CHARACTERIZATION OF MAXIMAL OPERATORS IN ORLICZ-MORREY SPACES OF HOMOGENEOUS TYPE

. In this paper,the characterization of boundedness of Hardy-Littlewood maximal operators in Orlicz-Morrey spaces LΦφ（X,μ） of homogeneous type is founded.

A MAXIMAL FUNCTION CHARACTERIZATION 0F HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE

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.

INTERPOLATION SPACES BETWEEN H^1 AND L~∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces, we study the interpolation spaces between H1 and L∞ on spaces of homogeneous type.

WEIGHTED ESTIMATES FOR COMMUTATORS OF POTENTIAL OPERATORS ON SPACES OF HOMOGENEOUS TYPE

We derive some strong type and weak type weighted norm estimates which re- late the commutators of potential integral operators to the corresponding maximal operators in the context of spaces of homogeneous type.

FRACTIONAL INTEGRAL OPERATOR ON SPACES OF HOMOGENEOUS TYPE

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.

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

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.

Composition of Maximal Operators on Space of Homogeneous Type

Let Ф be a Young function and MФ be the maximal operator associated with in the space of homogeneous type. In this paper, the composition of the maximal operators of type MФ is considered, and the result established by Carrozza and Passarelli Di Napoli is generalized to the space of homogeneous type.

Weighted Boundedness for Generalized Maximal andSingular Integral Operators in Orlicz-Morrey Spaces

In this paper,the weighted boundedness for the generalised maximal and singular integral operator in Orlicz-Morrey spaces are obtained.

Several equivalent characterizations of fractional Hajlasz-Morrey-Sobolev spaces

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.

Interpolation Spaces between L^1 and BMO on Spaces of Homogeneous Type

We study the interpolation spaces between L^1 and BMO on spaces of homogeneous type. For 0 〈 θ 〈 1, 1≤q ≤∞, we obtain （L^1,BMO）θ,q =Lpq, where θ=1-1/p.

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrdinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.

Atomic Decomposition and Boundedness Criterion of Operators on Multi-parameter Hardy Spaces of Homogeneous Type

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.

Geometry and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss

It is known that the space of homogeneous type introduced by Coifman and Weiss(1971) provides a very natural setting for establishing a theory of Hardy spaces. This paper concentrates on how the geometrical conditions of the space of homogeneous type play a crucial role in building a theory of Hardy spaces via the Littlewood-Paley functions.

Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type

In this paper, the author establishes the embedding theorems for different metrics of inhomoge-neous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. As an application, the author obtainssome estimates for the entropy numbers of the embeddings in the limiting cases between some Besov spacesand some logarithmic Lebesgue spaces.

Composition of Fractional Orlicz Maximal Operators and A1-weights on Spaces of Homogeneous Type

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.

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type:Littlewood–Paley Characterizations with Applications to Boundedness of Calderón–Zygmund Operators

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.

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(χ,ρ,μ)_d,θ be a space of homogeneous type,∈∈(0,θ],|s|<∈andmax{d/(d+∈),d/(d+s+∈)}<q≤∞.The author introduces the new Triebel-Lizorkin spaces F_∞q^s(X)and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces F_∞q^s(X).The frame characterizations of the Besovspace B_pq^s(X)with |s|<∈,max{d/(d+∈),d/(d+s+∈)}<p≤∞ and 0<q ≤∞and the Triebel-Lizorkin space F_spq^s(X)with |s|<∈,max{d/(d+∈),d/(d+s+∈)}<p<∞ and max{d/(d+∈),d/(d+s+∈)}<q≤∞ are also presented.Moreover,the au-thor introduces the new Triebel-Lizorkin spaces bF_∞q^s(X)and HF_∞q^s(X)associated to agiven para-accretive function b.The relation between the space bF_∞q^s(X)and the spaceHF_∞q^s(X)is also presented.The author further proves that if s=0 and q=2,thenHF_∞q^s(X)=F_∞q^s(X),which also gives a new characterization of the space BMO(X),since F_∞q^s(X)=BMO(X).

Commutators of maximal functions on spaces of homogeneous type and their weighted,local versions

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.

Difference Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

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.

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

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.