A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associa...Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.展开更多
An equivalent definition of fractional integral on spaces of homogeneous type is given. The behavior of the fractional integral operator in Triebel-Lizorkin space is discussed.
The main purpose of this paper is to derive a new (p,q)-atomic decomposition on the multi-parameter Hardy space HP(X1 × X2) for 0 〈 po 〈 P ≤ 1 for some po and all 1 〈 q 〈 ∞, where X1 ×X2 is the pro...The main purpose of this paper is to derive a new (p,q)-atomic decomposition on the multi-parameter Hardy space HP(X1 × X2) for 0 〈 po 〈 P ≤ 1 for some po and all 1 〈 q 〈 ∞, where X1 ×X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L^q(X1 × X2) (for 1 〈 q 〈 ∞) and Hardy space HP(X1× X2) (for 0 〈 p _〈 1). As an application, we prove that an operator T, which is bounded on Lq(X1× X2) for some 1 〈 q 〈 ∞, is bounded from H^p(X1 × X2) to L^p(X1 × X2) if and only if T is bounded uniformly on all (p, q)-product atoms in LP(X1 × X2). The similar boundedness criterion from HP(X1 × X2) to HP(X1 × X2) is also obtained.展开更多
For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, μ) by Mα,θf(x) = supx∈(B)α ||f||θ,B, where...For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, μ) by Mα,θf(x) = supx∈(B)α ||f||θ,B, where ||f||θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈(0,1) then (Mα,θf)δ is an A1-weight.展开更多
In this article,we obtain some weighted estimates for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type.The weightωconsidered here belongs to the Muckenhoupt’s class A_(∞).Moreover,weigh...In this article,we obtain some weighted estimates for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type.The weightωconsidered here belongs to the Muckenhoupt’s class A_(∞).Moreover,weighted estimates for commutators of BMO functions and Marcinkiewicz integrals are also given.展开更多
Let(X,ρ,μ)d,θ be a space of homogeneous type,ε∈ (0,θ],|s|<εand max{d/(d +ε),d/(d+s+ε)}<q≤∞.The author introduces the new Triebel-Lizorkin spaces Fs∞q(X) and establishes the frame characterizations of the...Let(X,ρ,μ)d,θ be a space of homogeneous type,ε∈ (0,θ],|s|<εand max{d/(d +ε),d/(d+s+ε)}<q≤∞.The author introduces the new Triebel-Lizorkin spaces Fs∞q(X) and establishes the frame characterizations of these spaces by first establishing a Plancherel-Polya-type inequality related to the norm of the spaces Fs∞q(X).The frame characterizations of the Besov space Bspq(X) with |s|<ε,max{d/(d+ε),d/(d+s+ε)}<p≤∞ and 0<q≤∞ and the Triebel-Lizorkin space Fspq(X) with |s|<ε,max {d/(d+ε),d/(d+s+ε)}<p<∞ and max{d/(d+ε),d/(d+s+ε)}<q≤∞ are also presented.Moreover,the author introduces the new Triebel-Lizorkin spaces bFs∞q(X) and HFs∞q(X) associated to a given para-accretive function b.The relation between the space bFs∞q(X) and the space 0 and q=2,then resented.The author further proves that if s=HFs∞q(X) is also pHFs∞q(X) = Fs∞q(X),which also gives a new characterization of the space BMO(X),since Fs∞q(X)=BMO(X).展开更多
Let(X,ρ,μ)be a space of homogeneous type in the sense of Coifman and Weiss,and Y(X)a ball quasi-Banach function space on X,which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness ...Let(X,ρ,μ)be a space of homogeneous type in the sense of Coifman and Weiss,and Y(X)a ball quasi-Banach function space on X,which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space.The authors first introduce the Hardy space H_(Y)(X)associated with Y(X),via the Lusin-area function,and then establish its various equivalent characterizations,respectively,in terms of atoms,molecules,and Littlewood–Paley g-functions and g_(λ)^(*)-functions.As an application,the authors obtain the boundedness of Calderón–Zygmund operators from H_(Y)(X)to Y(X),or to H_(Y)(X)via first establishing a boundedness criterion of linear operators on H_(Y)(X).All these results have a wide range of generality and,particularly,even when they are applied to variable Hardy spaces,the obtained results are also new.The major novelties of this article exist in that,to escape the reverse doubling condition ofμand the triangle inequality ofρ,the authors subtly use the wavelet reproducing formula,originally establish an admissible molecular characterization of H_(Y)(X),and fully apply the geometrical properties of X expressed by dyadic reference points or dyadic cubes.展开更多
We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theo...We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.展开更多
In this article,the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss,and discuss their relations with Besov and Triebel-Lizorkin spaces.As an application,t...In this article,the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss,and discuss their relations with Besov and Triebel-Lizorkin spaces.As an application,the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying spaceχvia using the geometrical property ofχexpressed by its dyadic reference points,dyadic cubes,and the(local)lower bound.Moreover,some results when p≤1 but near to 1 are new even whenχis an RD-space.展开更多
Let Φ be a Young function and MΦ be the maximal operator associated with Φ in the space of homogeneous type.In this paper,the composition of the maximal operators of type MΦ is considered,and the result establishe...Let Φ be a Young function and MΦ be the maximal operator associated with Φ in the space of homogeneous type.In this paper,the composition of the maximal operators of type MΦ is considered,and the result established by Carrozza and Passarelli Di Napoli[2] is generalized to the space of homogeneous type.展开更多
In this article, the authors establish several equivalent characterizations of fractional Hajlasz-Morrey-Sobolev spaces on spaces of homogeneous type in the sense of Coifman and Weiss.
