Let A be a symmetric expansive matrix and H^p(R^n) be the anisotropic Hardy space associated with A. For a function m in L∞(R^n), an appropriately chosen function η in Cc^∞(R^n) and j ∈ Z define mj(ξ) = m...Let A be a symmetric expansive matrix and H^p(R^n) be the anisotropic Hardy space associated with A. For a function m in L∞(R^n), an appropriately chosen function η in Cc^∞(R^n) and j ∈ Z define mj(ξ) = m(A^jξ)η(ξ). The authors show that if 0 〈 p 〈 1 and mj belongs to the anisotropic nonhomogeneous Herz space K1^1/P^-1,p(R^n), then m is a Fourier multiplier from H^p(R^n) to L^V(R^n). For p = 1, a similar result is obtained if the space K1^0.1(R^n) is replaced by a slightly smaller space K(w). Moreover, the authors show that if 0 〈 p 〈 1 and if the sequence {(mj)^v} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from H^p(R^n) to L^v(R^n).展开更多
Let (.the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces (w2) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces,...Let (.the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces (w2) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show that , the class of C(Rn)-functions with compactly support, is dense in and there is a subsequence, which converges in distrbutional sense to some distribution of , of any bounded sequence in In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness.展开更多
In this paper, we study the boundedness of higher order commutators of gen- eralized fractional integral operators on weighted Lp spaces and Herz-type Hardy spaces.
In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from HKq^a、P(ω1;ω2) into Kq^a、p (ω1; ω2).
In this paper, the authors study the multilinear operators with Dini Kernel and obtain their boundedness from Herz spaces to Herz-type Hardy spaces. Moreover, the authors also consider the corresponding fractional ope...In this paper, the authors study the multilinear operators with Dini Kernel and obtain their boundedness from Herz spaces to Herz-type Hardy spaces. Moreover, the authors also consider the corresponding fractional operators.展开更多
This paper establishes some strong type and weak type estimates for commutator [b,I1] on Herz-type spaces, where b E BMO(Rn) and I1 is a fractional integration with O < l < n.
In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ<SUP> n </SUP>are introduced, and the central atomic and molecular decomposition characte...In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ<SUP> n </SUP>are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces.展开更多
In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also...In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.展开更多
In this paper,the authors first establish some new real-variable characterizations of Herz- type Hardy spaces H<sub>q</sub><sup>α,p</sup>(ω<sub>1</sub>;ω<sub>2</sub>...In this paper,the authors first establish some new real-variable characterizations of Herz- type Hardy spaces H<sub>q</sub><sup>α,p</sup>(ω<sub>1</sub>;ω<sub>2</sub>)and HK<sub>q</sub><sup>α,P</sup>(ω<sub>1</sub>;ω<sub>2</sub>),where ω<sub>1</sub>,ω<sub>2</sub> ∈A<sub>1</sub>-weight,1【q【∞, n(1-1/q)≤α【∞ and 0【p【∞.Then,using these new characterizations,they investigate the convergence of a bounded set in these spaces,and study the boundedness of some potential operators on these spaces.展开更多
Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^(p,q)_A(R^n) associated with A via the non-tangential grand maximal function and then establish...Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^(p,q)_A(R^n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H^(p1,q1)_A(Rn) and H^(p2,q2)_A(R^n) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈(0, ∞], and also between H^(p,q1)_A(Rn) and H^(p,q2)_A(R^n) with p ∈(0, ∞)and 0 < q1 < q < q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H^(p,q)_A(R^n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^(p,∞)_A(R^n) to the weak Lebesgue space L^(p,∞)(R^n)(or to H^p_A(R^n)) in the ln λcritical case, from H^(p,q)_A(R^n) to L^(p,q)(R^n)(or to H^(p,q)_A(R^n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^(p,q)_A(R^n) to L^(p,∞)(R^n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.展开更多
Basic properties of the Herz-type Hardy spaces HK<sub>q</sub><sup>a,p</sup>, such as the boundedness of singular integral operators and the fractional integration operators, atomic decompositio...Basic properties of the Herz-type Hardy spaces HK<sub>q</sub><sup>a,p</sup>, such as the boundedness of singular integral operators and the fractional integration operators, atomic decomposition, dense subspaces, etc., are established in the full range 0【q【∞.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
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
基金Supported by NSP of China (Grant No. 10571015)RFDP of China (Grant No. 20050027025).
