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 \(0<p\le u<\infty \), 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.

中文翻译：

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域上的Morrey空间：不同的方法和增长信封。

我们处理通过不同方法获得的有界域\（\ Omega \）上的Morrey空间。特别地，我们考虑三个设置\（\ mathcal {M} _ {u，p}（\ Omega）\），\（\ mathbb {M} _ {u，p}（\ Omega）\）和\（\ mathfrak {M} _ {u，p}（\ Omega）\），其中\（0 <p \ le u <\ infty \）在文献中经常使用，并研究它们的联系和多样性。此外，我们确定增长包络\（\ mathfrak {E} _ {\ mathsf {G}}（\ mathcal {M} _ {u，p}（\ Omega））\）以及\（\ mathfrak {E } _ {\ mathsf {G}}（\ mathfrak {M} _ {u，p}（\ Omega））\），并在最佳嵌入方面获得了一些应用。令人惊讶的是，事实证明p之间的相互作用和ü在感是否\（\压裂{N} {ù} \ GE \压裂{1} {P} \）或\（\压裂{N} {U】<\压裂{1} {P} \ ）在这些空间的行为方面起着决定性的作用。