Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a loca...Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a local Musielak-Orlicz Hardy space h￠（Nn） by the local grand maximal function, and a local BMO-type space bmo￠（Nn） which is further proved to be the dual space of h￠（Nn）. As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo￠（Nn）, characterized by Nakai and Yabuta, is just the dual of LI（Rn） ＋ hФ0 （Rn）, where Ф is an increasing function on （0, co） satisfying some additional growth conditions and Ф0 a Musielak-Orlicz function induced by Ф. Characterizations of h￠（Rn）, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic char- acterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h￠（Rn）, from which, the authors further deduce some criterions for the boundedness on h￠（Rn） of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h￠（Rn）.展开更多
Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz ...Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.展开更多
Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathemat...Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics,can be expressed by a fairly general discrete group of dilations where A is a real matrix with all its elgenvalues A satisfy . The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let be an anisotropic p-growth function with . The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to and the boundedness of the anisotropic Calder6n-Zygmund operator from. It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of and the interpolation theorem.展开更多
Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal f...Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ（R^n）. The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case （namely, A ：= 2In×n） coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space HP（R^n） with p∈[（0, 1] （in this case, A ：= 2In×n,φ（x, t） ：= t^p for all x ∈ R^n and t∈[0,∞））, this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Caldeon-Zygmund operators from HA^φ（Hn） to L^φ（R^n） or from HA^φ（Hn） to itself.展开更多
In this article,the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardy-type spaces,including(weak)Musielak-Orlicz Hardy spaces,va...In this article,the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardy-type spaces,including(weak)Musielak-Orlicz Hardy spaces,variable(weak)Hardy spaces,and Hardy spaces associated with ball quasi-Banach function spaces.The authors also present some open problems.展开更多
Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 ...Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t） ：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn） （the class of Muckenhoupt weights on Rn）.展开更多
In this article,we introduce the martingale Musielak-Orlicz Hardy spaces H_φ^*(■),Pφ(?),H_φ^S(■),Qφ(?)and H_φ^s(■),respectively,via the maximal function,the quadratic variation and the conditional quadratic va...In this article,we introduce the martingale Musielak-Orlicz Hardy spaces H_φ^*(■),Pφ(?),H_φ^S(■),Qφ(?)and H_φ^s(■),respectively,via the maximal function,the quadratic variation and the conditional quadratic variation of martingales.We then establish the atomic characterizations of H_φ^s(■),Pφ(■)and Qφ(■).As applications,we obtain the dual space of H_φ^s(■)and several martingale inequalities which further clarify the relations among H_φ^*(■),Pφ(■),H_φ^S(■),Qφ(■)and H_φ^s(■).Especially,as special cases,the results on atomic characterizations of H_φ^s(■),Pφ(?)and Qφ(■)as well as on the dual space of H_φ^s(■)in the weighted case are also new.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11171027)Program for Changjiang Scholars and Innovative Research Team in University of China
文摘Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a local Musielak-Orlicz Hardy space h￠（Nn） by the local grand maximal function, and a local BMO-type space bmo￠（Nn） which is further proved to be the dual space of h￠（Nn）. As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo￠（Nn）, characterized by Nakai and Yabuta, is just the dual of LI（Rn） ＋ hФ0 （Rn）, where Ф is an increasing function on （0, co） satisfying some additional growth conditions and Ф0 a Musielak-Orlicz function induced by Ф. Characterizations of h￠（Rn）, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic char- acterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h￠（Rn）, from which, the authors further deduce some criterions for the boundedness on h￠（Rn） of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h￠（Rn）.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671414, 11271091, 11471040, 11461065, 11661075, 11571039 and 11671185)
文摘Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.
基金Li Baode is supported by NSFC(No.11461065,No.11161044)Scientific Research Projects in Colleges and Universities in Xinjiang Uyghur Autonomous Region(No.XJEDU2014S001)
文摘Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics,can be expressed by a fairly general discrete group of dilations where A is a real matrix with all its elgenvalues A satisfy . The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let be an anisotropic p-growth function with . The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to and the boundedness of the anisotropic Calder6n-Zygmund operator from. It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of and the interpolation theorem.
基金supported by the National Natural Science Foundation of China(11461065)Scientific Research Projects in Colleges and Universities in Xinjiang Uyghur Autonomous Region(XJEDU2014S001)
基金partially supported by National Natural Science Foundation of China(Grant Nos.11461065,11161044,11571039 and 11361020)supported by Scientific Research Projects in Colleges and Universities in Xinjiang Uyghur Autonomous Region(Grant No.XJEDU2014S001)+2 种基金supported by National Natural Science Foundation of China(Grant No.11271175)partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120003110003)the Fundamental Research Funds for Central Universities of China(Grant Nos.2013YB60and 2014KJJCA10)
文摘Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ（R^n）. The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case （namely, A ：= 2In×n） coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space HP（R^n） with p∈[（0, 1] （in this case, A ：= 2In×n,φ（x, t） ：= t^p for all x ∈ R^n and t∈[0,∞））, this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Caldeon-Zygmund operators from HA^φ（Hn） to L^φ（R^n） or from HA^φ（Hn） to itself.
基金supported by the National Natural Science Foundation of China(Grant Nos.11971058,12071197 and 11871100)the National Key Research and Development Program of China(Grant No.2020YFA0712900).
文摘In this article,the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardy-type spaces,including(weak)Musielak-Orlicz Hardy spaces,variable(weak)Hardy spaces,and Hardy spaces associated with ball quasi-Banach function spaces.The authors also present some open problems.
文摘Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t） ：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn） （the class of Muckenhoupt weights on Rn）.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471337, 11722114, 11571039 and 11671185)
文摘In this article,we introduce the martingale Musielak-Orlicz Hardy spaces H_φ^*(■),Pφ(?),H_φ^S(■),Qφ(?)and H_φ^s(■),respectively,via the maximal function,the quadratic variation and the conditional quadratic variation of martingales.We then establish the atomic characterizations of H_φ^s(■),Pφ(■)and Qφ(■).As applications,we obtain the dual space of H_φ^s(■)and several martingale inequalities which further clarify the relations among H_φ^*(■),Pφ(■),H_φ^S(■),Qφ(■)and H_φ^s(■).Especially,as special cases,the results on atomic characterizations of H_φ^s(■),Pφ(?)and Qφ(■)as well as on the dual space of H_φ^s(■)in the weighted case are also new.