文摘Let A be a symmetric expansive matrix and H^p(R^n) be the anisotropic Hardy space associated with A. For a function m in L∞(R^n), an appropriately chosen function η in Cc^∞(R^n) and j ∈ Z define mj(ξ) = m(A^jξ)η(ξ). The authors show that if 0 〈 p 〈 1 and mj belongs to the anisotropic nonhomogeneous Herz space K1^1/P^-1,p(R^n), then m is a Fourier multiplier from H^p(R^n) to L^V(R^n). For p = 1, a similar result is obtained if the space K1^0.1(R^n) is replaced by a slightly smaller space K(w). Moreover, the authors show that if 0 〈 p 〈 1 and if the sequence {(mj)^v} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from H^p(R^n) to L^v(R^n).
基金Supported by the NECF and the NECF and the NNSF of China
文摘Let (.the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces (w2) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show that , the class of C(Rn)-functions with compactly support, is dense in and there is a subsequence, which converges in distrbutional sense to some distribution of , of any bounded sequence in In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness.
基金Supported by the NSF of China (10371087)NSF of Anhui Province (07021019)+2 种基金Education Committee ofAnhui Province (KJ2007A009Kj2008B244)the Grant for Younth of Anhui Normal University (2009xqn58)
文摘In this paper, we study the boundedness of higher order commutators of gen- eralized fractional integral operators on weighted Lp spaces and Herz-type Hardy spaces.
文摘In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from HKq^a、P(ω1;ω2) into Kq^a、p (ω1; ω2).
文摘In this paper, the authors study the multilinear operators with Dini Kernel and obtain their boundedness from Herz spaces to Herz-type Hardy spaces. Moreover, the authors also consider the corresponding fractional operators.
文摘This paper establishes some strong type and weak type estimates for commutator [b,I1] on Herz-type spaces, where b E BMO(Rn) and I1 is a fractional integration with O < l < n.
基金NSF of China (Grant Nos.10571014 and 10571015)SRFDP of China (Grant No.20050027025)
文摘In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ<SUP> n </SUP>are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces.
基金supported by the National Natural Science Foundation of China (Grant No. 10571015)Specialized Research Foundation for Doctor Programme (Grant No. 20050027025)
文摘In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.
文摘In this paper,the authors first establish some new real-variable characterizations of Herz- type Hardy spaces H<sub>q</sub><sup>α,p</sup>(ω<sub>1</sub>;ω<sub>2</sub>)and HK<sub>q</sub><sup>α,P</sup>(ω<sub>1</sub>;ω<sub>2</sub>),where ω<sub>1</sub>,ω<sub>2</sub> ∈A<sub>1</sub>-weight,1【q【∞, n(1-1/q)≤α【∞ and 0【p【∞.Then,using these new characterizations,they investigate the convergence of a bounded set in these spaces,and study the boundedness of some potential operators on these spaces.
基金supported by National Natural Science Foundation of China (Grant Nos. 11571039, 11361020 and 11471042)
文摘Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^(p,q)_A(R^n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H^(p1,q1)_A(Rn) and H^(p2,q2)_A(R^n) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈(0, ∞], and also between H^(p,q1)_A(Rn) and H^(p,q2)_A(R^n) with p ∈(0, ∞)and 0 < q1 < q < q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H^(p,q)_A(R^n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^(p,∞)_A(R^n) to the weak Lebesgue space L^(p,∞)(R^n)(or to H^p_A(R^n)) in the ln λcritical case, from H^(p,q)_A(R^n) to L^(p,q)(R^n)(or to H^(p,q)_A(R^n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^(p,q)_A(R^n) to L^(p,∞)(R^n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.
基金Partly supported by the Grants-in-Aid for Scientific Research (A)(1) 11304009, (B)(1)10440046, Japan Society for the Promotion of Science.
文摘Basic properties of the Herz-type Hardy spaces HK<sub>q</sub><sup>a,p</sup>, such as the boundedness of singular integral operators and the fractional integration operators, atomic decomposition, dense subspaces, etc., are established in the full range 0【q【∞.
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(11461065)Scientific Research Projects in Colleges and Universities in Xinjiang Uyghur Autonomous Region(XJEDU2014S001)
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.