A Survey on Function Spaces of John--Nirenberg Type

In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $\mathbb{R}^n$ or a given cube $Q_0\subset\mathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.


Introduction
In this article, a cube Q means that it has finite side length and all its sides parallel to the coordinate axes, but Q is not necessary to be open or closed. Moreover, we always let X be R n or a given cube of R n . Recall that the Lebesgue space L q (X) with q ∈ [1, ∞] is defined to be the set of all measurable functions f on X such that In what follows, we use 1 E to denote the characteristic function of a set E ⊂ R n , and, for any given q ∈ [1, ∞), L q loc (X) the set of all measurable functions f on X such that f 1 E ∈ L q (X) for any bounded measurable set E ⊂ X.
It is well known that L p (X) with p ∈ [1, ∞] plays a leading role in the modern analysis of mathematics. In particular, when p ∈ (1, ∞), the space L p (X) enjoys some elegant properties, such as the reflexivity and the separability, which no longer hold true in L ∞ (X). Thus, many studies related to L p (X) need some modifications when p = ∞; for instance, the boundedness of Calderón-Zygmund operators. Recall that the Calderón-Zygmund operator T is bounded on L p (R n ) for any given p ∈ (1, ∞), but not bounded on L ∞ (R n ). Indeed, T maps L ∞ (R n ) into which was introduced by John and Nirenberg [58] in 1961 to study the functions of bounded mean oscillation, here and thereafter, and the supremum is taken over all cubes Q of R n . This implies that BMO (X) is a fine substitute of L ∞ (X). Also, it should be mentioned that, in the sense modulo constants, BMO (X) is a Banach space, but, for simplicity, we regard f ∈ BMO (X) as a function rather than an equivalent class f + C := { f + c : c ∈ C} if there exists no confusion. Moreover, the space BMO (X) and its numerous variants as well as their vanishing subspaces have attracted a lot of attentions since 1961. For instance, Fefferman and Stein [41] proved that the dual space of the Hardy space H 1 (R n ) is BMO (R n ); Coifman et al. [28] showed an equivalent characterization of the boundedness of Calderón-Zygmund commutators via BMO (R n ); Coifman and Weiss introduced the space of homogeneous type and studied the Hardy space and the BMO space in this context; Sarason [81] obtained the equivalent characterization of VMO (R n ), the closure in BMO (R n ) of uniformly continuous functions, and used it to study stationary stochastic processes satisfying the strong mixing condition and the algebra H ∞ + C; Uchiyama [99] established an equivalent characterization of the compactness of Calderón-Zygmund commutators via CMO (R n ) which is defined to be the closure in BMO (R n ) of infinitely differentiable functions on R n with compact support; Nakai and Yabuta [77] studied pointwise multipliers for functions on R n of bounded mean oscillation; Iwaniec [52] used the compactness theorem in Uchiyama [99] to study linear complex Beltrami equations and the L p (C)-theory of quasiregular mappings. All these classical results have wide generalizations as well as applications, and inspire a myriad of further studies in recent years; see, for instance, the references [54,25,11,9] for their applications in singular integral operators as well as their commutators, the references [75,74,76,73,72,63] for their applications in pointwise multipliers, the references [24,78,89] for their applications in partial differential equations, and the references [23,6,18,31,32,33] for more variants and properties of BMO (R n ). In particular, we refer the reader to Chang and Sadosky [26] for an instructive survey on functions of bounded mean oscillation, and also Chang et al. [23] for BMO spaces on the Lipschitz domain of R n . Naturally, BMO (X) extends L ∞ (X), in the sense that L ∞ (X) BMO (X) and · BMO (X) ≤ 2 · L ∞ (X) . Similarly, such extension exists as well for any L p (X) with p ∈ (1, ∞). Indeed, John and Nirenberg [58] also introduced a generalized version of the BMO condition which was subsequently used to define the so-called John-Nirenberg space JN p (Q 0 ) with exponent p ∈ (1, ∞) and Q 0 being any given cube of R n . Recall that, for any given p ∈ (1, ∞) and any given cube Q 0 of R n , the John-Nirenberg space JN p (Q 0 ) is defined to be the set of all f ∈ L 1 (Q 0 ) such that where the supremum is taken over all collections of interior pairwise disjoint cubes {Q i } i of Q 0 . It is easy to see that the limit of JN p (Q 0 ) when p → ∞ is just BMO (Q 0 ); see also Corollary 3.6 below. Moreover, the John-Nirenberg space is closely related to the Lebesgue space L p (Q 0 ) and the weak Lebesgue space L p,∞ (Q 0 ) which is defined as in Definition 2.1 below. Precisely, let p ∈ (1, ∞). On one hand, the inequality obtained in [58,Lemma 3] (see also Theorem 2.11 below) implies that JN p (Q 0 ) ⊂ L p,∞ (Q 0 ); also, by [1, Example 3.5], we further know that JN p (Q 0 ) L p,∞ (Q 0 ). On the other hand, it is obvious that L p (Q 0 ) ⊂ JN p (Q 0 ) with · JN p (Q 0 ) ≤ 2 · L p (Q 0 ) , but the striking nontriviality was showed very recently by Dafni et al. [34,Proposition 3.2 and Corollary 4.2] which says that L p (Q 0 ) JN p (Q 0 ). Combining these facts, we conclude that L p (Q 0 ) JN p (Q 0 ) L p,∞ (Q 0 ). (1.2) 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-Nirenbergtype spaces have attracted a lot of attentions as well in recent years; see, for instance, [51,12,69,34,13,84,92] for the Euclidean space case, [42,65,1,66] for the metric measure space case. It should be mentioned that the mean oscillation truly makes a difference in both BMO and JN p ; for instance, (i) via the characterization of distribution functions, we know that BMO is closely related to the space L exp whose definition [see (2.4) below] is similar to an equivalent expression of BMO but with f − f Q replaced by f (see Proposition 2.6 below); (ii) there exists an interesting observation presented by Riesz [80], which says that, in (1.1), if we replace f − f Q i by f , then JN p (Q 0 ) turns to be L p (Q 0 ). Moreover, this conclusion also holds true when Q 0 is replaced by R n ; see Proposition 4.3 below.
The main purpose of this article is to give a survey on some recent developments of both the John-Nirenberg space JN p and the space BMO, mainly including their several generalized (or related) spaces and some vanishing subspaces. We warm up in Section 2 by recalling some definitions and basic properties of BMO and JN p . Section 3 summarizes some recent developments of the John-Nirenberg-Campanato space, the localized John-Nirenberg-Campanato space, and the special John-Nirenberg-Campanato space via congruent cubes. Section 4 focuses on the Riesz-type space which differs from the John-Nirenberg space in subtracting integral means, and its congruent counterpart. In Section 5, we pay attention to some vanishing subspaces of aforementioned John-Nirenberg-type spaces, such as VMO, XMO, CMO, V JN p , and C JN p on R n or any given cube Q 0 of R n . In addition, several related open questions are also summarized in this survey.
More precisely, the remainder of this survey is organized as follows. Section 2 is split into two subsections. In Subsection 2.1, via recalling the definitions of distribution functions and some related function spaces (including the weak Lebesgue space, the Morrey space, and the space L exp ), we present the relation L ∞ (Q 0 ) BMO (Q 0 ) L exp (Q 0 ) in Proposition 2.5 below, which is a counterpart of (1.2) above, and also show two equivalent Orlicz-type norms on BMO (R n ) in Proposition 2.6 below; moreover, corresponding results for the localized BMO space are also obtained in Corollary 2.10 below. Subsection 2.2 is devoted to some significant results of JN p , including the famous John-Nirenberg inequality (see Theorem 2.11 below), and the accurate relations of JN p and L p as well as L p,∞ (see Remark 2.12 below). Furthermore, some recent progress of JN p is also briefly listed at the end of this subsection. Section 3 is split into three subsections. In Subsection 3.1, we first recall the notions of the John-Nirenberg-Campanato space (for short, JNC space), the corresponding Hardy-type space, and their basic properties which include the limit results and the relations with other classical spaces. Then we review the dual theorem between these two spaces, and the independence over the second sub-index of JNC spaces and Hardy-type spaces. Subsection 3.2 is devoted to the localized counterpart of Subsection 3.1. The aim of Subsection 3.3 is the summary of the special JNC space defined via congruent cubes (for short, congruent JNC space), including their basic properties corresponding to those in Subsection 3.1. Also, some applications about the boundedness of operators on congruent spaces are mentioned as well.
In Section 4, via subtracting integral means in the JNC space, we first give the definition of the Riesz-type space appearing in [92], and then present some basic facts about this space in Subsection 4.1. Moreover, the predual space (namely, the block-type space) and the corresponding dual theorem of the Riesz-type space are also displayed in this subsection. Subsection 4.2 is devoted to the congruent counterpart of the Riesz-type space and the boundedness of some important operators.
Section 5 is split into three subsections. Subsection 5.1 is devoted to several vanishing subspaces of BMO (R n ), including VMO (R n ), CMO (R n ), MMO (R n ), XMO (R n ), and X 1 MO (R n ). We first recall their definitions, and then review their [except MMO (R n )] mean oscillation characterizations, respectively, in Theorems 5.2, 5.3, and 5.4 below. Meanwhile, an open question on the corresponding equivalent characterization of MMO (R n ) is also listed in Question 5.6 below. Then we further review the compactness theorems of the Calderón-Zygmund commutators [b, T ] where b belongs to the vanishing subspaces CMO (R n ) as well as XMO (R n ), and propose an open question on [b, T ] with b ∈ XMO (R n ). Moreover, the characterizations via Riesz transforms of BMO (R n ), VMO (R n ), and CMO (R n ), as well as the localized results of these vanishing subspaces are presented. Also, some open questions are listed in this subsection. Subsection 5.2 devotes to the vanishing subspaces of JNC spaces. We first recall the definition of the vanishing JNC space on cubes in Definition 5.17, and then review its equivalent characterization as well as its dual result, respectively, in Theorems 5.18 and 5.19. Moreover, for the case of R n , we review the corresponding results for V JN p (R n ) and C JN p (R n ), which are, respectively, counterparts of VMO (R n ) and CMO (R n ) (see Theorems 5.22 and 5.24 below). As before, some open questions are also listed at the end of this subsection. Subsection 5.3 is devoted to the congruent counterpart of Subsection 5.2, some similar conclusions are listed in this subsection; meanwhile, some open questions in the JNC space have affirmative answers in the congruent setting; see Proposition 5.33 below.
Finally, we make some conventions on notation. Let N := {1, 2, . . .}, Z + := N ∪ {0}, and Z n + := (Z + ) n . We always denote by C and C positive constants which are independent of the main parameters, but they may vary from line to line. Moreover, we use C (γ, β, ...) to denote a positive constant depending on the indicated parameters γ, β, . . .. Constants with subscripts, such as C 0 and A 1 , do not change in different occurrences. Moreover, the symbol f g represents that f ≤ Cg for some positive constant C. If f g and g f , we then write f ∼ g. If f ≤ Cg and For any p ∈ [1, ∞], let p ′ be its conjugate index, that is, p ′ satisfies 1/p + 1/p ′ = 1. We use 1 E to denote the characteristic function of a set E ⊂ R n and |E| the Lebesgue measure when E ⊂ R n is measurable, and 0 the origin of R n . For any function f on R n , let supp ( f ) := {x ∈ R n : f (x) 0}. Let X be a normed linear space. We use (X) * to denote its dual space.

