A Survey on Function Spaces of John–Nirenberg Type

Tao, Jin; Yang, Dachun; Yuan, Wen

doi:10.3390/math9182264

Cited by 16 publications

(11 citation statements)

References 95 publications

(207 reference statements)

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“…Both JN p and BMO are function spaces based on mean oscillations of functions, and we refer the reader to [3,6,7,8,12,13,14,18,21,22] for more related researches. Obviously, we have JN 1 (Q 0 ) = L 1 (Q 0 ) and JN ∞ (Q 0 ) = BMO (Q 0 ); see, for instance, [27]. Moreover, an interesting result of Dafni et al [9] shows the non-triviality of JN p (Q 0 ) with p ∈ (1, ∞) via constructing a surprising function belonging to JN p (Q 0 ) but not an element of L p (Q 0 ).…”

Section: Introductionmentioning

confidence: 99%

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

Hu¹,

Jin²,

Yang³

2022

Preprint

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Let p, q ∈ [1, ∞], α ∈ R, and s be a non-negative integer. In this article, the authors introduce a new function space JN (p,q,s) α (X) of John-Nirenberg-Campanato type, where X denotes R n or any cube Q 0 of R n with finite edge length. The authors give an equivalent characterization of JN (p,q,s) α (X) via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case s = 0, this new space can be equivalently characterized by both maximal functions and their commutators. Additionally, the authors give some basic properties, a good-λ inequality, and a John-Nirenberg type inequality for JN (p,q,s) α (X).

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Section: Introductionmentioning

confidence: 99%

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

Hu¹,

Jin²,

Yang³

2022

Preprint

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“…Observe that the John-Nirenberg space has attracted a lot of attention in recent years. For example, Dafni et al [10] showed the nontriviality of JN p (Q 0 ) and found the predual space of JN p (Q 0 ); Aalto et al [1] introduced the John-Nirenberg space in the context of doubling metric measure spaces; Hurri-Syrjänen et al [23] established a local-to-global result for the space JN p (Ω) on an open subset Ω of R n (see also [18,19,32,34,35,51] for more studies on John-Nirenberg-type spaces).…”

Section: Introductionmentioning

confidence: 99%

“…Brudnyi and Brudnyi [6] introduced a BMO-type space V κ ([0, 1] n ) which coincides with JN (p,q,s)α ([0, 1] n ) when α ∈ [0, s+1 n ] (see [48,Proposition 2.9] for more details). In addition, Sun et al [46] introduced and studied the localized John-Nirenberg-Campanato space (see also [51] for more studies on those spaces).…”

Section: Introductionmentioning

confidence: 99%

Special John-Nirenberg-Campanato spaces via congruent cubes

et al. 2021

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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.

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“…Moreover, Tao et al [42] studied the John-Nirenberg-Campanato space, which is a generalization of the space J N p (Q 0 ), and Sun et al [41] studied the localized John-Nirenberg-Campanato space. We refer the reader to [13,14,16,25,27,28,44,45] for more studies on John-Nirenberg-type spaces.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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

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A Survey on Function Spaces of John–Nirenberg Type

Cited by 16 publications

References 95 publications

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

Special John-Nirenberg-Campanato spaces via congruent cubes

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

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