Inhomogeneous Besov Spaces Research Articles

The Tb theorem for some inhomogeneous Besov and Triebel-Lizorkin spaces over space of homogeneous type

In this paper, the authors give some new characterizations of the inhomogeneous Besov spaces and inhomogeneous Triebel-Lizorkin spaces respectively over space of homogeneous type, and establish the Tb theorems for the boundedness of the Calderón-Zygmund singular integral operator with non-convolution kernel on these new inhomogeneous Besov and Triebel-Lizorkin spaces.

Local existence for the d‐dimensional magneto‐micropolar equations with fractional dissipation in Besov spaces

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations ( or ) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case.

Boussinesq system with measure forcing

The paper analyzes the Navier–Stokes system coupled with the convective-diffusion equation responsible for thermal effects. It is a version of the Boussinesq approximation with a heat source. The problem is studied in the two dimensional plane and the heat source is a measure transported by the flow. For arbitrarily large initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructed solutions and it requires a non-standard framework of inhomogeneous Besov spaces of the L^infty (0,T;B^s_{p,infty })-type. The uniqueness of solutions is proved by using the Lagrangian coordinates.

Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces.

Unique weak solutions of the magnetohydrodynamic equations with fractional dissipation

AbstractThis paper examines the existence and uniqueness of weak solutions to the d‐dimensional magnetohydrodynamic (MHD) equations with fractional dissipation and fractional magnetic diffusion . The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting and when , and . The case when with and has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for , which allows us to close the estimates in the aforementioned Besov spaces.

A classification of anisotropic Besov spaces

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A∈GL(d,R), with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out that the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices A,B that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of A,B.

Oscillatory hyper-Hilbert transform along curves on modulation spaces

We consider the boundedness of the n-dimension oscillatory hyper-Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems signicantly improve some known results.

Strongly singular integrals along curves on \u03b1-modulation spaces

In this paper, we study the strongly singular integrals \t\t\tTn,β,γf(x)=p.v.∫−11f(x−Γθ(t))e−2πi|t|−βt|t|γdt\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$T_{n, \\beta, \\gamma}f(x)=\\mathrm{p.v.} \\int_{-1}^{1}f\\bigl(x-\\Gamma_{\\theta}(t) \\bigr)\\frac {e^{-2\\pi i \\vert t \\vert ^{-\\beta}}}{t \\vert t \\vert ^{\\gamma}}\\,dt $$\\end{document} along homogeneous curves Gamma_{theta}(t). We prove that T_{n, beta, gamma} is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces.

Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates

Let $(X, d, \mu)$ be a space of homogeneous type equipped with a distance $d$ and a measure $\mu$. Assume that $L$ is a closed linear operator which generates an analytic semigroup $e^{-tL}, t > 0$. Also assume that $L$ has a bounded $H_\infty$-calculus on $L^2(X)$ and satisfies the $L^p-L^q$ semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel bounds). The aim of this paper is to establish a theory of inhomogeneous Besov spaces associated to such an operator $L$. We prove the molecular decompositions for the new Besov spaces and obtain the boundedness of the fractional powers $(I+L)^{-\gamma}, \gamma > 0$ on these Besov spaces. Finally, we carry out a comparison between our new Besov spaces and the classical Besov spaces.

Shearlets and pseudo-differential operators

Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hormander definition with x-dependent parameter $$\delta $$ for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO.

A unified approach for Littlewood-Paley decomposition of abstract Besov spaces

We apply spectral theoretic methods to obtain a Littlewood-Paley decomposition of abstract inhomogeneous Besov spaces in terms of “smooth” and “bandlimited” functions. Well-known decompositions in several contexts are shown as special examples and are unified under the spectral theoretic approach.

Estimation of the transition density of a Markov chain

Nous présentons deux procédures pour estimer la densité de transition d’une chaîne de Markov homogène. Dans la première procédure, nous construisons un estimateur constant par morceaux sur une partition aléatoire bien choisie. Nous établissons des bornes de risque non-asymptotiques pour une perte de type Hellinger lorsque la racine carrée de la densité de transition appartient à un espace de Besov inhomogène dont l’indice de régularité peut être petit. Nous illustrons ces résultats par des simulations numériques. La deuxième procédure est d’intérêt théorique. Elle permet d’obtenir un théorème de sélection de modèle à partir duquel nous déduisons des vitesses de convergence sur des espaces de Besov inhomogènes anisotropes. Nous étudions finalement les vitesses qui peuvent être atteintes sous des hypothèses structurelles sur la densité de transition.

The Cauchy problem for the generalized Camassa–Holm equation in Besov space

In this paper we consider the Cauchy problem for the generalized Camassa–Holm equation ut+uQux+∂x(1−∂x2)−1[2ku+Q2+3Q2(Q+1)uQ+1+Q2uQ−1ux2]=0 in Besov space. First, we prove that the solutions to the Cauchy problem for the generalized Camassa–Holm equation do not depend uniformly continuously on the initial data in Hs(R) with s<3/2 when k=0. Second, combining the real interpolations among inhomogeneous Besov spaces with Lemma 5.2.1 of [6] which is called Osgood Lemma (a substitute for Gronwall inequality), we show that the Cauchy problem for the generalized Camassa–Holm equation is locally well-posed in B2,13/2. Finally, we give a blow-up criterion.

Generalized Besov Spaces and Their Applications

We define and study the Bessel potential and inhomogeneous Besov spaces associated with the Dunkl operators on $\mathbf{R}^d$. As applications on these spaces we construct the Sobolev type embedding theorem and the paraproduct operators associated with the Dunkl operators, as similar to those defined by Bony. We also establish Strichartz type estimates for the Dunkl-Schrodinger equation and finally we study the problem of well posedness of the generalized heat equation.

Pointwise multipliers of Besov spaces on Carnot–Carathéodory spaces

Let (X,d,μ) be a Carnot–Carathéodory space, namely, X is a smooth manifold, d is a control, or Carnot–Carathéodory, metric induced by a collection of vector fields of finite type. μ is a nonnegative Borel regular measure on X satisfying that there exists constant C0∈[1,∞) such that for all x∈X and 0<r<diamX, μ(B(x,2r)):=μ({y∈X:d(x,y)<2r})≤C0μ(B(x,r))<∞(doublingproperty). Using the discrete Calderón reproducing formula and the Plancherel–Pôlya characterization of the inhomogeneous Besov spaces developed by Han et al. [12], and Han et al. (2008) [10], pointwise multipliers of inhomogeneous Besov spaces are obtained.

Uniqueness of weak solutions to the Cauchy problem for the 3-D time-dependent Ginzburg-Landau model for superconductivity

We prove some uniqueness results for the Cauchy problem for the 3-D time-dependent Ginzburg-Landau (TDGL) model for superconductivity with the choice of the Lorentz gauge in the multiplier spaces (Morrey spaces) and in the inhomogeneous Besov spaces, respectively.

A Theory of Besov and Triebel‐Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot‐Carathéodory Spaces

We work on RD‐spaces 𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝒳. An important example is the Carnot‐Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel‐Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spaces Hp(𝒳) and local Hardy spaces hp(𝒳) on RD‐spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces when p ≥ 1. The relations among homogeneous Besov spaces and Triebel‐Lizorkin spaces, inhomogeneous Besov spaces and Triebel‐Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel‐Lizorkin spaces for classes of singular integral operators, which include non‐isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn‐Laplacian on certain classes of model domains in ℂN. Our theory applies in a wide range of settings.