BMO and JN p
It is well known that the space BMO has played an important role in harmonic analysis, partial differential equations, and other mathematical fields since it was introduced by John and Nirenberg in the celebrated article [58]. However, in the same article [58], another mysterious space appeared as well, which is nowadays called the John-Nirenberg space JN p . Indeed, BMO can be viewed as the limit space of JN p as p → ∞; see Proposition 3.5 and Corollary 3.6 below with α := 0. To establish the relations of BMO and JN p , and also summarize some recent works of John-Nirenberg-type spaces, we first recall some basic properties of BMO and JN p in this section.
This section is devoted to some well-known results of BMO (X) and JN p (X), respectively, in Subsections 2.1 and 2.2. In addition, it is trivial to find that all the results in Subsection 2.1 also hold true with the cube Q 0 replaced by the ball B 0 of R n .

(Localized) BMO and L exp
This subsection is devoted to several equivalent norms of the spaces BMO and localized BMO. To this end, we begin with the distribution function where f ∈ L 1 loc (X) and t ∈ (0, ∞). Recall that the distribution function is closely related to the following weak Lebesgue space.
where, for any measurable function f on X, Moreover, the distribution function also features BMO (X), which is exactly the famous result obtained by John and Nirenberg [58, Lemma 1']: there exist positive constants C 1 and C 2 , depending only on the dimension n, such that, for any given f ∈ BMO (X), any given cube Q ⊂ X, and any t ∈ (0, ∞), (2. 2) The main tool used in the proof of (2.2) is the following well-known As an application of (2.2), we find that, for any given q ∈ (1, ∞), f ∈ BMO (R n ) if and only if f ∈ L 1 loc (R n ) and (this equivalence is proved in Proposition 2.6 below), here and thereafter, for any given cube Q of R n , and any measurable function g, the locally normalized Orlicz norm g L exp (Q) is defined by setting Moreover, for any given cube Q of R n , the space L exp (Q) is defined by setting The space L exp (Q) was studied in the interpolation of operators (see, for instance, [7, p. 243]) and it is closely related to the space BMO (Q) (see Proposition 2.6 below). On the Orlicz function in (2.4), we have the following properties.
Before showing the equivalent Orlicz-type norms of BMO (X), we first prove the following equivalent characterizations of BMO (X). These characterizations might be well known. But, to the best of our knowledge, we did not find a complete proof. For the convenience of the reader, we present the details here.
Next, we show the implication (ii) =⇒ (iii). Suppose that f satisfies (ii). Then there exist positive constants C 3 and C 4 such that, for any cube Q ⊂ X and any t ∈ (0, ∞), which implies that f satisfies (iii). This shows the implication (ii) =⇒ (iii). Finally, we show the implication (iii) =⇒ (i). Suppose that f satisfies (iii). Then there exists a λ ∈ (0, ∞) such that From this and the basic inequality x ≤ e x − 1 for any x ∈ R, we deduce that which implies that f satisfies (i), and hence the implication (iii) =⇒ (i) holds true. This finishes the proof of Proposition 2.4.
In what follows, for any normed space Y(X), equipped with the norm · Y(X) , whose elements are measurable functions on X, let Proposition 2.5. Let Q 0 be a cube of R n . Then Proof. Indeed, on one hand, from for any c ∈ C, we deduce that On the other hand, by Proposition 2.4(iii), we easily find that BMO (Q 0 ) ⊂ [L exp (Q 0 )/C]. Moreover, without loss of generality, we may assume that Q 0 := (−1, 1) and let x ∈ (0, 1).
Proof. To prove this proposition, we only need to prove that, for any f ∈ L 1 loc (X), We first show that, for any f ∈ L 1 loc (X), f BMO (X) ≤ f BMO(X) and f BMO (X) ≤ f L exp (X) . To this end, let f ∈ L 1 loc (X). For any cube Q ⊂ X and any λ ∈ (0, ∞), by t ≤ e t − 1 for any t ∈ (0, ∞), we have Moreover, to show f BMO (X) ≤ f L exp (X) , it suffices to assume that f ∈ L exp (X), otherwise f L exp (X) = ∞ and hence the desired inequality automatically holds true. Then, by t ≤ e t − 1 for any t ∈ (0, ∞), we conclude that, for any n ∈ N and any cube Q ⊂ X, From the definition of · L exp (X) , we deduce that, for any n ∈ N, there exists a By this, (2.7), and the monotonicity of e (·) − 1, we conclude that, for any n ∈ N and any cube Q ⊂ X, and hence Letting n → ∞, we then obtain To sum up, we have, for any f ∈ L 1 loc (X), Next, we show that the reverse inequalities hold true for any f ∈ L 1 loc (X), respectively. Actually, we may assume that f ∈ BMO (X) because, otherwise, the desired inequalities automatically hold true. Now, let f ∈ BMO (X). Then, for any cube Q ⊂ X and any λ ∈ (C −1 2 f BMO (X) , ∞), by (2.2) and the calculation of (2.6), we obtain where C 1 ∈ (1, ∞) is as in (2.2). From this and Lemma 2.3(i) with s replaced by 1/C 1 , we deduce that On one hand, by (2.9) and we conclude that and hence On the other hand, by (2.9), we conclude that From this and we deduce that Combining this with (2.10), we have, for any f ∈ BMO (X), This, together with (2.8), then finishes the proof of Proposition 2.6.
Remark 2.7. There exists another norm on L exp (Q 0 ), defined by the distribution functions as follows. Let f be a measurable function on Q 0 . The decreasing rearrangement f * of f is defined by setting, for any u ∈ [0, ∞), Moreover, for any v ∈ (0, ∞), let Then f ∈ L exp (Q 0 ) if and only if f is measurable on Q 0 and meanwhile, · L * exp (Q 0 ) is a norm of L exp (Q 0 ); see [7, p. 246, Theorem 6.4] for more details. Furthermore, from [7, p. 7, Corollary 1.9], we deduce that · L * exp (Q 0 ) and · L exp (Q 0 ) are equivalent. Notice that f * and f * * are fundamental tools in the theory of Lorentz spaces; see [47, p. 48] for more details.
