In this paper, boundedness and compactness of the composition operator on the generalized Lipschitz spaces Λα （α 〉 1） of holomorphic functions in the unit disk are characterized.
The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory.As an application,we derive a boundedness criterion for singular integral operators i...The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory.As an application,we derive a boundedness criterion for singular integral operators in a mixed Journéclass on mixed Lipschitz spaces.Key elements of the paper are the development of the Littlewood-Paley theory for a special mixed Besov spaces,and a density argument for the mixed Lipschitz spaces in the weak sense.展开更多
Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f ...Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y） is a cubic function for all x∈G, then there exists a cubic function C：G→E such that f？C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y）=0 on Lipschitz spaces.展开更多
Abstract The aim of this paper is to prove duality and reflexivity of generalized Lipschitz spaces ∧κα,p,q（R）, α ∈R and 1≤〈 p, q ≤∞, in the context of Dunkl harmonic analysis.
In this paper, we define β-Hausdorff operator on the unit polydisk and study the boundedness of the operator on Lipschitz space. Firstly, we translate the problem of coefficient into integral of weighted composition ...In this paper, we define β-Hausdorff operator on the unit polydisk and study the boundedness of the operator on Lipschitz space. Firstly, we translate the problem of coefficient into integral of weighted composition operator, then give the sufficient conditions of boundedness, and also obtain an upper bound for the operator norm on Lipschitz space.展开更多
We prove that the weak Morrey space W M_(q)^(p) is contained in the Morrey space M_(q1)^(p) for 1 ≤ q1<q ≤ p < ∞. As applications, we show that if the commutator [b, T ] is bounded from L^(p) to L^(p,∞) for ...We prove that the weak Morrey space W M_(q)^(p) is contained in the Morrey space M_(q1)^(p) for 1 ≤ q1<q ≤ p < ∞. As applications, we show that if the commutator [b, T ] is bounded from L^(p) to L^(p,∞) for some p ∈(1, ∞), then b ∈ BMO, where T is a Calderón-Zygmund operator. Also, for 1 < p ≤ q < ∞, b ∈ BMO if and only if [b, T ] is bounded from M_(q)^(p) to WM_(q)^(p). For b belonging to Lipschitz class, we obtain similar results.展开更多
In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the author...In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.展开更多
In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
Let Un be the unit polydisc of Cn and φ= (φ1,...,φn) a holomorphic self-map of Un. Let0 ≤α＜ 1. This paper shows that the composition operator Cφ is bounded on the Lipschitz space Lipα(Un)if and only if there e...Let Un be the unit polydisc of Cn and φ= (φ1,...,φn) a holomorphic self-map of Un. Let0 ≤α＜ 1. This paper shows that the composition operator Cφ is bounded on the Lipschitz space Lipα(Un)if and only if there exists M ＞ 0 such that∑nk,l=1|φl/zk(z)|( 1-|zk|2/1-|φl(z)|2)1-α≤ Mfor z ∈ Un. Moreover Cφ is compact on Lipα(Un) if and only if Cφ is bounded on Lipα(Un) and for everyε＞ 0, there exists a δ＞ 0 such that∑nk,l=1|φl/zk(z)|( 1-|zk|2/1-|φl(z)|2)1-α≤εwhenever dist(φ(z), Un) ＜δ.展开更多
If f(z) =Σ∞ n=0 anzn and g(z) =Σ∞n=0bnzn for functions f, g are analytic in the unit disc, the Hadamard products of f and g is defined by f * g = ∞ n=0 a n b n z n . In this paper, the Lipschitz spaces Λ(s, α) ...If f(z) =Σ∞ n=0 anzn and g(z) =Σ∞n=0bnzn for functions f, g are analytic in the unit disc, the Hadamard products of f and g is defined by f * g = ∞ n=0 a n b n z n . In this paper, the Lipschitz spaces Λ(s, α) and QK type spaces are studied in terms of the Hadamard products.展开更多
In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These res...In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These results improve and generalize several wellknown comparable results in the literature. Moreover, our results are supported by some examples.展开更多
The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of...The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of such spaces in the case|x|<1 is given also.展开更多
We shall introduce 1-type Lipschitz multifunctions from R into generalized 2-normed spaces, and give some results about their 1-type Lipschitz selections.
