Morrey Spaces on Domains: Different Approaches and Growth Envelopes |
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Authors: | " target="_blank">Dorothee D Haroske Cornelia Schneider Leszek Skrzypczak |
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Institution: | 1.Institute of Mathematics,Friedrich Schiller University Jena,Jena,Germany;2.Mathematics Department,Friedrich-Alexander University Erlangen-Nüremberg,Erlangen,Germany;3.Faculty of Mathematics and Computer Science,Adam Mickiewicz University,Poznań,Poland |
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Abstract: | 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. |
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