Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains [Formula: see text] obtained by different approaches. In particular, we consider three settings [Formula: see text] , [Formula: see text] and [Formula: see text] , where [Formula: see text] , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes [Formula: see text] as well as [Formula: see text] , 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 [Formula: see text] or [Formula: see text] plays a decisive role when it comes to the behaviour of these spaces.