Let X = (X, d,μ) The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P(X) and prove that the generalized Hardy spaces have...Let X = (X, d,μ) The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P(X) and prove that the generalized Hardy spaces have the same property as H^P(X). Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let (X, δ, μ) be the normalized space of (X, d, μ) in the sense of Macias and Segovia. We also study the relations of our function spaces for (X, d, μ) and (X, δ,μ).展开更多
In this paper we use the Calderón-Zygmund operator theory to provide a discrete Calderón- type reproducing formula.Since translation,dilation and,in particular,the Fourier transform are never used in the pro...In this paper we use the Calderón-Zygmund operator theory to provide a discrete Calderón- type reproducing formula.Since translation,dilation and,in particular,the Fourier transform are never used in the proofs,all results still hold on spaces of homogeneous type introduced by Coifman and Weiss. As a consequence,we obtain a class of frames with the minimum regularity properties.展开更多
In this paper we explore conditions on variable symbols with respect to Haar systems,defining Calderón–Zygmund type operators with respect to the dyadic metrics associated to the Haar bases. We show that Petermi...In this paper we explore conditions on variable symbols with respect to Haar systems,defining Calderón–Zygmund type operators with respect to the dyadic metrics associated to the Haar bases. We show that Petermichl’s dyadic kernel can be seen as a variable kernel singular integral and we extend it to dyadic systems built on spaces of homogeneous type.展开更多
文摘A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
文摘Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.
文摘An equivalent definition of fractional integral on spaces of homogeneous type is given. The behavior of the fractional integral operator in Triebel-Lizorkin space is discussed.
文摘The main purpose of this paper is to derive a new (p,q)-atomic decomposition on the multi-parameter Hardy space HP(X1 × X2) for 0 〈 po 〈 P ≤ 1 for some po and all 1 〈 q 〈 ∞, where X1 ×X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L^q(X1 × X2) (for 1 〈 q 〈 ∞) and Hardy space HP(X1× X2) (for 0 〈 p _〈 1). As an application, we prove that an operator T, which is bounded on Lq(X1× X2) for some 1 〈 q 〈 ∞, is bounded from H^p(X1 × X2) to L^p(X1 × X2) if and only if T is bounded uniformly on all (p, q)-product atoms in LP(X1 × X2). The similar boundedness criterion from HP(X1 × X2) to HP(X1 × X2) is also obtained.
文摘For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, μ) by Mα,θf(x) = supx∈(B)α ||f||θ,B, where ||f||θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈(0,1) then (Mα,θf)δ is an A1-weight.
文摘In this article,we obtain some weighted estimates for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type.The weightωconsidered here belongs to the Muckenhoupt’s class A_(∞).Moreover,weighted estimates for commutators of BMO functions and Marcinkiewicz integrals are also given.
基金supported by the National Natural Science Foundation of China(Grant No.10271015)the Research Fund for the Doctoral Program of Higher Education(Grant No.20020027004)of China.
