Let L be a one-to-one operator of type w having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k C N. In this paper, the authors introduce the Hardy space HPL（Rn） with p ∈（0, ...Let L be a one-to-one operator of type w having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k C N. In this paper, the authors introduce the Hardy space HPL（Rn） with p ∈（0, 1] associated with L in terms of square functions defined via {e-t2kL}t〉O and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schr6dinger type operator L2 ：= （-△）k ＋ Vk, where A is the Laplacian and 0≤V C Llkoc（Rn）. Moreover, as an application, for i E {1, 2}, the authors prove that the associated Riesz transform Vk（Li-1/2） p n HP（Rn） for @ （n/（n ＋ k）, 1] and establish the Riesz transform characterizations is bounded from HLI（IR ） to p of HPL1（]Rn） for p C （rn/（n ＋ kr）, 1] if {e-tL1 }t〉o satisfies the Lr - L2 k-off-diagonal estimates with r C （1, 2]. These results when k ：= I and L ：= L1 are known.展开更多
In this paper,some properties of Hardy-Sobolev spaces are obtained. The multipliers on these spaces are defined,and our results show that the multiplier algebra is more complex than that on the classical Hardy spaces....In this paper,some properties of Hardy-Sobolev spaces are obtained. The multipliers on these spaces are defined,and our results show that the multiplier algebra is more complex than that on the classical Hardy spaces. In addition,the spectrum theorem is obtained for some special multiplier.展开更多
In this paper,we introduce the work done on Hardy-Sobolev spaces and Fock-Sobolev spaces and their operators and operator algebras,including the study of boundedness,compactness,Fredholm property,index theory,spectrum...In this paper,we introduce the work done on Hardy-Sobolev spaces and Fock-Sobolev spaces and their operators and operator algebras,including the study of boundedness,compactness,Fredholm property,index theory,spectrum and essential spectrum,norm and essential norm,Schatten-p classes,and the C^(∗) algebras generated by them.展开更多
The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on t...The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions.It is proved that,for n/(n+α)<p<∞,T_~Ω,α is a bounded operator from the Hardy-Sobolev space Hp_α to the Hardy space Hp.The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden's paper(1969).The proof is based on a new atomic decomposition of the space Hp_α by Han,Paluszynski and Weiss(1995).By using the same proof,the singluar integral operators with variable kernels are also studied.展开更多
Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.
In this paper, we prove that for -1/2≤β≤0, suppose M is an invariant subspaces of the Hardy Sobolev spaces Hβ^2（D） for Tz^β, then M zM is a generating wandering subspace of M, that is, M = [M zM]Tz^β. More...In this paper, we prove that for -1/2≤β≤0, suppose M is an invariant subspaces of the Hardy Sobolev spaces Hβ^2（D） for Tz^β, then M zM is a generating wandering subspace of M, that is, M = [M zM]Tz^β. Moreover, any non-trivial invariant subspace M of Hβ^2（D） is also generated by the quasi-wandering subspace PMTz^βM^⊥, that is, M = [PMTz^βM^⊥]Tz^β.展开更多
In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,...,...In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,..., M_(z_n)|S) is doubly commuting, then for any non-empty subset α = {α_1,..., α_k} of {1,..., n}, W_α~S is a generating wandering subspace for M_α|_S =(M_(z_(α_1))|_S,..., M_(z_(α_k))|_S), that is, [W_α~S]_(M_(α |S))= S, where W_α~S=■(S ■ z_(α_i)S).展开更多
Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be poin...Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.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 L be a one-to-one operator of type w having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k C N. In this paper, the authors introduce the Hardy space HPL（Rn） with p ∈（0, 1] associated with L in terms of square functions defined via {e-t2kL}t〉O and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schr6dinger type operator L2 ：= （-△）k ＋ Vk, where A is the Laplacian and 0≤V C Llkoc（Rn）. Moreover, as an application, for i E {1, 2}, the authors prove that the associated Riesz transform Vk（Li-1/2） p n HP（Rn） for @ （n/（n ＋ k）, 1] and establish the Riesz transform characterizations is bounded from HLI（IR ） to p of HPL1（]Rn） for p C （rn/（n ＋ kr）, 1] if {e-tL1 }t〉o satisfies the Lr - L2 k-off-diagonal estimates with r C （1, 2]. These results when k ：= I and L ：= L1 are known.
基金supported by National Natural Science Foundation of China(Grant No.11271092)Doctoral Fund of Ministry of Education of China(Grant No.20114410110001)
文摘In this paper,some properties of Hardy-Sobolev spaces are obtained. The multipliers on these spaces are defined,and our results show that the multiplier algebra is more complex than that on the classical Hardy spaces. In addition,the spectrum theorem is obtained for some special multiplier.
基金G.Cao was supported by the NNSF of China(Grant No.12071155)L.He was supported by the NNSF of China(Grant No.11871170).
文摘In this paper,we introduce the work done on Hardy-Sobolev spaces and Fock-Sobolev spaces and their operators and operator algebras,including the study of boundedness,compactness,Fredholm property,index theory,spectrum and essential spectrum,norm and essential norm,Schatten-p classes,and the C^(∗) algebras generated by them.
文摘The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions.It is proved that,for n/(n+α)<p<∞,T_~Ω,α is a bounded operator from the Hardy-Sobolev space Hp_α to the Hardy space Hp.The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden's paper(1969).The proof is based on a new atomic decomposition of the space Hp_α by Han,Paluszynski and Weiss(1995).By using the same proof,the singluar integral operators with variable kernels are also studied.
基金Supported by the National Natural Science Foundation of China(1057115610871173)
文摘Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.
基金Supported by National Natural Science Foundation of China(Grant No.11671152)the key research project of Nanhu College of Jiaxing University(Grant.No.N41472001-18)
文摘In this paper, we prove that for -1/2≤β≤0, suppose M is an invariant subspaces of the Hardy Sobolev spaces Hβ^2（D） for Tz^β, then M zM is a generating wandering subspace of M, that is, M = [M zM]Tz^β. Moreover, any non-trivial invariant subspace M of Hβ^2（D） is also generated by the quasi-wandering subspace PMTz^βM^⊥, that is, M = [PMTz^βM^⊥]Tz^β.
基金supported by the Natural Science Foundation of China(11271092,11471143)the key research project of Nanhu College of Jiaxing University(N41472001-18)
文摘In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,..., M_(z_n)|S) is doubly commuting, then for any non-empty subset α = {α_1,..., α_k} of {1,..., n}, W_α~S is a generating wandering subspace for M_α|_S =(M_(z_(α_1))|_S,..., M_(z_(α_k))|_S), that is, [W_α~S]_(M_(α |S))= S, where W_α~S=■(S ■ z_(α_i)S).
基金supported in part by National Natural Foundation of China (Grant Nos. 11071250 and 11271162)
文摘Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.
