Morrey Spaces on Domains: Different Approaches and Growth Envelopes.

Abstract

We deal with Morrey spaces on bounded domains $\Omega$ obtained by different approaches. In particular, we consider three settings $M_{u,p}\left(\Omega \right)$ , $M_{u,p}\left(\Omega \right)$ and $M_{u,p}\left(\Omega \right)$ , where $0<p\le u<\infty$ , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes $E_G\left(M_{u,p}\left(\Omega \right)\right)$ as well as $E_G\left(M_{u,p}\left(\Omega \right)\right)$ , and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether $\frac{n}{u}\ge \frac{1}{p}$ or $\frac{n}{u}<\frac{1}{p}$ plays a decisive role when it comes to the behaviour of these spaces.