Recently, Izuki et al. [53] obtained both the John-Nirenberg inequality and the equivalent characterization of BMO (R n ) on the ball Banach function space which contains Morrey spaces, (weighted, mixed-norm, variable) Lebesgue spaces, and Orlicz-slice spaces as special cases; see [53,Definition 2.8] and also [94] for the related definitions. Precisely, let X be a ball Banach function space satisfying the additional assumption that the Hardy-Littlewood maximal operator M is bounded on X ′ (the associate space of X; see [53, Definition 2.9] for its definition), and, for where the supremum is taken over all balls B of R n . It is obvious that · BMO L 1 (R n ) = · BMO (R n ) . Moreover, in [53, Theorem 1.2], Izuki et al. showed that, under the above assumption of X, b ∈ BMO (R n ) if and only if b ∈ L 1 loc (R n ) and b BMO X < ∞; meanwhile, Furthermore, the John-Nirenberg inequality on X was also obtained in [53, Theorem 3.1] which shows that there exists some positive constant C such that, for any ball B ⊂ R n and any τ ∈ [0, ∞), where M X ′ →X ′ denotes the operator norm of M on X ′ . Later, these results were applied in [94] to establish the compactness characterization of commutators on ball Banach function spaces. Now, we come to the localized counterpart. The local space BMO (R n ), denoted by bmo (R n ), was originally introduced by Goldberg [46]. In the same article, Goldberg also introduced the localized Campanato space Λ α (R n ) with α ∈ (0, ∞), which proves the dual space of the localized Hardy space. Later, Jonsson et al. [59] constructed the localized Hardy space and the localized Campanato space on the subset of R n ; Chang [22] studied the localized Campanato space on bounded Lipschitz domains; Chang et al. [24] studied the localized Hardy space and its dual space on smooth domains as well as their applications to boundary value problems; Dafni and Liflyand [35] characterized the localized Hardy space in the sense of Goldberg, respectively, by means of the localized Hilbert transform and localized molecules. In what follows, for any cube Q of R n , we use ℓ(Q) to denote its side length, and let ℓ(R n ) := ∞. Recall that bmo (X) : for some given c 0 ∈ (0, ℓ(X)), and the supremum taken over all cubes Q of X. Also, a well-known fact is that bmo (X) is independent of the choice of c 0 ; see, for instance, [36, Lemma 6.1].
Proposition 2.8. Let X be R n or a cube Q 0 of R n . Then and Moreover, and, for any cube Q 0 of R n , Proof. First, we prove (2.13). To this end, let f ∈ L 1 loc (X). Then, for any c ∈ C and any cube Q of X, From this and the definitions of · BMO (X) and · bmo (X) , it follows that (2.13) holds true, which further implies (2.12).
We next prove (2.15). By the above example g 2 , we conclude that L ∞ (Q 0 ) bmo (Q 0 ). Meanwhile, BMO (Q 0 ) [L exp (Q 0 )/C] was obtained in Proposition 2.5. Moreover, for any given Combining this with the observations that [ bmo (Q 0 )/C] ⊂ BMO (Q 0 ) and that, for any c ∈ C, we find that [ bmo (Q 0 )/C] = BMO (Q 0 ) and To sum up, we obtain (2.15). This finishes the proof of Proposition 2.8.
Let f ∈ L 1 loc (X). Similarly to Proposition 2.6, let and where c 0 ∈ (0, ℓ(X)) and f Q,c 0 is as in (2.11). To show that they are equivalent norms of bmo (X), we first establish the following John-Nirenberg inequality for bmo (X), namely, Proposition 2.9 below. In what follows, for any given cube Q of R n , (a 1 , . . . , a n ) denotes the left and lower vertex of Q, which means that, for any (x 1 , . . . , x n ) ∈ Q, x i ≥ a i for any i ∈ {1, . . . , n}. Recall that, for any given cube Q of R n , the dyadic system D Q of Q is defined by setting Proposition 2.9. Let f ∈ bmo (X) and c 0 ∈ (0, ℓ(X)). Then there exist positive constants C 5 and C 6 such that, for any given cube Q ⊂ X, and any t ∈ (0, ∞), Proof. Indeed, this proof is a slight modification of the proof of [58, Lemma 1] or [39,Theorem 6.11]. We give some details here again for the sake of completeness. Let f ∈ bmo (X). Then, from Proposition 2.8, we deduce that f ∈ BMO (X) and f BMO (X) ≤ 2 f bmo (X) , which further implies that, for any cube Q ⊂ X with ℓ(Q) < c 0 , and any t ∈ (0, ∞), where C 1 and C 2 are as in (2.2), and the distribution function D is defined as in (2.1). Therefore, to show (2.19), it remains to prove that, for any given cube Q with ℓ(Q) ≥ c 0 , and any t ∈ (0, ∞), Notice that, in this case, there exists a unique m 0 ∈ Z + such that 2 −(m 0 +1) ℓ(Q) < c 0 ≤ 2 −m 0 ℓ(Q). Moreover, since inequality (2.19) is not altered when we multiply both f and t by the same constant, without loss of generality, we may assume that f bmo (X) = 1. Let Q 0 be any given dyadic subcube of Q with level m 0 , namely, (2.20) From the Calderón-Zygmund decomposition (namely, Theorem 2. 2) of f with height λ := 2, we deduce that there exists a family {Q 1, j } j ⊂ D (1) Q 0 such that, for any j, and | f (x)| ≤ 2 when x ∈ Q \ j Q 1, j . By this and (2.20), we conclude that and, for any j, Moreover, for any j, from the Calderón-Zygmund decomposition of f − f Q 1, j with height 2, we deduce that there exists a family Thus, we obtain, for any j, and, for any k, Repeating this process, then, for any T ∈ N, we obtain a family Notice that, for any t ∈ [2 n+1 , ∞), there exists a unique T ∈ N such that T 2 n+1 ≤ t < (T + 1)2 n+1 ≤ T 2 n+2 . Therefore, we obtain where C 5 := √ 2. By this, (2.21), and the arbitrariness of Q 0 ∈ D (m 0 ) Q , we conclude that, for any t ∈ (0, ∞), and hence (2.19) holds true. This finishes the proof of Proposition 2.9.
As a corollary of Proposition 2.9, we have the following result, namely, · bmo 1 (X) in (2.16) and · bmo 2 (X) in (2.17) are equivalent norms of bmo (X). The proof of Corollary 2.10 is just a repetition of the proof of Proposition 2.6 with (2.2) replaced by (2.19); we omit the details here.