文摘Let(X,ρ,μ)d,θ be a space of homogeneous type,ε∈ (0,θ],|s|<εand max{d/(d +ε),d/(d+s+ε)}<q≤∞.The author introduces the new Triebel-Lizorkin spaces Fs∞q(X) and establishes the frame characterizations of these spaces by first establishing a Plancherel-Polya-type inequality related to the norm of the spaces Fs∞q(X).The frame characterizations of the Besov space Bspq(X) with |s|<ε,max{d/(d+ε),d/(d+s+ε)}<p≤∞ and 0<q≤∞ and the Triebel-Lizorkin space Fspq(X) with |s|<ε,max {d/(d+ε),d/(d+s+ε)}<p<∞ and max{d/(d+ε),d/(d+s+ε)}<q≤∞ are also presented.Moreover,the author introduces the new Triebel-Lizorkin spaces bFs∞q(X) and HFs∞q(X) associated to a given para-accretive function b.The relation between the space bFs∞q(X) and the space 0 and q=2,then resented.The author further proves that if s=HFs∞q(X) is also pHFs∞q(X) = Fs∞q(X),which also gives a new characterization of the space BMO(X),since Fs∞q(X)=BMO(X).
基金Supported by the National Key Research and Development Program of China(Grant No.2020YFA0712900)the National Natural Science Foundation of China(Grant Nos.11971058,12071197 and 11871100)the Fundamental Research Funds for the Central Universities(Grant Nos.500421359 and 500421126)。
文摘Let(X,ρ,μ)be a space of homogeneous type in the sense of Coifman and Weiss,and Y(X)a ball quasi-Banach function space on X,which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space.The authors first introduce the Hardy space H_(Y)(X)associated with Y(X),via the Lusin-area function,and then establish its various equivalent characterizations,respectively,in terms of atoms,molecules,and Littlewood–Paley g-functions and g_(λ)^(*)-functions.As an application,the authors obtain the boundedness of Calderón–Zygmund operators from H_(Y)(X)to Y(X),or to H_(Y)(X)via first establishing a boundedness criterion of linear operators on H_(Y)(X).All these results have a wide range of generality and,particularly,even when they are applied to variable Hardy spaces,the obtained results are also new.The major novelties of this article exist in that,to escape the reverse doubling condition ofμand the triangle inequality ofρ,the authors subtly use the wavelet reproducing formula,originally establish an admissible molecular characterization of H_(Y)(X),and fully apply the geometrical properties of X expressed by dyadic reference points or dyadic cubes.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171221,12071197)the Natural Science Foundation of Shandong Province(Grant Nos.ZR2019YQ04,2020KJI002).
文摘We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.
基金partially supported by the National Key Research and Development Program of China(Grant No.2020YFA0712900)the National Natural Science Foundation of China(Grant Nos.11971058,12071197 and 11871100).
文摘In this article,the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss,and discuss their relations with Besov and Triebel-Lizorkin spaces.As an application,the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying spaceχvia using the geometrical property ofχexpressed by its dyadic reference points,dyadic cubes,and the(local)lower bound.Moreover,some results when p≤1 but near to 1 are new even whenχis an RD-space.
基金Supported by the NSF from the Education Department of Henan Province(2007110006)
文摘Let Φ be a Young function and MΦ be the maximal operator associated with Φ in the space of homogeneous type.In this paper,the composition of the maximal operators of type MΦ is considered,and the result established by Carrozza and Passarelli Di Napoli[2] is generalized to the space of homogeneous type.
基金supported by the National Natural Science Foundation of China(11471042,11361020 and 11571039)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20120003110003)the Fundamental Research Funds for Central Universities of China(2014KJJCA10)
文摘In this article, the authors establish several equivalent characterizations of fractional Hajlasz-Morrey-Sobolev spaces on spaces of homogeneous type in the sense of Coifman and Weiss.
基金Grant-in-Aid for Exploratory Research,No.17654033 the Ministry of Education,Culture,Sports,Science and Technology,JapanGrant-in-Aid for Scientific Research (C),No.20540167,Japan Society for the Promotion of Science
文摘Let X = (X, d,μ) The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P(X) and prove that the generalized Hardy spaces have the same property as H^P(X). Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let (X, δ, μ) be the normalized space of (X, d, μ) in the sense of Macias and Segovia. We also study the relations of our function spaces for (X, d, μ) and (X, δ,μ).
文摘In this paper we use the Calderón-Zygmund operator theory to provide a discrete Calderón- type reproducing formula.Since translation,dilation and,in particular,the Fourier transform are never used in the proofs,all results still hold on spaces of homogeneous type introduced by Coifman and Weiss. As a consequence,we obtain a class of frames with the minimum regularity properties.
基金Supported by Consejo Nacional de Investigaciones Científicas y Técnicas,Universidad Nacional del Litoral and Universidad Nacional del Comahue,Argentina
文摘In this paper we explore conditions on variable symbols with respect to Haar systems,defining Calderón–Zygmund type operators with respect to the dyadic metrics associated to the Haar bases. We show that Petermichl’s dyadic kernel can be seen as a variable kernel singular integral and we extend it to dyadic systems built on spaces of homogeneous type.