On embeddings of Morrey type spaces between weighted Lebesgue or Stummel spaces with application to Herz spaces

We study embeddings of Morrey type spaces M^p,q,ω(Rⁿ) , 1 ⩽ p< ∞, 1 ⩽ q< ∞, both local and global, into weighted Lebesgue spaces L^p(Rⁿ, w) , with the main goal to better understand the local behavior of functions f∈ M^p,q,ω(Rⁿ) and also their behavior at infinity. Under some assumptions on the function ω, we prove that the local Morrey type space is embedded into L^p(Rⁿ, w) , where w(r) = ω(r) if q= 1 , and w(r) is “slightly distorted” in comparison with ω(r) if q> 1. In the case q> p we show that the embedding, in general, cannot hold with ω= 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 L^p(Rⁿ, 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.