John-Nirenberg space JN p
Although there exist a lot of fruitful studies of the space BMO in recent years, but, as was mentioned before, the structure of JN p is largely a mystery and there still exist many unsolved problems on JN p . The first well-known property of JN p is the following John-Nirenberg inequality obtained in [58,Lemma 3] which says that JN p (Q 0 ) is embedded into the weak Lebesgue space L p,∞ (Q 0 ) (see Definition 2.1).
and there exists a positive constant C (n,p) , depending only on n and p, but independent of f , such that It should be mentioned that the proof of Theorem 2.11 relies on the Calderón-Zygmund decomposition (namely, Theorem 2.2) as well. Moreover, as an application of Theorem 2.11, Dafni et al. recently showed in [34, Proposition 5.1] that, for any given p ∈ (1, ∞) and q ∈ [1, p), f ∈ JN p (Q 0 ) if and only if f ∈ L 1 (Q 0 ) and where the supremum is taken in the same way as in (1.1); meanwhile, · JN p (Q 0 ) ∼ · JN p,q (Q 0 ) . Furthermore, in [34, Proposition 5.1], Dafni et al. also showed that, for any given p ∈ (1, ∞) and q ∈ [p, ∞), the spaces JN p,q (Q 0 ) and L q (Q 0 ) coincide as sets.
(i) As a counterpart of Proposition 2.5, for any given p ∈ (1, ∞) and any given  Dafni et al. showed that, for any given p ∈ (1, ∞) and any given interval I 0 ⊂ R which is no matter bounded or not, monotone functions are in JN p (I 0 ) if and only if they are also in L p (I 0 ). Thus, JN p (X) may be very "close" to L p (X) for any given p ∈ (1, ∞).
(ii) JN 1 (Q 0 ) coincides with L 1 (Q 0 ). To be precise, let Q 0 be any given cube of R n , and where f JN 1 (Q 0 ) is defined as in (1.1) with p replaced by 1. Then we claim that JN 1 (Q 0 ) = [L 1 (Q 0 )/C] with equivalent norms. Indeed, for any f ∈ JN 1 (Q 0 ), by the definition of Conversely, for any given f ∈ L 1 (Q 0 ) and any c ∈ C, we have which implies that f JN 1 (Q 0 ) ≤ f L 1 (Q 0 )/C and hence the above claim holds true. Moreover, the relation between JN 1 (R) and L 1 (R) was studied in [13, Proposition 2].
(iii) Garsia and Rodemich in [45,Theorem 7.4] showed that, for any given where the supremum is taken in the same way as in (1.1); meanwhile, Recall that the predual space of BMO (X) is the Hardy space H 1 (X); see, for instance, [29,Theorem B]. Similarly to this duality, Dafni et al. [34] also obtained the predual space of JN p (Q 0 ) for any given p ∈ (1, ∞), which is denoted by the Hardy kind space HK p ′ (Q 0 ), here and thereafter, 1/p + 1/p ′ = 1. Later, these properties, including equivalent norms and duality, were further studied on several John-Nirenberg-type spaces, such as John-Nirenberg-Campanato spaces, localized John-Nirenberg-Campanato spaces, and congruent John-Nirenberg-Campanato spaces (see Section 3 for more details), and Riesz-type spaces (see Section 4 for more details).
Finally, let us briefly recall some other related studies concerning the John-Nirenberg space JN p , which would not be stated in details in this survey while all of them are quite instructive: • Stampacchia [82] introduced the space N (p,λ) , which coincides with JN (p,1,0) α (Q 0 ) in Definitions 3.3 if we write λ = pα with p ∈ (1, ∞) and α ∈ (−∞, ∞), and applied them to the context of interpolation of operators.
• Campanato [20] also used the John-Nirenberg spaces to study the interpolation of operators.
• In the context of doubling metric spaces, JN p and median-type JN p were studied, respectively, by Aalto et al. in [1] and Myyryläinen in [71].
• Hurri-Syrjänen et al. [51] established a local-to-global result for the space JN p (Ω) on an open subset Ω of R n . More precisely, it was proved that the norm · JN p (Ω) is dominated by its local version · JN p,τ (Ω) modulus constants; here, τ ∈ [1, ∞), for any open subset Ω of R n , the related "norm" · JN p (Ω) is defined in the same way as · JN p (Q 0 ) in (1.1) with Q 0 replaced by Ω, and · JN p,τ (Ω) is defined in the same way as · JN p (Ω) with an additional requirement τQ ⊂ Ω for all chosen cubes Q in the definition of · JN p (Ω) .
• Marola and Saari [66] studied the corresponding results of Hurri-Syrjänen et al. [51] on metric measure spaces, and obtained the equivalence between the local and the global JN p norms. Moreover, in both articles [51,66], a global John-Nirenberg inequality for JN p (Ω) was established.
• Berkovits et al. [12] applied the dyadic variant of JN p (Q 0 ) in the study of self-improving properties of some Poincaré-type inequalities. Later, the dyadic JN p (Q 0 ) was further studied by Kinnunen and Myyryläinen in [60].
• Blasco and Espinoza-Villalva [13] computed the concrete value of 1 A JN p (R) for any given p ∈ [1, ∞] and any measurable set A ⊂ R of positive and finite Lebesgue measure, where JN ∞ (R) := BMO (R).

