Operator-valued Fourier Multipliers on Periodic Triebel Spaces

We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.     

算子值傅里叶乘子与向量值边值问题最大正则性

向量值L^p空间上的算子值傅里叶乘子由于L．Weis在2000年的重要工作而成为泛函分析的热点之一，其对R-有界性创造性的应用使这个领域的研究耳目一新，新的结果层出不穷．本文的目的是介绍算子值傅里叶乘子的这些最新进展，以及它们在向量值边值问题最大正则性方面的应用．包括N．J．Kalton和G．Lancien给出的关于L^p-最大正则性的反例．Besov空间和Triebel空间上的算子值傅里叶乘子以及在Besov空问和Triebel空间意义下的最大正则性也是我们要介绍的内容．     

关于Triebel空间上的算子值傅里叶乘子

利用Str6mberg-Torchinsky分解,给出了Triebel空间FEp.q(Rn,X)上算子值傅里叶乘子的一个充分条件.在rl相似文献   

关于Triebei空间上的算子值傅里叶乘子

利用Stroemberg-Torchinsky分解,给出了Triebel空间Fp-q（R^n,X）上算子值傅里叶乘子的一个充分条件．在n〈min（p,q）情形下,这里给出的充分条件改进了之前已知的结果．     

Pointwise multiplication by the characteristic function of the half-space on anisotropic vector-valued function spaces

We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel–Lizorkin type.     

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

Wavelets and local Triebel–Lizorkin spaces with the Lorentz index

In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.     

Periodic solutions of a fractional neutral equation with finite delay

In this paper, we prove the maximal regularity property of an abstract fractional differential equation with finite delay on periodic Besov and Triebel–Lizorkin spaces and use these results to guarantee the existence and uniqueness of periodic solution of a neutral fractional differential equation with finite delay. The main tool used to achieve our goal is an operator-valued version of Miklhin’s Fourier multiplier theorem and fixed-point argument.     

小波与函数空间

Triebe利用Littlewood Paley分解将大多数函数空间分类成两类三指标的函数空间：Besov空间和Triebel Lizorkin空间；但Littlewood Paley 分解很难直接分析Sobolev空间L^p的插值空间Lorentz空间，也很难分析Triebel Lizorkin空间F^{α,q}_1的预备对偶空间和对偶空间.运用小波，作者给出这些空间一个统一刻画：Triebel Lizorkin Lorentz 空间，Besov Lorentz空间和F^{α,q}_1的预备对偶空间和对偶空间；另外也研究这些空间的三个性质.     

Some Remarks on the Boundedness of Pseudodifferential Operators

Some methods are described for reducing the problem of the boundedness of pseudodifferential operators (ΨDOs) to the theory of Fourier multipliers. Special attention is given to the boundedness of ΨDOs in Besov - Triebel - Lizorkin spaces.     

Harmonic Besov and Triebel–Lizorkin Spaces on the Ball

Harmonic Besov and Triebel–Lizorkin spaces on the unit ball in \({\mathbb R}^d\) with full range of parameters are introduced and studied. It is shown that these spaces can be identified with respective Besov and Triebel–Lizorkin spaces of distributions on the sphere. Frames consisting of harmonic functions are also developed and frame characterization of the harmonic Besov and Triebel–Lizorkin spaces is established.     

Anisotropic Lizorkin–Triebel spaces with mixed norms — traces on smooth boundaries

This article deals with trace operators on anisotropic Lizorkin–Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self‐contained exposition of Lizorkin–Triebel spaces on manifolds and extend these results to mixed‐norm Lizorkin–Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right‐inverse of the trace is given explicitly and proved to be continuous in the scale of mixed‐norm Lizorkin–Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.     

Decomposition for Morrey type Besov–Triebel spaces

In this paper, the author establishes the decomposition of Morrey type Besov–Triebel spaces in terms of atoms and molecules concentrated on dyadic cubes, which have the same smoothness and cancellation properties as those of the classical Besov–Triebel spaces. The results extend those of M. Frazier, B. Jawerth for Besov–Triebel spaces and those of A. L. Mazzucato for Besov–Morrey spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.     

双线性Fourier乘子算子与Morrey空间

In this paper, by establishing a result concerning the mapping properties for bi(sub)linear operators on Morrey spaces, and the weighted estimates with general weights for the bilinear Fourier multiplier, the author establishes some results concerning the behavior on the product of Morrey spaces for bilinear Fourier multiplier operator with associated multiplierσ satisfying certain Sobolev regularity.     

Atomic and Subatomic Decompositions in Anisotropic Function Spaces

This work deals with decompositions in anisotropic function spaces. Defining anisotropic atoms as smooth building blocks which are the counterpart of the atoms from the works of M. Frazier and B. Jawerth , it is shown that the study of anisotropic function spaces can be done with the help of some sequence spaces in a similar way as it is done in the isotropic case. It is also shown that the subatomic decomposition theorem for isotropic function spaces, recently proved by H. Triebel , can be extended to the anisotropic case if the mean smoothness parameter is sufficiently large.     

Distributional Dimension of Fraetal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension     

On the coincidence of certain approaches to smoothness spaces related to Morrey spaces

In this paper, we compare the recent approach of Hans Triebel to introduce smoothness spaces related to Morrey‐Campanato spaces with Besov type and Triebel‐Lizorkin type spaces. These two scales have been introduced some years ago and represent a further variant to measure smoothness by using Morrey spaces.