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 between
weak Morrey spaces $ wL^{p,\lambda}\left( \mathbb{R}^{n} \right) $ and Stummel
classes $ 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}