John-Nirenberg-Campanato space
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 [93,84,55]. Moreover, at the end of each part, we list some open questions which are still unsolved so far. Now, we first recall some definitions of some basic function spaces.
• Let q ∈ [1, ∞] and Q 0 be a cube of R n . For any measurable function f , let • For any given v ∈ [1, ∞] and s ∈ Z + , and any measurable subset E ⊂ R n , let Let Q be any given cube of R n . It is well known that P (0) Q ( f ) = f Q and, for any s ∈ Z + , there exists a constant C (s) ∈ [1, ∞), independent of f and Q, such that |γ| ≤ s} on the cube Q with respect to the weight 1/|Q|, namely, for any γ, ν, µ ∈ Z n + with |γ| ≤ s, |ν| ≤ s, and |µ| ≤ s, ϕ (γ) Q ∈ P s (Q) and Then

John-Nirenberg-Campanato spaces
In this subsection, we recall the definitions of Campanato spaces, John-Nirenberg-Campanato spaces (for short, JNC spaces), and Hardy-type spaces, respectively, in Definitions 3.1, 3.3, and 3.11 below. Moreover, we review some properties of JNC spaces and Hardy-type spaces, including their limit spaces (Proposition 3.5 and Corollary 3.6 below), relations with the Lebesgue space (Propositions 3.8 and 3.13 below), the dual result (Theorem 3.12 below), the monotonicity over the first sub-index (Proposition 3.14 below), the John-Nirenberg-type inequality (Theorem 3.15 below), and the equivalence over the second sub-index (Propositions 3.17 and 3.18 below).
A general dual result for Hardy spaces was given by Coifman and Weiss [29] who proved that, for any given p ∈ (0, 1], q ∈ [1, ∞], and s being the non-negative integer not smaller than n( 1 p − 1), the dual space of the Hardy space H p (R n ) is the Campanato space C 1 p −1, q, s (R n ) which was introduced by Campanato [19] and coincides with BMO (R n ) when p = 1.
and the supremum is taken over all cubes Q of X. In addition, the "norm" · C α,q,s (X) is defined modulo polynomials and, for simplicity, the space C α,q,s (X) is regarded as the quotient space C α,q,s (X)/P s (X).
(ii) The dual space (C α,q,s (X)) * of C α,q,s (X) is defined to be the set of all continuous linear functionals on C α,q,s (X) equipped with the weak- * topology.
From Campanato [19,Theorem 6.II], it follows that, for any given q ∈ [1, ∞) and α ∈ [− 1 q , 0), and any f ∈ C q,α,0 (X), where the positive equivalence constants are independent of f , and see also Nakai [73, Theorem 2.1 and Corollary 2.3] for this conclusion on spaces of homogeneous type. In addition, a surprising result says that, in the definition of supremum · M p q (R n ) , if "cubes" were changed into "measurable sets", then the Morrey norm · M p q (R n ) becomes an equivalent norm of the weak Lebesgue space (see Definition 2.1). To be precise, for any given 0 < q < p < ∞, f ∈ L p,∞ (R n ) if and only if f ∈ L q loc (R n ) and see, for instance, [44,p. 485,Lemma 2.8]. Another interesting JN p -type equivalent norm of the weak Lebesgue space was presented in Remark 2.12(iii).
Inspired by the relation between BMO and the Campanato space, as well as the relation between BMO and JN p , Tao el al. [93] 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 . Definition 3.3. Let p, q ∈ [1, ∞), s ∈ Z + , and α ∈ R.
for any i is as in (3.1) with Q replaced by Q i , and the supremum is taken over all collections of interior pairwise disjoint cubes {Q i } i of X. Also, the "norm" · JN (p,q,s)α (X) is defined modulo polynomials and, for simplicity, the space JN (p,q,s) α (X) is regarded as the quotient space JN (p,q,s) α (X)/P s (X).
(ii) 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 the weak- * topology.
Remark 3.4. In [93], the JNC space was introduced only for any given α ∈ [0, ∞) to study its relation with the Campanato space in Definition 3.1, and for any given p ∈ (1, ∞) due to Remark 2.12(ii). However, many results in [93] also hold true when α ∈ R and p = 1 just with some slight modifications of their proofs. Thus, in this survey, we introduce the JNC space for any given α ∈ R and p ∈ [1, ∞), and naturally extend some related results with some identical proofs omitted.
The following proposition, which is just [93, Proposition 2.6], means that the classical Campanato space serves as a limit space of JN (p,q,s) α (X), similarly to the Lebesgue spaces L ∞ (X) and L p (X) when p → ∞. In Proposition 3.5, if we take X = Q 0 , we then have following corollary which is just [93, Corollary 2.8].
Corollary 3.6. Let q ∈ [1, ∞), α ∈ [0, ∞), s ∈ Z + , and Q 0 be a cube of R n . Then and, for any f ∈ C α,q,s (Q 0 ), ∞) and Q 0 be a cube of R n . It is easy to show that However, we claim that BMO (R n ) JN p (R n ).
Indeed, for the simplicity of the presentation, without loss of generality, we may show this claim only in R. Let g(x) := log(|x|) for any x ∈ R \ {0}, and g(0) := 0. Then g ∈ BMO (R) due to [48, Example 3.1.3], and hence it suffices to prove that g JN p (R) for any given p ∈ (1, ∞). To do this, let I t := (0, t) for any t ∈ (0, ∞). Then, by some simple calculations, we obtain and hence as t → ∞. But, the John-Nirenberg inequality of JN p (I t ) in Theorem 2.11 implies that, for any t ∈ (0, ∞), with the implicit positive constants depending only on p. Thus, g JN p (R) and hence the above claim holds true.
(ii) The predual counterpart of Corollary 3.6 is still unclear so far; see Question 3.21 below for more details.
It is a very interesting open question to find a counterpart of Proposition 3.8 when α ∈ R \ {0}; see Question 3.20 below for more details. Now, we review the predual of the John-Nirenberg-Campanato space via introducing atoms, polymers, and Hardy-type spaces in order, which coincide with the same notation as in [34] when u ∈ (1, ∞), v ∈ (u, ∞], and α = 0 = s; see [93, Remarks 3.4 and 3.8] for more details. In particular, when α = 0, the (u, v, s) 0 -atom below is just the classic atom of the Hardy space; see [93,Remark 3.2].
In what follows, for any u ∈ [1, ∞], let u ′ denote its conjugate index, namely, 1/u + 1/u ′ = 1, and, for any {λ j } j ⊂ C, let Moreover, any g ∈ HK (u,v,s) α (X) is called a (u, v, s) α -polymer with its norm g HK (u,v,s)α (X) defined by setting where the infimum is taken over all decompositions of g as above.
Definition 3.11. Let u, v ∈ [1, ∞], s ∈ Z + , and α ∈ R. The Hardy-type space HK (u,v,s) α (X) is defined by setting and, for any g ∈ HK (u,v,s) α (X), let where the infimum is taken over all decompositions of g as above. Moreover, the finite atomic Hardy-type space HK fin (u,v,s) α (X) is defined to be the set of all finite summations M m=1 λ m a m , where M ∈ N, {λ m } M m=1 ⊂ C, and {a m } M m=1 are (u, v, s) α -atoms. The significant dual relation between JN (p,q,s) α (X) and HK (p ′ ,q ′ ,s) α (X) reads as follows, which is just [93, Theorem 3.9] with α ∈ [0, ∞) replaced by α ∈ R (this makes sense because the crucial lemma, [93, Lemma 3.12], still holds true with the corresponding replacement).
Proposition 3.14. Let s ∈ Z + and Q 0 be a cube of R n .
(ii) Let 1 < p 1 < p 2 < ∞. If q ∈ (1, ∞) and α ∈ R, or q = 1 and α ∈ [0, ∞), then and there exists some positive constant C such that Proof. (i) is a direct corollary of the fact that, for any (u 2 , v, s) α -atom a on the cube Q, Now, we consider the independence over the second sub-index, which strongly relies on the John-Nirenberg inequality as in the BMO case. The following John-Nirenberg-type inequality is just [93,Theorem 4.3], which coincides with Theorem 2.11 when α = 0 = s.
Q 0 ( f ) ∈ L p,∞ (Q 0 ) and there exists a positive constant C (n,p,s) , depending only on n, p, and s, but independent of f , such that It should be mentioned that the main tool used in the proof of Theorem 3.15 is the following good-λ inequality (namely, Lemma 3.16 below) which is just [93,Lemma 4.6]; see also [1,Lemma 4.5] when s = 0. Recall that, for any given cube Q 0 of R n , the dyadic maximal operator M (d) Q 0 is defined by setting, for any given g ∈ L 1 (Q 0 ) and any x ∈ Q 0 , where D Q 0 is as in (2.18) with Q replaced by Q 0 , and the supremum is taken over all dyadic cubes Q ∈ D Q 0 and Q ∋ x.
The last question in this subsection is on an interpolation result in [82]. We first recall some notation in [82]. Let p ∈ (1, ∞), λ ∈ R, and Q 0 be a cube of R n . The space N (p,λ) (Q 0 ) is defined by setting Moreover, for any t ∈ [0, 1], The theorem holds true also in the limit case p 1 = ∞ and 1 q 1 = µ 1 = 0.
Therefore for some simple function v ∈ F (Q 0 ), where 1/q t + 1/q ′ t = 1. To sum up, we need to find a simple function v such that both (3.5) and (3.6) hold true, which seems unreasonable because T u may behave so bad even though both u and u are simple functions. Thus, the proof of Theorem 3.23 in [82] seems problematic. It is interesting to check whether or not Theorem 3.23 is really true.

