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.     

<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.     

Global well-posedness for Navier–Stokes equations in critical Fourier–Herz spaces

We prove the global well-posedness for the 3D Navier–Stokes equations in critical Fourier–Herz spaces, by making use of the Fourier localization method and the Littlewood–Paley theory. The advantage of working in Fourier–Herz spaces lies in that they are more adapted than classical Besov spaces, for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero. Our result is an improvement of a recent theorem by Lei and Lin (2011) [10].     

A remark on the regularity criterion of Boussinesq equations with zero heat conductivity

In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012)  [6].     

Besov spaces via wavelets on metric spaces endowed with doubling measure,singular integral,and the T1 type theorem

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Using this we derive several useful characterizations from the one-dimensional case to the d-dimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.     

Besov type function spaces defined on metric-measure spaces

The purpose of this article is to study the Besov type function spaces for maps which are defined on abstract metric-measure spaces. We extend some of the embedding theorems of the classical Besov spaces to the setting of abstract 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)     

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.     

Some characterizations of the Besov space and the α-Bloch space

In this paper we obtain some characterizations of Besov spaces and α-Bloch spaces on the unit ball of Cn, which extend some results in the settings of the unit disk.     

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.     

Wavelet Characterizations for Anisotropic Besov Spaces

The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L_p-spaces with 0<p<∞ are derived.     

Analysis on Besov spaces II: Embedding and duality theorems

This paper gives several results on Besov spaces of holomorphic functions on a very large class of domains D in Cn. They include duality theorem, embedding theorem, best growth estimate, and boundedness of multiplication operators on Besov spaces.     

Convergence Rates of Adaptive Methods,Besov Spaces,and Multilevel Approximation

This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations.     

Interpolation inequality of logarithmic type in abstract Besov spaces and an application to semilinear evolution equations

An abstract version of Besov spaces is introduced by using the resolvent of nonnegative operators. Interpolation inequalities with respect to abstract Besov spaces and generalized Lorentz spaces are obtained. These inequalities provide a generalization of Sobolev inequalities of logarithmic type. Uniqueness problems to abstract semilinear evolution equations are also discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Besov Spaces with General Weights

We introduce Besov spaces with general smoothness. These spaces unify and generalize the classical Besov spaces. We establish the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. We establish the smooth atomic, molecular and wavelet decomposition of these function spaces. A characterization of these function spaces in terms of the difference relations is given.     

Ideal density‐dependent incompressible viscoelastic flow in the critical Besov spaces

In this paper, we consider the well‐posedness issue for the density‐dependent incompressible viscoelastic fluids of the Oldroyd model for the ideal case in space dimension greater than 2. We obtain the local well‐posedness of this model under the assumption that the initial density is bounded away from zero in the critical Besov spaces by means of the Littlewood‐Paley theory and Bony's paradifferential calculus. In particular, we obtain a Beale‐Kato‐Majida–type regularity criterion.     

多维双极Euler-Poisson方程的C~1解的整体存在性(英文)

本文研究了出现在半导体器件或者等离子中的多维双极Euler-Poisson方程,证明了它的初值问题的C1解在Besov空间的整体存在性,同时也得到了在二维和三维情形下,速度的璇度以指数的速率收敛到零.