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

Let(X,p,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 (?)_∞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 (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~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 TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

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 p0 p ≤ 1 for some p0 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 Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (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.     

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.     

Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces and the Triebel-Lizorkin spaces on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where . Moreover, using these inequalities, we can easily show that the Littlewood-Paley -function and -function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.

     

Maximal operators on spaces of homogeneous type

We avoid the assumption given in the work of C. Pérez and R. Wheeden (2001) to prove boundedness properties of certain maximal functions in a general setting of the spaces of homogeneous type with infinite measure. In addition, an example shows that the result can be false if the space has finite measure.

     

Bilinear operators on Herz-type Hardy spaces

The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on are bounded from into if and only if they have vanishing moments up to a certain order dictated by the target space. Here are homogeneous Herz-type Hardy spaces with and . As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.

     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Commutators of singular integrals on spaces of homogeneous type

In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on L ^p (w) when w belongs to the Muckenhoupt’s class A _p , p > 1. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.     

Duals of Hardy spaces on homogeneous groups

Hardy spaces on homogeneous groups were introduced and studied by Folland and Stein [3]. The purpose of this note is to show that duals of Hardy spaces H^p , 0 < p ≤ 1, on homogeneous groups can be identified with Morrey–Campanato spaces. This closes a gap in the original proof of this fact in [3]. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New abstract Hardy spaces

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H¹. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L¹ and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.     

Muckenhoupt weights with singularities on closed lower dimensional sets in spaces of homogeneous type

We give sufficient conditions on a real number β and on a closed set F   in a general space of homogeneous type (X,d,μ)$(X,d,\mu )$ in such a way that μ(B(x,d(x,F)))^β$\mu {(B(x,d(x,F)))}^{\beta }$ becomes a Muckenhoupt weight. In order to prove our result, we modify the underlying space so that it becomes 1-Ahlfors regular.     

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.     

带齐次权的Hardy型不等式

在采用Hoffmann-Ostenhof和Laptev构造权函数的思想,进行加权推广,给出了一类带齐次权的Hardy型不等式.利用Avkhadiev和Wirths得到的一维Hardy型不等式,运用放缩法,得到一类带余项的加权Hardy型不等式.获得的结论将HoffmannOstenhof和Laptev中的相关结论推广至加权与带余项的情形.     

Weighted BMO and Carleson measures on spaces of homogeneous type

In this paper we characterize the weighted BMO(ω)(X), with X a space of homogeneous type, through an adequate weighted Carleson measure. As a byproduct we can define the weighted Triebel-Lizorkin space and obtain the identification with the above space.     

Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type

Let 0<γ<1, b be a BMO function and the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for in the context of the spaces of homogeneous type. The first one establishes that, for A_∞ weights, the operator is bounded in the weighted Lp norm by the maximal operator Mγ(Mm), where Mγ is the fractional maximal operator and Mm is the Hardy-Littlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator is bounded from to , where [(m+1)p] is the integer part of (m+1)p and no condition on the weight w is required. From the first inequality we also obtain weighted Lp-Lq estimates for generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.     

Generalized Besov type spaces on the dual of the Laguerre hypergroup

In this paper we introduce the homogeneous Besov type spaces ∧p,q^γ（K） on the dual of Laguerre hypergroup K and we establish some new harmonic analysis results. We give some character- izations of these spaces using equivalent seminorms. Also we study the non-homogeneous Besov type spaces ∧p,q^γ（K）. We give some properties of these spaces and embeddings results with respect to their parameters p, q and γ.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Besov spaces with non-doubling measures

Suppose that is a Radon measure on which may be non-doubling. The only condition on is the growth condition, namely, there is a constant 0$"> such that for all and 0,$">

where In this paper, the authors establish a theory of Besov spaces for and , where 0$"> is a real number which depends on the non-doubling measure , , and . The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.

     

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from L^p(v) to L^q,∞(u), 1<p?q<∞, provided the pair of weights (u,v) verifies a Muckenhoupt condition with a “power-bump” on the weight u.