Morrey Spaces are Embedded Between Weak Morrey Spaces and Stummel Classes - Tumalun, Nicky; Gunawan, Hendra (Journal of the Indonesian Mathematical Society, 2019)
In this paper, we show that the Morrey spaces $ L^{1,\left( \frac{λ}{p} - \frac{n}{p} + n \right) } \left( \mathbb{R}^{n} \right) $ are embedded between weak Morrey spaces $ wL^{p,λ}\left( \mathbb{R}^{n} \right) $ and Stummel classes $ S_{α}\left( \mathbb{R}^{n} \right) $ under some conditions on $ p, λ $ and $ α $. More precisely, we prove that $ wL^{p,λ}\left( \mathbb{R}^{n} \right) \subseteq L^{1,\left( \frac{λ}{p} - \frac{n}{p} + n \right) } \left( \mathbb{R}^{n} \right) \subseteq S_{α}\left( \mathbb{R}^{n} \right) $ where $ 1<p<∞, 0<λ<n $ and $ \frac{n-λ}{p}<α<n $. We also show that these inclusion relations under the above conditions are proper. Lastly, we present an inequality of Adams' type \cite{A}