Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

Some characterizations of weighted Herz-type Hardy spaces and their applications

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-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. Supported by the NNSF of China     

A maximal function characterization of Hardy spaces on spaces of homogeneous type

A new maximal funtion 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. This work is supported by NSF.     

Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms

Let p?1 be near to 1 and X be an RD-space, which includes any Carnot-Carathéodory space with a doubling measure. In this paper, the authors prove that a sublinear operator T extends to a bounded sublinear operator from Hardy spaces Hp(X) to some quasi-Banach space B if and only if T maps all (p,2)-atoms into uniformly bounded elements of B.     

Herz-type Hardy spaces on Vilenkin groups and their applications

Let 1＜q＜∞, 1-1/q≤α＜∞, 0＜p＜∞ and G be a locally compact Vilenkin group.The authors first introduce the general Herz-type Hardy spaces HKα,pq(G) and hKα,pq(G), then present the central atom or the central block and atom decompositions of these spaces. Using this characterization, they discuss some properties of these spaces and investigate the boundedness on the spaces HKα,pq(G) of the fractional integral operators and the boundedness on the spaces hKα,pq(G) of the pseudo-differential operators of order zero.     

Some characterizations of weighted Hardy spaces

In this paper, we will characterize the weighted Hardy spaces by the intrinsic square functions including the Lusin area function, Littlewood-Paley G-function and -function.     

Local Hardy spaces of Musielak-Orlicz type and their applications

Letφ:R n × [0,∞) → [0,∞) be a function such that φ(x,·) is an Orlicz function and (·,t) ∈ A ∞loc (Rn) (the class of local weights introduced by Rychkov).In this paper,the authors introduce a local Musielak-Orlicz Hardy space hφ(Rn) by the local grand maximal function,and a local BMO-type space bmoφ(Rn) which is further proved to be the dual space of hφ(Rn).As an application,the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (Rn),characterized by Nakai and Yabuta,is just the dual of L 1 (Rn) + h Φ 0 (Rn),where φ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ 0 a Musielak-Orlicz function induced by φ.Characterizations of hφ(Rn),including the atoms,the local vertical and the local nontangential maximal functions,are presented.Using the atomic characterization,the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hφ(Rn),from which,the authors further deduce some criterions for the boundedness on hφ(Rn) of some sublinear operators.Finally,the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on hφ(Rn).     

Hardy spaces H~p over non-homogeneous metric measure spaces and their applications

Let(X,d,)be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈(1,∞),0p≤1≤q≤∞,p≠q,γ∈[1,∞)and ∈∈(0,∞).In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ∈(μ)via the discrete coefficient K(ρ),p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ(μ)(or Hp,q,γatb,ρ(μ))into Lp(μ),and from Hp,q,γ+1atb,ρ(ρ+1)(μ)into H p,q,γ,12(δ-νp+ν)mb,ρ(μ).The boundedness of the generalized fractional integral Tβ(β∈(0,1))from Hp1,q,γ,θmb,ρ(μ)(or Hp1,q,γatb,ρ(μ))into Lp2(μ)with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈(1,∞),of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ(μ)and show that Eα,qρ,η,γ(μ)is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ(μ),which coincide with each other;the authors finally prove that Hp,q,γatb,ρ(μ)is the predual of E1/p-1,1ρ,ρ,1(μ).Moreover,if is doubling,the authors show that Eα,qρ,η,γ(μ)and the Lipschitz space Lipα,q(μ)(q∈[1,∞)),or Hp,q,γatb,ρ(μ)and the atomic Hardy space Hp,q at(μ)(q∈(1,∞])of Coifman and Weiss coincide.Finally,if(X,d,)is an RD-space(reverse doubling space)with(X)=∞,the authors prove that Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ)and Hp,q at(μ)coincide for any q∈(1,2].In particular,when(X,d,):=(RD,||,dx)with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ),Hp,q,γatb,ρ(μ)and Hp,q,γ,mb,ρ(μ)all coincide with Hp(RD)for any q∈(1,∞).     

Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A ^k : k ∈ ?}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ| > 1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let φ: ? ⁿ ×[0,∞) → [0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type H _A ^φ,∞ (? ⁿ ) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to H _A ^φ,∞ (? ⁿ ) and the boundedness of the anisotropic Calderón-Zygmund operator from H _A ^φ,∞ (? ⁿ ) to L _A ^φ,∞ (? ⁿ ). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of H _A ^φ,∞ (? ⁿ ) and the interpolation theorem.     

Anisotropic Herz-type Hardy spaces with variable exponent and their applications

A class of anisotropic Herz-type Hardy spaces with variable exponent associated with a non-isotropic dilation on \({\mathbb{R}^{n}}\) are introduced, and characterizations of these spaces are established in terms of atomic and molecular decompositions. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of a linear operator on the anisotropic Herz-type Hardy spaces with variable exponent.     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

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.     

Maximal characterizations of Herz type Hardy spaces on homogeneous groups

In this paper, the authors establish some characterizations of Herz-type Hardy spaces HKα,p q(G) and HKq,p q(G), where 1 ＜ q ＜∞, Q(1 - 1/q) ≤α＜∞, 0 ＜ p ＜∞ and G denotes a graded homogeneous Lie group.     

The maximal conjugate and Hilbert operators on real Hardy spaces

Summary We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H¹ to L¹, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T² and R².     

Convolutions, multipliers and maximal regularity on vector-valued Hardy spaces

We extend the recently developed L^p-theory for the maximal regularity of the abstract Cauchy problem and the related Fourier multiplier techniques to the real-variable Hardy space H¹. Some results for H^p, 0 < p < 1, are also proved.     

On characterizations of isometries on function spaces

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball B_n in C~n.     

New properties of Besov and Triebel-Lizorkin spaces on RD-spaces

An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$ . In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on ${\mathcal X}$ are independent of the choice of the regularity ${\epsilon\in (0,1)}$ ; as a result of this, the Besov and Triebel-Lizorkin spaces on ${\mathcal X}$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.     

New characterizations of Morrey spaces and their preduals with applications to fractional Laplace equations

In this article, the authors characterize the Morrey spaces as well as their preduals via quadratic functions related to the Taylor remainder of the kernel of the Riesz potential. As applications, the authors obtain some strong capacitary inequalities, which are then used to study the regularity of the duality/weak solution to the fractional Laplace equation with measure data.     

Weighted Lipschitz spaces and their analytic characterizations

Weighted Lipschitz spaces are studied where the growth function varies, i.e., $$|f(x + h) - f(x)| \leqslant C\rho (h)$$ and where the growth function may also depend on x. Connections between a functionf defined on the complex unit circle and its analytic extensionF to the inside are given. In particular, theory relating the Lipschitz growth of the boundary function and derivative growth of the analytic function is developed. An atomic decomposition of the boundary values of a weighted Besov space is also obtained.