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.     

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.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces on domains

The purpose of this paper is to develop a theory of the Besov‐Morrey spaces and the Triebel‐Lizorkin‐Morrey spaces on domains in R ⁿ. We consider the pointwise multiplier operator, the trace operator, the extension operator and the diffeomorphism operator. Not only to domains in R ⁿ we extend our definition of function spaces to compact oriented Riemannian manifolds. Among the properties above, the result for the trace operator is in particular interesting, which reflects the property of the parameters p, q in the Morrey space ??^p_q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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)     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

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.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures

We define Morrey type Besov‐Triebel spaces with the underlying measure non‐doubling. After defining the function spaces, we investigate boundedness property of some class of the singular integral operators (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Atomic decomposition for weighted Besov and Triebel‐Lizorkin spaces

Rychkov defined weighted Besov spaces and weighted Triebel‐Lizorkin spaces coming with a weight in the class $A_p^{\rm loc}$, which is even wider than the class A_p due to Muckenhoupt. In the present paper we are concerned with the atomic decomposition of these spaces. As an application, we obtain some key theorems in the theory of function spaces. Finally we conclude this paper with some helpful examples showing the difference between weighted spaces and unweighted spaces.     

Regular wavelets and Triebel‐Lizorkin type oscillation spaces

In this paper, we apply wavelets to study the Triebel‐Lizorkin type oscillation spaces and identify them with the well‐known Triebel‐Lizorkin‐Morrey spaces. Further, we prove that Calderón‐Zygmund operators are bounded on .     

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.     

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.     

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.     

Relations among Besov-type spaces,Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.     

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].     

On Stein's extension operator preserving Sobolev–Morrey spaces

We prove that Stein's extension operator preserves Sobolev–Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n‐dimensional Euclidean space.     

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.     

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)     

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.