Morrey Spaces on Domains: Different Approaches and Growth Envelopes.

We deal with Morrey spaces on bounded domains Ω obtained by different approaches. In particular, we consider three settings M u , p ( Ω ) , M u , p ( Ω ) and M u , p ( Ω ) , where 0 < p ≤ u < ∞ , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes E G ( M u , p ( Ω ) ) as well as E G ( M u , p ( Ω ) ) , 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 n u ≥ 1 p or n u < 1 p plays a decisive role when it comes to the behaviour of these spaces.