We study embeddings of Morrey type spaces Mp,q,ω(Rn)\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) $$\end{document}, 1⩽p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1 \leqslant p<\infty $$\end{document}, 1⩽q<∞\documentclass[12pt]{minimal}
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\begin{document}$$1 \leqslant q<\infty $$\end{document}, both local and global, into weighted Lebesgue spaces Lp(Rn,w)\documentclass[12pt]{minimal}
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\begin{document}$$ {\varvec{L}}^p({\mathbb {R}}^n ,w) $$\end{document}, with the main goal to better understand the local behavior of functions f∈Mp,q,ω(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$ f \in {\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) $$\end{document} and also their behavior at infinity. Under some assumptions on the function ω\documentclass[12pt]{minimal}
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\begin{document}$$ \omega $$\end{document}, we prove that the local Morrey type space is embedded into Lp(Rn,w)\documentclass[12pt]{minimal}
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\begin{document}$$ {\varvec{L}}^{p } ({\mathbb {R}}^n ,w) $$\end{document}, where w(r)=ω(r)\documentclass[12pt]{minimal}
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\begin{document}$$ w(r)=\omega (r) $$\end{document} if q=1\documentclass[12pt]{minimal}
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\begin{document}$$ q=1 $$\end{document}, and w(r) is “slightly distorted” in comparison with ω(r)\documentclass[12pt]{minimal}
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\begin{document}$$ \omega (r) $$\end{document} if q>1.\documentclass[12pt]{minimal}
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\begin{document}$$ q>1.$$\end{document} In the case q>p\documentclass[12pt]{minimal}
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\begin{document}$$ q>p $$\end{document} we show that the embedding, in general, cannot hold with ω=w\documentclass[12pt]{minimal}
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\begin{document}$$ \omega =w $$\end{document}. 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 Lp(Rn,w)\documentclass[12pt]{minimal}
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\begin{document}$$ {\varvec{L}}^{p } ({\mathbb {R}}^n , w) $$\end{document} 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.
