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.     

Characterization of multiplier spaces by wavelets and logarithmic Morrey spaces

In this paper, we consider multipliers from Sobolev spaces to Lebesgue spaces. We establish some wavelet characterization of multiplier spaces without using capacity. Further, we give a sharp logarithmic Morrey space condition for multipliers which lessens Fefferman’s Morrey space condition to the logarithm level and generalizes Lemarié’s counter-example to non-integer cases and expresses his results in a more precise way.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

Holomorphic Sobolev spaces associated to compact symmetric spaces

Using Gutzmer's formula, due to Lassalle, we characterise the images of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.     

Fourier–Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier–Jacobi expansions in Morrey spaces.     

Composition operators on Dirichlet spaces and Bloch space

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.     

Boundaries of Asplund spaces

We study the relationship between the classical combinatorial inequalities of Simons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a characterization of strong boundaries for Asplund spaces using the new concept of finitely self-predictable set. Strong properties for w^∗-K-analytic boundaries are established as well as a sup-lim sup theorem for Baire maps.     

Haarlet analysis of Lipschitz regularity in metric measure spaces

In this note we shall give a characterization of Lipschitz spaces on spaces of homogeneous type via Haar coefficients.     

On the boundedness of rough oscillatory singular integrals on Triebel-Lizorkin spaces

We obtain appropriate sharp bounds on Triebel-Lizorkin spaces for rough oscillatory integrals with polynomial phase. By using these bounds and using an extrapolation argument we obtain some new and previously known results for oscillatory integrals under very weak size conditions on the kernel functions.     

Interpolation theorem on Lorentz spaces over weighted measure spaces

In 1997 Ferreyra proved that it is impossible to extend the Stein-Weiss theorem in the context of Lorentz spaces. In this paper we obtain an interpolation theorem on Lorentz spaces over weighted measure spaces.

     

Besov spaces and Herz spaces on local fields

The relationship of Besov spaces and Herz spaces on local fields is given. As an application, one multiplier theorem is obtained. And the decompositional characterization of the weighted Besov spaces is established.     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

赋范线性空间中的列收敛

本文证明了Ｓｃｈｕｒ空间与有限维Ｂａｎａｃｈ空间的两个特征定理，并对赋范线性空间中的列收敛进行了讨论，得到了一些有趣的结果。     

Basic functional properties of certain scale of rearrangement-invariant spaces

We define a new scale of function spaces governed by a norm-like functional based on a combination of a rearrangement-invariant norm, the elementary maximal function, and powers. A particular instance of such spaces surfaced recently in connection with optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures; however, a thorough analysis of these structures has not been carried out. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings, and, in a particular situation, a characterization of their associate structures. We discover a new one-parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.     

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and Rauhut (2005) [24] that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.     

CWC映射和度量化定理

利用CWC映射,本文获得了对称度量空间和g可度量空间的特征,建立了几个度量化定理,改进了一些已知结果．主要的定理是证明正则空间X是可度量化空间当且仅当存在X上的CWC映射g满足如下条件: (Ⅰ)若序列{xn}和{yn}对于每一n∈N有xn∈g(n,yn)且xn→x,则yn→x．(Ⅱ)若序列{xn}和{yn}对于每一n∈N有yn∈g(n,yn)且xn→x,则yn→x．     

Changes of variables in modulation and Wiener amalgam spaces

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of such spaces and, as a consequence, we obtain several versions of local and global Beurling–Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on ${\mathscr F}L^q$. Finally, counterparts of these results are discussed for spaces on the torus.