Localized John-Nirenberg-Campanato spaces
As a combination of the JNC space and the localized BMO space in Subsection 2.1, Sun et al. [84] 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 [46,Theorem 5]. In what follows, for any s ∈ Z + and c 0 ∈ (0, ℓ(X)), let Definition 3.25. Let q ∈ [1, ∞), s ∈ Z + , and α ∈ [0, ∞). Fix c 0 ∈ (0, ℓ(X)). The local Campanato space Λ (α,q,s) (X) is defined to be the set of all functions f ∈ L q loc (X) such that where the supremum is taken over all cubes Q of X.
Fix the constant c 0 ∈ (0, ℓ(X)). In Definition 3.3, if P (s) Q j ( f ) were replaced by P (s) Q j ,c 0 ( f ), then we obtain the following localized John-Nirenberg-Campanato space. As was mentioned in Remark 3.4, we naturally extend the ranges of α and p similarly to Subsection 3.1; we omit some identical proofs.
with equivalent norms.
However, the case q ∈ [p, ∞) in Proposition 3.30 is unclear so far; see Question 3.45 below.
As an application of Propositions 3.29(ii) and 3.30, we have the following result.
, and Q 0 be a cube of R n . Then Proof. Let p, q, s, α, and Q 0 be as in this proposition. Then, by Propositions 3.29(ii) and 3.30, we obtain and

This implies that JN
which completes the proof of Proposition 3.31.  In other range of indices, namely, q ≥ p, the following relation between jn (p,q,s) α (X) and the Lebesgue space is just [84,Proposition 3.4].

Moreover,
Proposition 3.35. Let s ∈ Z + and Q 0 be a cube of R n .
Using the localized atom, Sun el al. [84] introduced the localized Hardy-type space and showed that this space is the predual of the localized John-Nirenberg-Campanato space. First, recall the definitions of localized atoms, localized polymers, and localized Hardy-type spaces in order as follows.
Definition 3.36. Let v, w ∈ [1, ∞], s ∈ Z + , and α ∈ R. Fix c 0 ∈ (0, ℓ(X)) and let Q denote a cube of R n . Then a function a on R n is called a local (v, w, s) (iii) when ℓ(Q) < c 0 , Q a(x)x β dx = 0 for any β ∈ Z n + and |β| ≤ s. (3.4) for the definition of · ℓ v ]. Any g ∈ hk (v,w,s) α,c 0 (X) is called a local (v, w, s) α,c 0 -polymer on X and let where the infimum is taken over all decompositions of g as above.
Correspondingly, hk (v,w,s) α,c 0 (X) is independent of the choice of the positive constant c 0 as well, which is just [84,Proposition 4.7].
The corresponding dual theorem (namely, Theorem 3.40 below) is just [84,Theorem 4.11]. In what follows, the space hk fin (v,w,s) α (X) is defined to be the set of all finite linear combinations of local (v, w, s) α -atoms supported, respectively, in cubes of X.
(ii) Any bounded linear functional L on hk (v,w,s) α (X) can be represented by a function f ∈ jn (v ′ ,w ′ ,s) α (X) in the following sense: Moreover, there exists a positive constant C, depending only on s, such that f jn (v ′ ,w ′ ,s)α (X) ≤ C L (hk (v,w,s)α (X)) * .
As a corollary of Theorem 3.40 as well as a counterpart of Proposition 3.34, for any admissible (v, s, α), Proposition 3.41, which is just [84,Proposition 5.1], shows that hk (v,w,s) α (X) is invariant on w ∈ (v, ∞].
with equivalent norms.
The following proposition, which is just [84,Proposition 5.6], might be viewed as a counterpart of Proposition 3.35.
Finally, the following relation between hk (v,w,s) α (X) and the atomic localized Hardy space is just [84,Proposition 5.7].
We also list some open questions at the end of this subsection. Then it is still unknown whether or not holds true, namely, it is still unknown whether or not there exists some non-constant function h such that h ∈ JN p (R) but h jn p (R). Moreover, it is still unknown whether or not The following question is on the case q > p corresponding to Proposition 3.30.
Also, the corresponding localized cases of Questions 3.20 and 3.21 are listed as follows. The following Question 3.46 is a modification of [84, Remark 3.5], and Question 3.47 is just [84,Remark 5.8].
. Then the relation between jn (p,q,s) α (R n ) and the Riesz-Morrey space RM p,q,α (R n ) (see Subsection 4.1 for its definition) is still unclear so far, except the identity jn (p,p,s) 0 (R n ) = L p (R n ) = RM p,p,0 (R n ) due to Proposition 3.35(ii) and Theorem 4.4(ii), and the inclusion due to (3.2) and their definitions, where the implicit positive constant is independent of functions under consideration. (i) It is interesting to clarify the relation between v∈(1,∞) hk (v,w,0) 0 (Q 0 ) and h 1,w at (Q 0 ), and to find the condition on g such that g h 1,w at (Q 0 ) = lim v→1 + g hk (v,w,0) 0 (Q 0 ) . (ii) Let α ∈ (0, ∞) and s ∈ Z + . As v → 1 + , the relation between the atomic localized Hardy space (see [46] for the definition) and hk (v,w,s) α (Q 0 ) is still unknown.

