In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces Fp^a,q（w;R^n1×R^n2） based on the Frazier and Jawerth＇ method in [11]. This space was firstly introduced in [18]. Then we estab...In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces Fp^a,q（w;R^n1×R^n2） based on the Frazier and Jawerth＇ method in [11]. This space was firstly introduced in [18]. Then we establish its dual space and get that （Fp＇q）＊ = CMOp^-a,q＇ for 0 ~p≤ 1.展开更多
In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class A q and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.
Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish ...Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ R, p ∈ （0, 1] and q ∈ [p, ∞）. The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in （0, 1]. Finite atomic decompositions for smooth functions in Fs,w p,q（Rn） are also obtained, which further implies that a （sub）linear operator that maps smooth atoms of Fs,w p,q（Rn） uniformly into a bounded set of a （quasi-）Banach space is extended to a bounded operator on the whole Fs,w p,q（Rn） As an application, the baundedness of the local Riesz operator on the space Fs,w p,q（Rn） is obtained.展开更多
It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to ...It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).展开更多
In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^＊ on the w...In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^＊ on the weighted amalgam spaces （Lω^q,L^p）^α（R^n）as 1〈q≤α〈p≤∞.展开更多
In this paper, we will obtain the weak type estimates of intrinsic square func- tions including the Lusin area integral, Littlewood-Paley g-function and g^-function on the weighted Morrey spaces L^1,k （w） for 0〈k〈...In this paper, we will obtain the weak type estimates of intrinsic square func- tions including the Lusin area integral, Littlewood-Paley g-function and g^-function on the weighted Morrey spaces L^1,k （w） for 0〈k〈 1 and w ∈ A1.展开更多
Let L = -△＋V be a Schrodinger operator acting on L^2（R^n）, n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp （w） associated wit...Let L = -△＋V be a Schrodinger operator acting on L^2（R^n）, n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp （w） associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform △↓L^-1/2 associated with L is bounded from our new space HP（w） to the classical weighted Hardy sp ace HP （ w ） when n / （n ＋ 1 ） 〈 p 〈 1 and w ∈ A 1 ∩ R H（ 2 / p ）＇.展开更多
Let w be a Muckenhoupt weight and Hwp （JRn） be the weighted Hardy space. In this paper, by using the atomic decomposition of Hwp（Rn）, we will show that the Bochner-Riesz operators TRδ are bounded from Hwp（Rn） t...Let w be a Muckenhoupt weight and Hwp （JRn） be the weighted Hardy space. In this paper, by using the atomic decomposition of Hwp（Rn）, we will show that the Bochner-Riesz operators TRδ are bounded from Hwp（Rn） to the weighted weak Hardy spaces WHwp （Rn） for 0 〈 p 〈 1 and δ = n/p- （n ＋ 1）/2. This result is new even in the unweighted case.展开更多
Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the...Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon＇s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe＇s covering lemma in multiparameter setting.展开更多
In this paper, we introduce weighted vector-valued Morrey spaces and obtain some estimates for vector-valued commutators on these spaces. Applications to CalderSn-Zygmund singular integral operators, oscillatory singu...In this paper, we introduce weighted vector-valued Morrey spaces and obtain some estimates for vector-valued commutators on these spaces. Applications to CalderSn-Zygmund singular integral operators, oscillatory singular integral operators and parabolic difference equations are considered.展开更多
The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950'...The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper, we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces(of fractional order of smoothness),based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney extension operator.展开更多
基金Supported by NNSF of China grants(11501308,11271209,11371370)Jiangsu Government Scholarship for Overseas Studies
文摘In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces Fp^a,q（w;R^n1×R^n2） based on the Frazier and Jawerth＇ method in [11]. This space was firstly introduced in [18]. Then we establish its dual space and get that （Fp＇q）＊ = CMOp^-a,q＇ for 0 ~p≤ 1.
基金The projectsupported by NSF of China and the Foundation of Advanced Research Center of Zhongshan Universi-ty
文摘In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class A q and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.
基金supported by the National Natural Science Foundation of China(Nos.11101425,11171027)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20120003110003)
文摘Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ R, p ∈ （0, 1] and q ∈ [p, ∞）. The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in （0, 1]. Finite atomic decompositions for smooth functions in Fs,w p,q（Rn） are also obtained, which further implies that a （sub）linear operator that maps smooth atoms of Fs,w p,q（Rn） uniformly into a bounded set of a （quasi-）Banach space is extended to a bounded operator on the whole Fs,w p,q（Rn） As an application, the baundedness of the local Riesz operator on the space Fs,w p,q（Rn） is obtained.
文摘It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).
基金supported in part by National Natural Foundation of China (Grant No. 11161042 and No. 11071250)
文摘In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^＊ on the weighted amalgam spaces （Lω^q,L^p）^α（R^n）as 1〈q≤α〈p≤∞.
文摘In this paper, we will obtain the weak type estimates of intrinsic square func- tions including the Lusin area integral, Littlewood-Paley g-function and g^-function on the weighted Morrey spaces L^1,k （w） for 0〈k〈 1 and w ∈ A1.
文摘Let L = -△＋V be a Schrodinger operator acting on L^2（R^n）, n ≥ 1, where V ≠ 0 is a nonnegative locally integrable function on R^n. In this article, we will introduce weighted Hardy spaces Hp （w） associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform △↓L^-1/2 associated with L is bounded from our new space HP（w） to the classical weighted Hardy sp ace HP （ w ） when n / （n ＋ 1 ） 〈 p 〈 1 and w ∈ A 1 ∩ R H（ 2 / p ）＇.
文摘Let w be a Muckenhoupt weight and Hwp （JRn） be the weighted Hardy space. In this paper, by using the atomic decomposition of Hwp（Rn）, we will show that the Bochner-Riesz operators TRδ are bounded from Hwp（Rn） to the weighted weak Hardy spaces WHwp （Rn） for 0 〈 p 〈 1 and δ = n/p- （n ＋ 1）/2. This result is new even in the unweighted case.
基金supported by the NSF of USA(Grant No.DMS0901761)supported by NNSF of China(Grant Nos.10971228and11271209)Natural Science Foundation of Nantong University(Grant No.11ZY002)
文摘Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon＇s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe＇s covering lemma in multiparameter setting.
基金Supported by National Natural Science Foundation of China(Grant Nos.10901076and11271175)National Natural Science Foundation of Shandong Province(Grant No.ZR2012AQ026)the Key Laboratory of Mathematics and Complex System (Beijing Normal University),Ministry of Education,China
文摘In this paper, we introduce weighted vector-valued Morrey spaces and obtain some estimates for vector-valued commutators on these spaces. Applications to CalderSn-Zygmund singular integral operators, oscillatory singular integral operators and parabolic difference equations are considered.
基金supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Grant No. 307333)
文摘The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper, we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces(of fractional order of smoothness),based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney extension operator.
