Embeddings of Generalized Weighted Morrey Spaces

Abstract

We study continuous embeddings from two classes of generalized weighted Morrey spaces into generalized Morrey spaces. The Komori--Shirai-type spaces use the weighted measure of cubes in their normalization, whereas the Samko-type spaces use the Lebesgue measure. We first consider generalized Morrey spaces on bounded domains. In the non-radial setting, we establish embedding results under local radial comparability assumptions on the Morrey functions. As a consequence, we characterize the corresponding embeddings in the radial setting for the full range $0<p_1,p_2<\infty$. We then study embeddings from generalized Komori--Shirai-type weighted Morrey spaces associated with piecewise power weights and general Muckenhoupt weights. For the Samko-type scale, we treat both generalized and classical weighted Morrey spaces, again considering piecewise power weights and general Muckenhoupt weights. Finally, assuming that the weight is comparable to a positive constant on some cube or ball, we show that the continuous embeddings under consideration are not compact.