

It provides a new perspective by describing coorbit spaces as decomposition spaces, enabling complete embedding characterizations and insights into shearlet group influences.

## Key findings

- Complete characterization of embeddings for certain integrability exponents
- Identification of dilation group features affecting embeddings
- Insights into shearlet coorbit spaces as smoothness spaces

## Abstract

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed $L^p$-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.