Morrey Spaces are Embedded Between Weak Morrey Spaces and Stummel Classes

Abstract

In this paper, we show that the Morrey spaces $ L^{1,\left( \frac{\lambda}{p} -\frac{n}{p} + n \right) } \left( \mathbb{R}^{n} \right) $ are embedded betweenweak Morrey spaces $ wL^{p,\lambda}\left( \mathbb{R}^{n} \right) $ and Stummelclasses $ S_{\alpha}\left( \mathbb{R}^{n} \right) $ under some conditions on$ p, \lambda $ and $ \alpha $. More precisely, we prove that $ wL^{p,\lambda}\left(\mathbb{R}^{n} \right) \subseteq L^{1,\left( \frac{\lambda}{p} - \frac{n}{p} + n\right) } \left( \mathbb{R}^{n} \right) \subseteq S_{\alpha}\left( \mathbb{R}^{n}\right) $ where $ 1<p<\infty, 0<\lambda<n $ and $ \frac{n-\lambda}{p}<\alpha<n $.We also show that these inclusion relations under the above conditions are proper.Lastly, we present an inequality of Adams' type \cite{A}