LAUSR

We study embeddings of Morrey type spaces $${\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) $$ , $$1 \leqslant p<\infty $$ , $$1 \leqslant q<\infty $$ , both local and global, into weighted Lebesgue spaces $$ {\varvec{L}}^p({\mathbb {R}}^n ,w) $$ , with the main goal to better understand the local behavior of functions $$ f \in {\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) $$ and also their behavior at infinity. Under some assumptions on the function $$ \omega $$ , we prove that the local Morrey type space is embedded into $$ {\varvec{L}}^{p } ({\mathbb {R}}^n ,w) $$ , where $$ w(r)=\omega (r) $$ if $$ q=1 $$ , and w(r) is “slightly distorted” in comparison with $$ \omega (r) $$ if $$ q>1.$$ In the case $$ q>p $$ we show that the embedding, in general, cannot hold with $$ \omega =w $$ . For global Morrey type spaces we also prove embeddings into Stummel spaces. Similar embeddings for complementary Morrey type spaces are obtained. We also study inverse embeddings of weighted Lebesgue spaces $$ {\varvec{L}}^{p } ({\mathbb {R}}^n , w) $$ into Morrey type and complementary Morrey type spaces. Finally, using our previous results on relations between Herz and Morrey type spaces, we obtain “for free” similar embeddings for Herz spaces.