# Some embeddings of Morrey spaces with critical smoothness

## TL;DR

This paper investigates the embeddings of Morrey-type function spaces at critical smoothness levels, characterizing their inclusion in local integrability spaces and Orlicz-Morrey spaces, thus extending understanding of these embeddings at boundary cases.

## Contribution

It provides a detailed analysis of embeddings of Besov-Morrey and Triebel-Lizorkin-Morrey spaces at critical smoothness, including characterizations in local and exponential-type Morrey spaces.

## Key findings

- Characterization of embeddings at critical smoothness levels
- Embeddings into Orlicz-Morrey and generalized Morrey spaces
- Extension of embedding theory for Morrey-type spaces

## Abstract

We study embeddings of Besov-Morrey spaces ${\cal N}^{s}_{u,p,q}}({\mathbb R}^d)$ and of Triebel-Lizorkin-Morrey spaces ${\cal E}^{s}_{u,p,q}}({\mathbb R}^d)$ in the limiting cases when the smoothness $s$ equals $s_0=d\max(1/u-p/u,0)$ or $s_{\infty}=d/u$, which is related to the embeddings in $L_1^{loc}({\mathbb R}^d)$ or in $L_\infty({\mathbb R}^d)$, respectively. When $s=s_0$ we characterise the embeddings in $L_1^{loc}({\mathbb R}^d)$ and when $s=s_{\infty}$ we obtain embeddings into Orlicz-Morrey spaces of exponential type and into generalised Morrey spaces.