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.

There still exist many unsolved problems on the study related to John–Nirenberg spaces. In this article, the authors introduce two new vanishing subspaces of the John–Nirenberg space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} denoted, respectively, by VJN p ⁢ ( ℝ n ) {\mathrm{VJN}_{p}(\mathbb{R}^{n})} and CJN p ⁢ ( ℝ n ) {\mathrm{CJN}_{p}(\mathbb{R}^{n})} , and establish their equivalent characterizations which are counterparts of those characterizations for the classic spaces VMO ⁢ ( ℝ n ) {\mathrm{VMO}(\mathbb{R}^{n})} and CMO ⁢ ( ℝ n ) {\mathrm{CMO}(\mathbb{R}^{n})} obtained, respectively, by D. Sarason and A. Uchiyama. All these results shed some light on the mysterious space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} . The approach strongly depends on the fine geometrical properties of dyadic cubes, which enable the authors to subtly classify any collection of interior pairwise disjoint cubes.

In this article, via combining Riesz norms with Morrey norms, the authors introduce and study the so-called Riesz-Morrey space, which differs from the John-Nirenberg-Campanato space in subtracting integral means. These spaces provide a bridge connecting both Lebesgue spaces and Morrey spaces which prove to be the endpoint spaces of Riesz-Morrey spaces. Moreover, the authors introduce a block-type space which proves to be the predual space of the Riesz-Morrey space.

and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg (1961) and the space B introduced by Bourgain et al. ( 2015), we introduce a special John-Nirenberg-Campanato space JN con (p,q,s)α over R n or a given cube of R n with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p → ∞ is just the Campanato space which coincides with the space BMO (the space of functions with bounded mean oscillations) when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO (the space of functions with vanishing mean oscillations) over R n or a given cube of R n with finite side length. Furthermore, some VMO-H 1 -BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.

Let CΓ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,CΓ] is bounded (resp, compact) on the Morrey space Lp,λfalse(double-struckRfalse) for any (or some) p  ∈  (1,∞) and λ  ∈  (0,1) if and only if b∈0.1emBMOfalse(double-struckRfalse) (resp, CMOfalse(double-struckRfalse)). As an application, a factorization of the classical Hardy space H1false(double-struckRfalse) in terms of CΓ and its adjoint operator is obtained.