Congruent John-Nirenberg-Campanato spaces
Inspired by the JNC space (see Subsection 3.1) and the space B (introduced and studied by Bourgain et al. [16]), Jia et al. [55] introduced the special John-Nirenberg-Campanato spaces via congruent cubes, which are of some amalgam features. This subsection is devoted to the main properties and some applications of congruent JNC spaces.
In what follows, for any m ∈ Z, D m (R n ) denotes the set of all subcubes of R n with side length 2 −m , D m (Q 0 ) the set of all subcubes of Q 0 with side length 2 −m ℓ(Q 0 ) for any given m ∈ Z + , and D m (Q 0 ) := ∅ for any given m ∈ Z \ Z + ; here and thereafter, ℓ(Q 0 ) denotes the side length of Q 0 . Definition 3.48. Let p, q ∈ [1, ∞), s ∈ Z + , and α ∈ R. The special John-Nirenberg-Campanato space via congruent cubes (for short, congruent JNC space) JN con (p,q,s) α (X) is defined to be the set of all f ∈ L 1 loc (X) such that f JN con (p,q,s)α (X) := sup where, for any m ∈ Z, [ f ] (m) (p,q,s) α ,X is defined to be with P (s) Q j ( f ) for any j as in (3.1) via Q replaced by Q j and the supremum taken over all collections of interior pairwise disjoint cubes {Q j } j ⊂ D m (X). In particular, let JN con p,q (X) := JN con (p,q,0) 0 (X).
(i) (non-dyadic side length) f ∈ JN con (p,q,s) α (X) if and only if f ∈ L 1 loc (X) and if and only if f ∈ L 1 loc (X) and f JN con (p,q,s)α (X) := sup where the suprema are taken over all collections of interior pairwise disjoint cubes {Q j } j of X with the same side length; moreover, · JN con (p,q,s)α (X) ∼ · JN con (p,q,s)α (X) ∼ · JN con (p,q,s)α (X) ; see [55, Remark 1.6(ii) and Propositions 2.6 and 2.7].
Then f ∈ JN con (p,q,s) α (R n ) if and only if f ∈ L 1 loc (R n ) and moreover, · JN con (p,q,s)α (R n ) ∼ · * ; see [55, Proposition 2.2] for this equivalence which plays an essential role when establishing the boundedness of operators on congruent JNC spaces; see [56,57] for more details.
Proposition 3.50. Let s ∈ Z + , α ∈ R, and Q 0 be any given cube of R n .
Moreover, when X = Q 0 , we further have the VMO-H 1 type duality for the congruent Hardytype space; see Theorem 5.34 below.
Recall that Essén et al. [40] introduced and studied the Q space on R n , which generalizes the space BMO (R n ). Later, the Q space proves very useful in harmonic analysis, potential analysis, partial differential equations as well as the closely related fields; see, for instance, [98,61,101]. Thus, it is natural to consider some "new Q space" corresponding to the John-Nirenberg space JN p . Based on Remark 3.49(ii), Tao et al. [95] introduced and studied the John-Nirenberg-Q space on R n via congruent cubes, which contains the congruent John-Nirenberg space on R n as special cases, and also sheds some light on the mysterious John-Nirenberg space.

Riesz-type space
Observe that, if we partially subtract integral means (or polynomials for high order cases) in f JN (p,q,s)α (X) , namely, drop P (s) Q i ( f )} i in the norm of the JNC space). As a bridge connecting Lebesgue and Morrey spaces via Riesz norms, it was called the "Riesz-Morrey space". For more studies on the well-known Morrey space, we refer the reader to, for instance, [49,68,67,50] and, in particular, the recent monographs by Sawano et al. [85,86].

Riesz-Morrey spaces
As a suitable substitute of L ∞ (X), the space BMO (X) proves very useful in harmonic analysis and partial differential equations. Recall that Indeed, the only difference between them exists in subtracting integral means, which is just the following proposition. In what follows, for any measurable function f , let if and only if f ∈ L 1 loc (X) and f L ∞ * (X) < ∞. Moreover, · L ∞ (X) = · L ∞ * (X) . Proof. On one hand, for any f ∈ L ∞ (X), it is easy to see that f ∈ L 1 loc (X) and On the other hand, for any f ∈ L 1 loc (X) and f L ∞ * (X) < ∞, let x be any Lebesgue point of f . Then, from the Lebesgue differentiation theorem, we deduce that which, together with the Lebesgue differentiation theorem again, further implies that and hence f ∈ L ∞ (X). Moreover, we have · L ∞ (X) = · L ∞ * (X) . This finishes the proof of Proposition 4.1.
Also, if we remove integral means in the JN p (Q 0 )-norm where the supremum is taken over all collections of cubes {Q i } i of Q 0 with pairwise disjoint interiors, then we obtain which coincides with f L p (Q 0 ) due to Riesz [80]. Corresponding to the JNC space, the following triple index Riesz-type space R p,q,α (X), called the Riesz-Morrey space, was introduced and studied in [92].
and the suprema are taken over all collections of subcubes {Q i } i of X with pairwise disjoint interiors. In addition, R p,q,0 (X) =: R p,q (X).
Observe that the Riesz-Morrey norm · RM p,q,α (X) differs from the John-Nirenberg-Campanato norm · JN (p,q,s)α (X) with s = 0 only in subtracting mean oscillations; see [92,Remark 2] for more details. It is easy to see that · R p,1,0 (Q 0 ) = · R p (Q 0 ) and, as a generalization of the above equivalence in Riesz [80], the following proposition is just [92,Proposition 1]. Moreover, L p (X) = R p,q (X) with equivalent norms, namely, for any f ∈ L q loc (X), f L p (X) = f R p,q (X) .
As for the case 1 ≤ p < q ≤ ∞, by [92,Remark 2.3], we know that  [100] give an affirmative answer to this question via constructing two nontrivial functions over R n and any given cube Q of R n . It should be pointed out that the nontrivial function on the cube Q is geometrically similar to the striking function constructed by Dafni   (i) Let p ∈ (1, ∞] and q ∈ [1, p). Then In particular, if α ∈ (− 1 q , 0), then RM ∞,q,α (R n ) = M −1/α q (R n ) which is just the Morrey space defined in Remark 3.2.
In particular, RM ∞,q,α (Q 0 , and Q 0 be any cube of R n . Then Recall that, by [14,Theorem 1], the predual space of the Morrey space is the so-called block space. Combining this with the duality of John-Nirenberg-Campanato spaces in [93,Theorem 3.9], the authors in [92] introduced the block-type space which proves the predual of the Riesz-Morrey space. Observe that every (∞, v, α)-block in Definition 4.5(i) is exactly a (v, α n )-block introduced in [14].
where {b j } j are (u, v, α)-blocks supported, respectively, in subcubes {Q j } of X with pairwise disjoint interiors, and {λ j } j ⊂ C with {λ j } j ℓ u < ∞ [see (3.4) for the definition of · ℓ u ]. Moreover, any h ∈ B u,v,α (X) is called a (u, v, α)-chain and its norm is defined by setting where the infimum is taken over all decompositions of h as above.
(iii) The block-type space B u,v,α (X) is defined by setting where {h i } i are (u, v, α)-chains. Moreover, for any g ∈ B u,v,α (X), where the infimum is taken over all decompositions of g as above.
(ii) If L ∈ (B p ′ ,q ′ ,α (X)) * , then there exists some f ∈ RM p,q,α (X) such that, for any g ∈ B fin p ′ ,q ′ ,α (X), with the positive equivalence constants independent of f .   and, for any f ∈ JN (p,q,s) α (X), with the positive equivalence constants independent of f , still hold true. This is still unclear so far.
Question 4.9. Recall that, for any given f ∈ L 1 loc (X) and any x ∈ X, the Hardy-Littlewood maximal function M( f )(x) is defined by setting is finite; moreover, · RM con p,q,α (R n ) ∼ · RM con p,q,α (R n ) ; see [56] for more details. Recall that, for any y ∈ R n and r ∈ (0, ∞), The following boundedness of the Hardy-Littlewood maximal operator on congruent Riesz-Morrey spaces was obtained in [56].
Finally, since a congruent Riesz-Morrey space is a ball Banach function space, we refer the reader to [94] for the equivalent characterizations of the boundedness and the compactness of Calderón-Zygmund commutators on ball Banach function spaces. It should be mentioned that, a crucial assumption in [94] is the boundedness of M, and hence Theorem 4.12 provides an essential tool when studying the boundedness of operators on congruent Riesz-Morrey spaces.

Vanishing subspace
In this section, we focus on several vanishing subspaces of aforementioned John-Nirenbergtype spaces. In what follows, C ∞ (R n ) denotes the set of all infinitely differentiable functions on R n ; 0 denotes the origin of R n ; for any α := (α 1 , . . . , α n ) ∈ Z n + := (Z + ) n , let ∂ α := ( ∂ ∂x 1 ) α 1 · · · ( ∂ ∂x n ) α n ; for any given normed linear space Y and any given its subset X, X Y denotes the closure of the set X in Y in terms of the topology of Y, and, if Y = R n , we then denote X Y simply by X.