In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator [b,TR^δ] by Bochner-Riesz operators and the function b on weighted Morrey spaces L^p,k （ω） under appro...In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator [b,TR^δ] by Bochner-Riesz operators and the function b on weighted Morrey spaces L^p,k （ω） under appropriate conditions on the weight w, where b belongs to Lipschitz space or weighted Lipschitz space.展开更多
In this paper, the authors study the boundedness of the operator [μΩ,b], the commutator generated by a function b ∈ Lipβ(Rn)(0 <β≤1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the...In this paper, the authors study the boundedness of the operator [μΩ,b], the commutator generated by a function b ∈ Lipβ(Rn)(0 <β≤1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω ∈ Lipα(Sn-1)(0 <α≤ 1).展开更多
In this paper, the authors study two classes of multilinear singular integrals and obtain their boundedness from Lebesgue spaces to Lipschitz spaces and from Herz type spaces to central Campanato spaces. Moreover, the...In this paper, the authors study two classes of multilinear singular integrals and obtain their boundedness from Lebesgue spaces to Lipschitz spaces and from Herz type spaces to central Campanato spaces. Moreover, the authors also consider the extreme cases.展开更多
Let b^→=（b1,…,bm）,bi∈∧°βi（R^n）,1≤i≤m,0〈βi〈β,0〈β〈1,[B^→,T]f（x）=∫R^n（b1（x）-b1（y））…（bm（x）-bm（y））K（x-y）f（y）dy,where K is a Calder6n-Zygmund kernel. In this paper, we show that ...Let b^→=（b1,…,bm）,bi∈∧°βi（R^n）,1≤i≤m,0〈βi〈β,0〈β〈1,[B^→,T]f（x）=∫R^n（b1（x）-b1（y））…（bm（x）-bm（y））K（x-y）f（y）dy,where K is a Calder6n-Zygmund kernel. In this paper, we show that [b^→,T] is bounded from L^p（R^n） to Fp^β,∞（R^n）,as well as [b^→,1α]form L^p （R^n） to Fp^β,∞（R^n）,where 1/q=1/p-α/n.展开更多
In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf（x） = p.v.∫R^n ^e^iP（x,y）Ω（x - y）/｜x - y｜^n＋M r∏s=1 Rms＋1（As;x,y）f（y）dy, n≥2 where P...In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf（x） = p.v.∫R^n ^e^iP（x,y）Ω（x - y）/｜x - y｜^n＋M r∏s=1 Rms＋1（As;x,y）f（y）dy, n≥2 where P（x,y） is a nontrivial and real-valued polynomial defined on R^n×R^n,Ω（x） is homogeneous of degree zero on R^n, As（x） has derivatives of order ms in ∧βs （0〈βs〈 1）, Rms＋1 （As;x, y） denotes the （ms＋1）-st remainder of the Taylor series of As at x expended about y （s = 1, 2, ..., r）, M = ∑s^r =1 ms, the author proves that if 0 〈=β1=∑s^r=1 βs〈1,and Ω∈L^q（S^n-1） for some q 〉 1/（1 -β）, then for any p∈（1, ∞）, and some appropriate 0 〈β〈 1, TA1,A2,...,Ar, is bounded on L^P（R^n）.展开更多
基金Supported in part by the National Natural Science Foundation of China (10971219)
文摘In this paper, boundedness and compactness of the composition operator on the generalized Lipschitz spaces Λα （α 〉 1） of holomorphic functions in the unit disk are characterized.
基金Supported by Zhejiang Provincial Natural ScienceFoundation of China(LQ22A010018)National Natural Science Foundation of China(12071437)。
文摘The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory.As an application,we derive a boundedness criterion for singular integral operators in a mixed Journéclass on mixed Lipschitz spaces.Key elements of the paper are the development of the Littlewood-Paley theory for a special mixed Besov spaces,and a density argument for the mixed Lipschitz spaces in the weak sense.
文摘Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y） is a cubic function for all x∈G, then there exists a cubic function C：G→E such that f？C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y）=0 on Lipschitz spaces.
文摘Abstract The aim of this paper is to prove duality and reflexivity of generalized Lipschitz spaces ∧κα,p,q（R）, α ∈R and 1≤〈 p, q ≤∞, in the context of Dunkl harmonic analysis.
文摘In this paper, we define β-Hausdorff operator on the unit polydisk and study the boundedness of the operator on Lipschitz space. Firstly, we translate the problem of coefficient into integral of weighted composition operator, then give the sufficient conditions of boundedness, and also obtain an upper bound for the operator norm on Lipschitz space.
基金supported by the National Natural Science Foundation of China (No.12101010)the Natural Science Foundation of Anhui Province (No.2108085QA19)
文摘We prove that the weak Morrey space W M_(q)^(p) is contained in the Morrey space M_(q1)^(p) for 1 ≤ q1<q ≤ p < ∞. As applications, we show that if the commutator [b, T ] is bounded from L^(p) to L^(p,∞) for some p ∈(1, ∞), then b ∈ BMO, where T is a Calderón-Zygmund operator. Also, for 1 < p ≤ q < ∞, b ∈ BMO if and only if [b, T ] is bounded from M_(q)^(p) to WM_(q)^(p). For b belonging to Lipschitz class, we obtain similar results.
