On embeddings of Morrey type spaces between weighted Lebesgue or Stummel spaces with application to Herz spaces - Rafeiro, Humberto (2021)
We study embeddings of Morrey type spaces M-p,M-q,M-omega(R-n), 1 <= p < infinity, 1 <= q < infinity, both local and global, into weighted Lebesgue spaces L-p(R-n, w), with the main goal to better understand the local behavior of functions f is an element of M-p,M-q,M-omega(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 L-p(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 L-p(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.