Inhomogeneous Besov and Triebel-Lizorking spaces on spaces of homogeneous type

The authors introduce the inhomogeneous Besov space and the inhomogeneous Triebel-Lizorkin space on spaces of homogeneous type: and present their atom and molecule decompositions, their dual spaces and the complex interpolation theorems. They also establishe the relation between the homogoeneous Besov space and the inhomogeneous one, and between the homogeneous Triebel-Lizorkin space and the inhomogeneous one. Moreover, they establish T1 theorems for these inhomogeneous spaces when a≠0, and apply these T1 theorems to give new characterizations of these spaces.     

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 obtains some estimates for the entropy numbers of the embeddings in the limiting cases between some Besov spaces and some logarithmic Lebesgue spaces.     

New characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type

In this paper we use the T1 theorem to prove a new characterization with minimum regularity and cancellation conditions for inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. These results are new even for R^n.     

New characterizations of Besov and Triebel-Lizorkin spaces over spaces of homogeneous type

Using the T1 theorem for the Besov and Triebel-Lizorkin spaces, we give new characterizations of Besov and Triebel-Lizorkin spaces with minimum regularity and cancellation conditions over spaces of homogeneous type.     

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.     

Characterizations of realized homogeneous Besov and Triebel-Lizorkin spaces via differences

Based on the role of the polynomial functions on the homogeneous Besov spaces, on the homogeneous Triebel-Lizorkin spaces and on their realized versions, we study and obtain characterizations of these spaces via difference operators in a certain sense.     

Tl theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the Plancherel-Polya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.     

T1 theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the PlancherelPolya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.     

Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals

Though the theory of one-parameter Triebel-Lizorkin and Besov spaces has been very well developed in the past decades, the multi-parameter counterpart of such a theory is still absent. The main purpose of this paper is to develop a theory of multi-parameter Triebel-Lizorkin and Besov spaces using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the recent work of Han and Lu in which they established a satisfactory theory of multi-parameter Littlewood-Paley-Stein analysis and Hardy spaces associated with the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. We also prove the boundedness of flag singular integral operators on Triebel-Lizorkin space and Besov space. Our methods here can be applied to develop easily the theory of multi-parameter Triebel-Lizorkin and Besov spaces in the pure product setting.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Oscillatory hyper-Hilbert transform along curves on modulation spaces

We consider the boundedness of the n-dimension oscillatory hyper-Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems signicantly improve some known results.     

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J.?Fourier Anal Appl. 1:311?C353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.     

Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces

Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces by atoms,molecules and wavelets are given.These results generalize the corresponding results for classical Besov and Triebel-Lizorkin spaces.     

<Emphasis Type="Italic">T</Emphasis>1 theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the Plancherel-Polya characterization for the Besov and Triebel-Lizorkin spaces, theT1 theorem for the Besov and Triebel-Lizorkin spaces was proved.     

Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón's identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting.     

Herz type Besov and Triebel-Lizorkin spaces with variable exponent

The Herz type Besov and Triebel-Lizorkin spaces with variable exponent are introduced. Then characterizations of these new spaces by maximal functions are given.     

Boundedness and continuity of Marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case.     

Compactly supported refinable distributions in Triebel-Lizorkin spaces and besov spaces

The aim of this article is to characterize compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces by projection operators on certain wavelet space and by some operators on a finitely dimensional space.Research partially supported by the National Natural Sciences Foundation of China # 69735020, the Tian Yuan Projection of the National Natural Sciences Foundation of China, the Doctoral Bases Promotion Foundation of National Educational Commission of China #97033519 and the Zhejiang Provincial Sciences Foundation of China # 196083, and by the Wavelets Strategic Research Program funded by the National Science and Technology Board and the Ministry of Education, Singapore.     

Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents

In this paper, Morrey type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Then equivalent quasi-norms of these new spaces in terms of Peetre?s maximal functions are obtained. Finally, applying those equivalent quasi-norms, the authors obtain the atomic, molecular and wavelet decompositions of these new spaces.     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.