Boundedness of Calderón–Zygmund operators on special John–Nirenberg–Campanato and Hardy-type spaces via congruent cubes

Abstract

Let p ∈ [1, ∞], q ∈ (1, ∞), s ∈ Z + := N ∪ {0}, and α ∈ R. In this article, the authors introduce a reasonable version T of the Calderón-Zygmund operator T on J N con ( p,q,s) α (R n ), the special John-Nirenberg-Campanato space via congruent cubes, which when p = ∞ coincides with the Campanato space C α,q,s (R n ). Then the authors prove that T is bounded on, which is a well-known assumption. To this end, the authors find an equivalent version of this assumption. Moreover, the authors show that T can be extended to a unique continuous linear operator on the Hardy-kind space H K con ( p,q,s) α (R n ), the predual space of J N con ( p ,q ,s) α (R n ) with 1 p + 1 p = 1 = 1 q + 1 q , if and only if, for any γ ∈ Z n + with |γ | ≤ s, T * (x γ ) = 0. The main interesting integrands in the latter boundedness are that, to overcome the difficulty caused by that • H K con ( p,q,s)α (R n ) is no longer concave, the authors first find an equivalent

Special John-Nirenberg-Campanato spaces via congruent cubes

“…(iii) We have obtained the boundedness of Calderón-Zygmund operators, fractional integrals and Littlewood-Paley operators on congruent JNC spaces, respectively, in [24][25][26], which is actually also one of our main motivations to study these spaces. One main tool used in those proofs in [24][25][26] is Proposition 2.2 below, but it cannot be applied into the John-Nirenberg-Campanato space, and hence the corresponding boundedness results on the John-Nirenberg-Campanato space are still unknown. In addition, if we require the number of cubes {Q j } j in the supremum of • JN con (p,q,s)α (X ) less then 2 m(n−1) as in • B((0,1) n ) , then Proposition 2.2 would not hold true.…”

“…Similar to the Riesz-Morrey space introduced in [49], the space (L q , p ) α (R n ) also provides a bridge connecting Lebesgue spaces and Morrey spaces. Indeed, this space is closely related to the congruent Riesz-Morrey space which is introduced and studied in two forthcoming articles [24,25].…”

“…Furthermore, several VMO-H 1 -BMO-type dual properties for congruent JNC spaces are also established. In another three articles [24][25][26], some applications of this new space in the study on the boundedness of Calderón-Zygmund operators and Littlewood-Paley operators are given. Observe that 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 which shed some light on the mysterious spaces JN p (R n ) and B(R n ).…”

John--Nirenberg-$Q$ Spaces via Congruent Cubes

“…Moreover, let JN con p (R n ) := JN con p,1 (R n ). Very recently, Jia et al [20,21,22] further showed that several important operators (such as the Hardy-Littlewood maximal operator, Calderón-Zygmund operators, fractional integrals, and Littlewood-Paley operators) are bounded on congruent John-Nirenberg spaces. Thus, it is meaningful to study and reveal more properties of the congruent John-Nirenberg space.…”

“…Indeed, we obtain a more general equivalence principle in Proposition 2.5 below via introducing the 'almost increasing' set function Φ (see Definition 2.3 below). It should be pointed out that Proposition 2.5 may have independent interest because, as a special case of this proposition with Φ replaced by the mean oscillation as in (1.1), [19,Proposition 2.2] proves extremely useful when studying the boundedness of the Hardy-Littlewood maximal operator, Calderón-Zygmund operators, fractional integrals, and Littlewood-Paley operators on congruent John-Nirenberg spaces; see [20,21,22] for more details. Moreover, we show that Q α (R n ) can be regarded as the limit space of JNQ α p,2 (R n ) as p → ∞ in Proposition 2.10 below, and also obtain an extension result over cubes (with finite edge length) in Proposition 2.11 below.…”