John–Nirenberg–Campanato Spaces

Abstract

Let p ∈ (1, ∞), q ∈ [1, ∞), α ∈ [0, ∞) and s be a non-negative integer. In this article, the authors introduce the John-Nirenberg-Campanato space JN (p,q,s) α (X), where X is R n or any closed cube Q 0 ⊂ R n , which when α = 0 and s = 0 coincides with the JN pspace introduced by F. John and L. Nirenberg in the sense of equivalent norms. The authors then give the predual space of JN (p,q,s) α (X) and a John-Nirenberg type inequality of John-Nirenberg-Campanato spaces. Moreover, the authors prove that the classical Campanato space serves as a limit space of JN (p,q,s) α (X) when p → ∞. , P (s)Q i ( f ) for any i is as in (1.1) with Q replaced by Q i , and the supremum is taken over all collections of pairwise disjoint cubes {Q i } i in X.The dual space (JN (p,q,s) α (X)) * of JN (p,q,s) α (X) is defined to be the set of all continuous linear functionals on JN (p,q,s) α (X) equipped with weak- * topology.Remark 1.3. Let X = Q 0 and α = 0 = s. In this case, it is obvious that JN (p,q,s) α (X) = JN p,q (X) with equivalent norms. Moreover, by [9, Proposition 5.1], we know that, when q ∈ [1, p), then JN (p,q,s) α (X) and JN p (X) coincide with equivalent norms and, when q ∈ [p, ∞), JN (p,q,s) α (X) and L q (X) coincide as sets.

A Survey on Function Spaces of John–Nirenberg Type

“…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.…”

“…Inspired by the relation between BMO and the Campanato space, as well as the relation between BMO and JN p , Tao et al [61] introduced a Campanato-type space JN (p,q,s) α (X) in the spirit of the John-Nirenberg space JN p (Q 0 ), which contains JN p (Q 0 ) as a special case. This John-Nirenberg-Campanato space is defined not only on any cube Q 0 but also on the whole space R n .…”

“…A. Brudnyi and Y. Brudnyi [60] introduced a class of function spaces V κ ([0, 1] n ) which coincides with JN (p,q,s) α ([0, 1] n ), defined below for suitable range of indices (see [61], Proposition 2.9, for more details). Very recently, Domínguez and Milman [62] further introduced and studied sparse Brudnyi and John-Nirenberg spaces.…”

The norm of the characteristic function of a set in the John‐Nirenberg space of exponent p

“…Although the spaces JN p were previously used by different authors 4-8 under several names and for different purposes, it was only recently that some results on the structure of the space have been achieved. 2,9 It is now known that for 1 < p < ∞ and any interval I 0 ⊂ R the following chain of inclusions hold:…”

“…It is elementary to check that if and one has The notation for the space is taken from Aalto et al 1 and Dafni et al, 2 but its first appearance goes back to the seminal work of John and Nirenberg, 3 where the space of functions of bounded mean oscillation was introduced, and the famous John‐Nirenberg inequality was given. Although the spaces were previously used by different authors 4–8 under several names and for different purposes, it was only recently that some results on the structure of the space have been achieved 2,9 …”