Vanishing BMO spaces
We now recall several vanishing subspaces of the space BMO (R n ).
• VMO (R n ), introduced by Sarason [81], is defined by setting where C u (R n ) denotes the set of all uniformly continuous functions on R n .
• CMO (R n ), announced in Neri [79], is defined by setting where C ∞ c (R n ) denotes the set of all infinitely differentiable functions on R n with compact support. In addition, by approximations of the identity, it is easy to find that where C c (R n ) denotes the set of all functions on R n with compact support, and C 0 (R n ) the set of all continuous functions on R n which vanish at the infinity.
• MMO (R n ), introduced by Torres and Xue [96], is defined by setting • XMO (R n ), introduced by Torres and Xue [96], is defined by setting • X 1 MO (R n ), introduced by Tao el al. [88], is defined by setting with C 1 (R n ) being the set of all functions f on R n whose gradients ∇ f := ( ∂ f ∂x 1 , . . . , ∂ f ∂x n ) are continuous.
The relation of these vanishing subspaces reads as follows.
and (5.6) holds true in this case.
and (5.6) holds true in this case.
Question 5.12. Since the Riesz transform is well defined on L ∞ (R n ), it is interesting to find the counterpart of Theorem 5.11 when f ∈ MMO (R n ). Moreover, since the Riesz transform characterization is useful when proving the duality of CMO-H 1 type, it is also interesting to find the dual spaces of MMO (R n ) and XMO (R n ).
When R n is replaced by some cube Q 0 with finite length, we have VMO (Q 0 ) = CMO (Q 0 ); see [30] for more details. Moreover, the vanishing subspace on the spaces of homogeneous type, denoted by X, was studied in Coifman et al. [28] and they proved (VMO(X)) * = H 1 (X), where VMO(X) denotes the closure in BMO (X) of continuous functions on X with compact support. Notice that, when X = R n , by (5.1), we have VMO(X) = VMO(R n ) = CMO (R n ).
Finally, we consider the localized version of these vanishing subspaces. The following characterization of local VMO (R n ) is a part of [15, Theorem 1].
Proposition 5.13. Let vmo (R n ) be the closure of C u (R n ) ∩ bmo (R n ) in bmo (R n ). Then f ∈ vmo (R n ) if and only if f ∈ bmo (R n ) and Moreover, the following localized result of CMO (R n ) is just Dafni [30,Theorem 6]; see also [15,Theorem 3].
Theorem 5.14. Let cmo (R n ) be the closure of C 0 (R n ) in bmo (R n ) . Then f ∈ cmo (R n ) if and only if f ∈ bmo (R n ) and In addition, the localized version of Theorem 5.11 can be found in [46, Corollary 1] for bmo (R n ), and in [15, Theorems 1 and 3] for vmo (R n ) and cmo (R n ), respectively. Question 5.15. Let mmo (R n ), xmo (R n ), and x 1 mo (R n ) be, respectively, the closure in bmo (R n ) of A ∞ (R n ), B ∞ (R n ), and B 1 (R n ). It is interesting to find the counterparts of (i) Theorem 5.14 with cmo (R n ) replaced by xmo (R n ); (ii) Theorem 5.4 with XMO (R n ) and X 1 MO (R n ) replaced, respectively, by xmo (R n ) and x 1 mo (R n ); (iii) Question 5.12 with MMO (R n ) replaced by mmo (R n ); (iv) the dual result ( cmo (R n )) * = h 1 (R n ), in [30,Theorem 9], with cmo (R n ) replaced by mmo (R n ) or xmo (R n ), where h 1 (R n ) is the localized Hardy space; (v) the equivalent characterizations for mmo (R n ) and xmo (R n ) via localized Riesz transforms.

Vanishing John-Nirenberg-Campanato spaces
Very recently, the vanishing subspaces of John-Nirenberg spaces were also studied in [91,17]. Indeed, as a counterpart of Subsection 5.1, the vanishing subspaces of JNC spaces enjoy similar characterizations which are summarized in this subsection.
It is obvious that Theorems 5.18 and 5.19 hold true with [0, 1] n replaced by any cube Q 0 of R n . As an application of the duality, Tao et al. [91,Proposition 5.7] showed that, for any p ∈ (1, ∞) and any given cube Q 0 of R n , namely, [91, (5.2)], directly from Theorems 5.19 and 3.12 because, in the statements of these dual theorems, q can not equal to 1. Indeed, (5.7) still holds true due to the equivalence of JN p,q (Q 0 ) with q ∈ [1, p). Precisely, let p ∈ (1, ∞) and q ∈ (1, p). By Theorems 5.19 and 3.12, we obtain Next, we consider the case X = R n . The following proposition indicates that the convolution is a suitable tool when approximating functions in JN p (R n ), which is a counterpart of [81, Lemma 1]. Indeed, the approximate functions in the proofs of both Theorems 5.22 and 5.24 are constructed via the convolution; see [91] for more details.  ∈ (1, ∞). Then the following three statements are mutually equivalent: and ∇ f denotes the gradient of f ; (ii) f ∈ JN p (R n ) and, for any given q ∈ [1, p), where the supremum is taken over all collections {Q i } i of interior pairwise disjoint subcubes of R n with side lengths no more than a; (iii) f ∈ JN p (R n ) and where the supremum is taken over all collections {Q i } i of interior pairwise disjoint subcubes of R n with side lengths no more than a. Now, we recall another vanishing subspace of JN p (R n ) introduced in [91], which is of CMO type.
Definition 5.23. Let p ∈ (1, ∞). The vanishing subspace C JN p (R n ) of JN p (R n ) is defined by setting where C ∞ c (R n ) denotes the set of all infinitely differentiable functions on R n with compact support. The following theorem is just [91,Theorem 4.3].
Theorem 5.24. Let p ∈ (1, ∞). Then f ∈ C JN p (R n ) if and only if f ∈ JN p (R n ) and f satisfies the following two conditions: where the supremum is taken over all collections {Q i } i of interior pairwise disjoint subcubes of R n with side lengths {ℓ(Q i )} i no more than a; (ii) lim a→∞ sup {Q⊂R n : ℓ(Q)≥a} where the supremum is taken over all cubes Q of R n with side lengths ℓ(Q) no less than a.
Moreover, Tao et al. [91,Theorem 4.4] showed that Theorem 5.24(ii) can be replaced by the following statement: where the supremum is taken over all collections {Q i } i of interior pairwise disjoint subcubes of R n with side lengths {ℓ(Q i )} i greater than a.
Furthermore, Tao  for any q ∈ [1, p). However, there still exist some unsolved questions on the vanishing John-Nirenberg space. The first question is on the case p = 1. (i) It is still unknown whether or not Theorems 5.22 and 5.24 hold true with JN p (R n ) replaced by JN (p,q,s) α (R n ) when p, q ∈ [1, ∞), s ∈ Z + , and α ∈ R \ {0}.
Obviously, [L p (R n )/C] ⊂ C JN p (R n ) ⊂ V JN p (R n ) ⊂ JN p (R n ). Then the last question naturally arises, which is just [91, Questions 5.6 and 5.8].
Question 5.27. Let p ∈ (1, ∞). It is interesting to ask whether or not L p (R n )/C C JN p (R n ) V JN p (R n ) JN p (R n ) holds true. This is still unclear so far.