Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications

Hou, Shaoxiong; Yang, Dachun; Yang, Sibei

doi:10.48550/arxiv.1201.1945

Cited by 5 publications

(17 citation statements)

References 30 publications

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“…We remark that the method used to obtain the atomic characterization of the Musielak-Orlicz-Hardy space H ϕ, L (X ) in this paper is different from that in [86], but more close to the method in [54,15,57]. More precisely, in [86], the atomic characterization of the weighted Hardy space H 1 L (R n ), associated with the Schrödinger operator L, was established by using the Calderón reproducing formula associated with L and a subtle decomposition of all dyadic cubes in R n .…”

Section: Below)mentioning

confidence: 89%

“…By the uniformly upper type p 1 property of ϕ with some p 1 ∈ [I(ϕ), 1], Theorem 3.1 and its proof, similar to the proof of [54,Corollary 3.4], we can show Corollary 3.5 and omit the details here.…”

Section: Some Basic Properties On Growth Functionsmentioning

confidence: 92%

“…Furthermore, some interesting applications of the spaces H ϕ (R n ) and BMO ϕ (R n ) were given in [11,13,14,63,64,65,66]. Moreover, the radial and the non-tangential maximal functions characterizations, the Littlewood-Paley function characterization and the molecular characterization of H ϕ (R n ) were obtained in [69,54]. As an application of the Lusin area function characterization of H ϕ (R n ), the ϕ-Carleson measure characterization of the space BMO ϕ (R n ) was obtained in [54].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the radial and the non-tangential maximal functions characterizations, the Littlewood-Paley function characterization and the molecular characterization of H ϕ (R n ) were obtained in [69,54]. As an application of the Lusin area function characterization of H ϕ (R n ), the ϕ-Carleson measure characterization of the space BMO ϕ (R n ) was obtained in [54]. Furthermore, the local Musielak-Orlicz-Hardy space and its dual space were studied in [97].…”

Section: Introductionmentioning

confidence: 99%

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Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Yang

2012

J Geom Anal

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Let X be a metric space with doubling measure and L a nonnegative selfadjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimates. Let ϕ : X × [0, ∞) → [0, ∞) be a function such that ϕ(x, ·) is an Orlicz function, ϕ(·, t) ∈ A ∞ (X ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(ϕ) ∈ (0, 1] and it satisfies the uniformly reverse Hölder inequality of order 2/[2 − I(ϕ)]. In this paper, the authors introduce a Musielak-Orlicz-Hardy space H ϕ, L (X ), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak-Orlicz BMO-type space BMO ϕ, L (X ), which is further proved to be the dual space of H ϕ, L (X ) and hence whose ϕ-Carleson measure characterization is deduced. Characterizations of H ϕ, L (X ), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize H ϕ, L (X ) in terms of the Littlewood-Paley g * λ -function g * λ, L and establish a Hörmandertype spectral multiplier theorem for L on H ϕ, L (X ). Moreover, for the Musielak-Orlicz-Hardy space H ϕ, L (R n ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R n ), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H ϕ, L (R n ) to the Musielak-Orlicz space L ϕ (R n ) when i(ϕ) ∈ (0, 1], and from H ϕ, L (R n ) to the Musielak-Orlicz-Hardy space H ϕ (R n ) when i(ϕ) ∈ ( n n+1 , 1], where i(ϕ) denotes the uniformly critical lower type index of ϕ.

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Section: Below)mentioning

confidence: 89%

Section: Some Basic Properties On Growth Functionsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Yang

2012

J Geom Anal

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“…Recently, Ky [26] introduced a new Musielak-Orlicz Hardy space H ϕ (R n ), which generalizes both the Orlicz-Hardy space (see, for example, [21,46]) and the weighted Hardy space (see, for example, [16,17,25,33,44]). Moreover, characterizations of H ϕ (R n ) in terms of Littlewood-Paley functions (see [19,30]) and the intrinsic ones (see [32]) were also obtained. As the dual space of H ϕ (R n ), the Musielak-Orlicz Campanato space L ϕ,q (R n ) with q ∈ [1, ∞) was introduced in [31], in which some characterizations of L ϕ,q (R n ) were also established.…”

Section: Introductionmentioning

confidence: 95%

Boundedness of intrinsic Littlewood--Paley functions on Musielak--Orlicz Morrey and Campanato spaces

Liang¹,

Nakai²,

Yang³

et al. 2014

Banach J. Math. Anal.

