New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.     

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.     

区域上Triebel-Lizorkin空间的分解

文中引进了区域上的Triebel-Lizorkin空间,以及原子和分子的概念,为了更好的理解这些空间,我们得到了这类Triebel-Lizorkin空间的原子分解和分子分解.这些结论是调和分析中函数空间分解理论的补充和完善.     

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.     

Decomposition of Besov and Triebel-Lizorkin spaces on the sphere

A discrete system of almost exponentially localized elements (needlets) on the n-dimensional unit sphere Sn is constructed. It shown that the needlet system can be used for decomposition of Besov and Triebel-Lizorkin spaces on the sphere. As an application of Besov spaces on Sn, a Jackson estimate for nonlinear m-term approximation from the needlet system is obtained.     

Function spaces of variable smoothness and integrability

In this article we introduce Triebel-Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.     

Atomic Decompositions of Triebel-Lizorkin Spaces with Local Weights and Applications

In this paper,the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fp,q s,w(Rn)with local weight w by using the Lusin-area functions for the full ranges of the indices,and then establish their atomic decompositions for s ∈ R,p ∈(0,1] and q ∈ [p,∞).The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in(0,1].Finite atomic decompositions for smooth functions in Fp,q s,w(Rn)are also obtained,which further implies that a(sub)linear operator that maps smooth atoms of Fp,q s,w(Rn)uniformly into a bounded set of a(quasi-)Banach space is extended to a bounded operator on the whole Fp,q s,w(Rn).As an application,the boundedness of the local Riesz operator on the space Fp,q s,w(Rn)is obtained.     

Embeddings of some classical Banach spaces into modulation spaces

We give sufficient conditions for a tempered distribution to belong to certain modulation spaces by showing embeddings of some Besov-Triebel-Lizorkin spaces into modulation spaces. As a consequence we have a new proof that the Hölder-Lipschitz space embeds into the modulation space when d$">. This embedding plays an important role in interpreting recent modulation space approaches to pseudodifferential operators.

     

A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces

Let s∈R, τ∈[0,∞), p∈(1,∞) and q∈(1,∞]. In this paper, we introduce a new class of function spaces which unify and generalize the Triebel-Lizorkin spaces with both p∈(1,∞) and p=∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel-Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where s∈R, p,q∈[1,∞), max{p,q}>1, , and t^′ denotes the conjugate index of t∈(1,∞); as an application of this, we further introduce certain Hardy-Hausdorff spaces and prove that the dual space of is just when p,q∈(1,∞).     

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...     

Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces

The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato. The first author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. The second author is supported by Fūjyukai foundation and the 21st century COE program at Graduate School of Mathematical Sciences, the University of Tokyo.     

Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of φ-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some related Sobolev-type embeddings and trace theorems of these spaces are also established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel–Lizorkin spaces.     

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.     

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 hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ ⁿ are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces. The research is supported by NSF of China (Grant Nos. 10571014 and 10571015) and SRFDP of China (Grant No. 20050027025)     

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     

Characterizations for Besov spaces and applications. Part I

The main theorem of this paper gives a characterization for holomorphic Besov space B^p(D) over a large class of bounded domains D in , which states that there is a bounded linear operator so that PV_D=I on B^p(D), where P is the Bergman projection, and is the biholomorphic invariant measure with K(z,z) being Bergman kernel function for D. Moreover, some application for characterizing Schatter von Neumann p-class small Hankel operation is given as a direct consequence of this theorem.     

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.     

Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means

In the present paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means. Further, by applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact.     

Singular integral operators on Triebel-Lizorkin spaces of para-accretive type

In this article, we study the boundedness of singular integral operators acting on Triebel-Lizorkin spaces of para-accretive type and show a Tb theorem on these spaces, which extends previous results in David, Journé and Semmes (1985) [3], Han (1994) [9], Han, Lee and Lin (2004) [10], Han and Sawyer (1990) [12], Lin and Wang (2009) [15], Wang (1999) [22].