Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains $$ obtained by different approaches. In particular, we consider three settings $$ , $$ and $$ , where $$ , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes $$ as well as $$ , and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether $$ or $$ plays a decisive role when it comes to the behaviour of these spaces.