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Let ϕ : R n × [0, ∞) → [0, ∞) be such that ϕ(x, ·) is nondecreasing, ϕ(x, 0) = 0, ϕ(x, t) > 0 when t > 0, lim t→∞ ϕ(x, t) = ∞ and ϕ(·, t) is a Muckenhoupt A ∞ (R n ) weight uniformly in t. Let φ : [0, ∞) → [0, ∞) be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space M ϕ,φ (R n ) and obtain the boundedness on M ϕ,φ (R n ) of the intrinsic Lusin area function S α , the intrinsic g-function g α , the intrinsic g * λ -function g * λ,α and their commutators with BMO(R n ) functions, where α ∈ (0, 1], λ ∈ (min{max{3, p 1 }, 3 + 2α/n}, ∞) and p 1 denotes the uniformly upper type index of ϕ. Let Φ : [0, ∞) → [0, ∞) be nondecreasing, Φ(0) = 0, Φ(t) > 0 when t > 0, and lim t→∞ Φ(t) = ∞, w ∈ A ∞ (R n ) and φ : (0, ∞) → (0, ∞) be nonincreasing. The authors also introduce the weighted Orlicz-Morrey space M Φ,φ w (R n ) and obtain the boundedness on M Φ,φ w (R n ) of the aforementioned intrinsic Littlewood-Paley functions and their commutators with BMO(R n ) functions. Finally, for q ∈ [1, ∞), the boundedness of the aforementioned intrinsic Littlewood-Paley functions on the Musielak Orlicz Campanato space L ϕ,q (R n ) is also established.

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

Hou

Yang

2014

Acta. Math. Sin.-English Ser.

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Let X be a space of homogenous type and ϕ : X × [0, ∞) → [0, ∞) a growth function such that ϕ(·, t) is a Muckenhoupt weight uniformly in t and ϕ(x, ·) an Orlicz function of uniformly upper type 1 and lower type p ∈ (0, 1]. In this article, the authors introduce a new Musielak-Orlicz BMO-type space BMO ϕ A (X ) associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces BMO ϕ A, max (X ) and BMO ϕ A (X ). Moreover, two variants of the John-Nirenberg inequality on BMO ϕ A (X ) are obtained. As an application, the authors further prove that the space BMO ϕ √ ∆ (R n ), associated with the Poisson semigroup of the Laplace operator ∆ on R n , coincides with the space BMO ϕ (R n ) introduced by L. D. Ky. A t f (x) := X a t (x, y)f (y) dµ(y). Duong and Yan [17] first introduced the suitable function set M(X ) such that, for all f ∈ M(X ) and all t, s ∈where the supremum is taken over all balls B in X and t B := r m B with r B being the radius of the ball B and m a positive constant. Duong and Yan [17] gave out some basic properties of BMO A (X ) including a variant of the John-Nirenberg inequality and further proved that the space BMO √ ∆ (R n ), associated with the Poisson semigroup of the Laplace operator ∆ on R n , and BMO(R n ) coincide with equivalent norms. Tang [45] introduced the Morrey-Campanato type spaces Lip A (α, X ) associated with the generalized approximation to the identity {A t } t>0 and established the John-Nirenberg inequality on these spaces. Furthermore, Deng, Duong and Yan [12] established a new characterization of the classical Morrey-Campanato space on R n by using an appropriate convolution to replace the minimizing polynomial of a function f in the Morrey-Campanato norm. Moreover, a similar characterization for the Morrey space on R n was also obtained by Duong, Xiao and Yan in [16]. Yang and Zhou [49] introduced some generalized approximations to the identity with optimal decay conditions in the sense that these conditions are sufficient and necessary for these generalized approximations to the identity to characterize BMO(X ), which was introduced by Long and Yang [36]. Furthermore, a new John-Nirenberg-type inequality associated with the generalized approximations to the identity on BMO(X ) was also established in [49]. Recently, Bui and Duong [6] introduced the weighted BMO space BMO A (X , ω) associated to the generalized approximations to the identity, {A t } t>0 , and also obtained the John-Nirenberg inequality on these spaces.Let X be a space of homogeneous type with degree (α 0 , n 0 , N 0 ) (see Remark 2.5 below for its definition), where α 0 , n 0 and N 0 are as in (2.7), (2.3) and (2.5) below, respectively. Let ϕ : X × [0, ∞) → [0, ∞) be a growth function such that ϕ(·, t) is a Muckenhoupt weight uniformly in t, and ϕ(x, ·) is an Orlicz function of uniformly upper type 1 and lower type p ∈ (0, 1]. Motivated by [28,17,45], in this article, we first introduce the...

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Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications

Cited by 5 publications

References 30 publications

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Boundedness of intrinsic Littlewood--Paley functions on Musielak--Orlicz Morrey and Campanato spaces

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

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