Morrey Spaces are Embedded Between Weak Morrey Spaces and Stummel Classes

Nicky Tumalun (Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung)
Hendra Gunawan (Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung)

Journal of the Indonesian Mathematical Society
Volume 25 Number 3 (November 2019)

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}