基金Supported by National Natural Science Foundation of China (Grant No. 10871024)
文摘In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.
文摘In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
基金This work was supported in part by the National Natural Science Foundation ofChina (Grant Nos. 19871081 & 10001030) LiuHui Center for Applied Mathematics, Nankai University and Tianjin University.
文摘Let Un be the unit polydisc of Cn and φ= (φ1,...,φn) a holomorphic self-map of Un. Let0 ≤α＜ 1. This paper shows that the composition operator Cφ is bounded on the Lipschitz space Lipα(Un)if and only if there exists M ＞ 0 such that∑nk,l=1|φl/zk(z)|( 1-|zk|2/1-|φl(z)|2)1-α≤ Mfor z ∈ Un. Moreover Cφ is compact on Lipα(Un) if and only if Cφ is bounded on Lipα(Un) and for everyε＞ 0, there exists a δ＞ 0 such that∑nk,l=1|φl/zk(z)|( 1-|zk|2/1-|φl(z)|2)1-α≤εwhenever dist(φ(z), Un) ＜δ.
基金supported by National Natural Science Foundation of China (Grant Nos.10371069, 10671115)National Science Foundation of Guangdong Province, China (Grant Nos. 04011000,06105648)
文摘If f(z) =Σ∞ n=0 anzn and g(z) =Σ∞n=0bnzn for functions f, g are analytic in the unit disc, the Hadamard products of f and g is defined by f * g = ∞ n=0 a n b n z n . In this paper, the Lipschitz spaces Λ(s, α) and QK type spaces are studied in terms of the Hadamard products.
基金Supported by the Foundation of Education Ministry of Hubei Province(D20102502)
文摘In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These results improve and generalize several wellknown comparable results in the literature. Moreover, our results are supported by some examples.
基金The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China.
文摘The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of such spaces in the case|x|<1 is given also.
文摘We shall introduce 1-type Lipschitz multifunctions from R into generalized 2-normed spaces, and give some results about their 1-type Lipschitz selections.
基金supported by NSFC (NO. 11201003)Education Committee of Anhui Province (NO. KJ2011A138 and NO. KJ2012A133)
文摘In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator [b,TR^δ] by Bochner-Riesz operators and the function b on weighted Morrey spaces L^p,k （ω） under appropriate conditions on the weight w, where b belongs to Lipschitz space or weighted Lipschitz space.
基金Supported by the National 973 Project (G.19990751) the SEDF (20010027002).
文摘In this paper, the authors study the boundedness of the operator [μΩ,b], the commutator generated by a function b ∈ Lipβ(Rn)(0 <β≤1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω ∈ Lipα(Sn-1)(0 <α≤ 1).
基金This project is supported by the National 973Project(G199907510)the SEDF of China(20010027002)
文摘In this paper, the authors study two classes of multilinear singular integrals and obtain their boundedness from Lebesgue spaces to Lipschitz spaces and from Herz type spaces to central Campanato spaces. Moreover, the authors also consider the extreme cases.
基金Supported by NSF of China (Grant: 10571015)NSF of China (Grant: 10371004)RFDP of China (Grant: 20050027025).
文摘Let b^→=（b1,…,bm）,bi∈∧°βi（R^n）,1≤i≤m,0〈βi〈β,0〈β〈1,[B^→,T]f（x）=∫R^n（b1（x）-b1（y））…（bm（x）-bm（y））K（x-y）f（y）dy,where K is a Calder6n-Zygmund kernel. In this paper, we show that [b^→,T] is bounded from L^p（R^n） to Fp^β,∞（R^n）,as well as [b^→,1α]form L^p （R^n） to Fp^β,∞（R^n）,where 1/q=1/p-α/n.
文摘In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf（x） = p.v.∫R^n ^e^iP（x,y）Ω（x - y）/｜x - y｜^n＋M r∏s=1 Rms＋1（As;x,y）f（y）dy, n≥2 where P（x,y） is a nontrivial and real-valued polynomial defined on R^n×R^n,Ω（x） is homogeneous of degree zero on R^n, As（x） has derivatives of order ms in ∧βs （0〈βs〈 1）, Rms＋1 （As;x, y） denotes the （ms＋1）-st remainder of the Taylor series of As at x expended about y （s = 1, 2, ..., r）, M = ∑s^r =1 ms, the author proves that if 0 〈=β1=∑s^r=1 βs〈1,and Ω∈L^q（S^n-1） for some q 〉 1/（1 -β）, then for any p∈（1, ∞）, and some appropriate 0 〈β〈 1, TA1,A2,...,Ar, is bounded on L^P（R^n）.
