Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity

Hou, Shao Xiong; Yang, Danni; Yang, Si Bei

doi:10.1007/s10114-014-3181-9

Cited by 7 publications

(6 citation statements)

References 57 publications

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“…A function φ : [0, ∞) → [0, ∞) is called an Orlicz function if it is non-decreasing and φ(0) = 0, φ(t) > 0 if t ∈ (0, ∞), and lim t→∞ φ(t) = ∞. Now, we recall the notions of uniformly lower (resp., upper) types on Musielak-Orlicz functions from [31].…”

Section: Preliminariesmentioning

confidence: 99%

“…We recall the notions of the uniformly Muckenhoupt condition and the uniformly reverse Hölder condition from [31].…”

Section: Definition 23mentioning

confidence: 99%

“…Based on these wavelet reproducing formulae, He et al [27] further established a complete real-variable theory of Hardy spaces H p (X ) on spaces of homogeneous type without having recourse to the reverse doubling condition (1.5). On another hand, Hou et al [31] investigated Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity on spaces of homogeneous type.…”

Section: Introductionmentioning

confidence: 99%

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Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Fu¹,

Ma²,

Yang³

2020

Ann. Acad. Sci. Fenn. Math.

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Let (X , d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak-Orlicz Hardy spaces on (X , d, µ). To be precise, the authors first introduce the atomic Musielak-Orlicz Hardy space H ϕ at (X) and then establish its various maximal function characterizations. The authors also investigate the Littlewood-Paley characterizations of H ϕ at (X) via Lusin area functions, Littlewood-Paley g-functions and Littlewood-Paley g * λ-functions. The authors further obtain the finite atomic characterization of H ϕ at (X) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H ϕ at (X) to a quasi-Banach space, which are also applied to the boundedness of Calderón-Zygmund operators. Moreover, the authors find the dual space of H ϕ at (X), namely, the Musielak-Orlicz BMO space BMO ϕ (X), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X). The main novelty of this article is that, throughout the article, except the last section, µ is not assumed to satisfy the reverse doubling condition.

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Section: Preliminariesmentioning

confidence: 99%

“…We recall the notions of the uniformly Muckenhoupt condition and the uniformly reverse Hölder condition from [31].…”

Section: Definition 23mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Fu¹,

Ma²,

Yang³

2020

Ann. Acad. Sci. Fenn. Math.

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“…(x, t)ˆB (x,t)ˆt 0 |s 2 Le −s 2 L e −s 2 L f | 2 ds s dµ(y) 1/f BMO L,w (X) .This, along with Theorem 3.21, implies the assertion. (b) It was proved in[44] thatf BMOw(R n ) ∼ sup x∈X,t>0|B(x, t)| w(B(x, t)) 2ˆB (x,t)ˆt 0 |s 2 ∆e s 2 ∆ (I − e s 2 ∆ )f | 2 ds s dy1/Using this and Theorem 3.21, we derive part (b). Coincidence with the weighted Sobolev spacesẆ s,L p,w .…”

mentioning

confidence: 73%

“…The unweighted BMO space BMO L (X) associated to operators L was first introduced by [35]. The weighted version was studied in [19,44].…”

Section: 3mentioning

confidence: 99%

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Bui

Duong

2020

Forum of Mathematics, Sigma

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Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L 2 (X) satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spacesḂ α,L p,q,w (X) and weighted Triebel-Lizorkin spacesḞ α,L p,q,w (X) associated to the operator L for the full range 0 < p, q ≤ ∞, α ∈ R and w being in the Muckenhoupt weight class A∞. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated to L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L and the spectral multiplier of L in our new function spaces.

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Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Yan

Yang

et al. 2023

Mathematische Nachrichten

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Let false(scriptX,ρ,μfalse)$({\mathcal {X}},\rho ,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Yfalse(scriptXfalse)$Y({\mathcal {X}})$ be a ball quasi‐Banach function space on X${\mathcal {X}}$, which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HY∗(X)$H_{Y}^*({\mathcal {X}})$, associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$, via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of HY∗(X)$H_{Y}^*({\mathcal {X}})$, which proves to be a ball Campanato‐type function space associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of X${\mathcal {X}}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of HY∗(X)$H_{Y}^*({\mathcal {X}})$, the authors further prove that Yfalse(scriptXfalse)$Y({\mathcal {X}})$ can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

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Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity

Cited by 7 publications

References 57 publications

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type

Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

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