Localized John–Nirenberg–Campanato spaces

A Survey on Function Spaces of John–Nirenberg Type

“…Proof. (i) is a direct corollary of the fact that for any (u 2 , v, s) α -atom a on the cube Q, [36], Remark 5.5, for more details).…”

“…The main target of this section is to summarize the main results of John-Nirenberg-Campanato spaces, localized John-Nirenberg-Campanato spaces, and congruent John-Nirenberg-Campanato spaces obtained, respectively, in [36,61,64]. Moreover, at the end of each part, we list some open questions which are still unsolved so far.…”

“…As a combination of the JNC space and the localized BMO space in Section 2.1, Sun et al [36] studied the localized John-Nirenberg-Campanato space, which is new even in a special case: localized John-Nirenberg spaces. Now, we recall the definition of the localized Campanato space, which was first introduced by Goldberg in [50] (Theorem 5).…”

“…Furthermore, in [36] (Proposition 2.5), Sun et al showed that jn (p,q,s) α,c 0 (X) in Definition 9 is independent of the choice of the positive constant c 0 . Therefore, in what follows, we write jn (p,q,s) α (X) := jn (p,q,s) α,c 0 (X).…”

“…Therefore, John-Nirenberg spaces are new spaces between Lebesgue spaces and weak Lebesgue spaces, which motivates us to study the properties of JN p . Furthermore, various John-Nirenberg-type spaces have also attracted a lot of attention in recent years (see, for instance, [31][32][33][34][35][36][37] for the Euclidean space case and [30,[38][39][40] for the metric measure space case).…”