Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains (Omega ) obtained by different approaches. In particular, we consider three settings (mathcal {M}_{u,p}(Omega )), (mathbb {M}_{u,p}(Omega )) and (mathfrak {M}_{u,p}(Omega )), where (0, commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes (mathfrak {E}_{mathsf {G}}(mathcal {M}_{u,p}(Omega ))) as well as (mathfrak {E}_{mathsf {G}}(mathfrak {M}_{u,p}(Omega ))), 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.