Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space. The second author was supported in part by a National Science Foundation grant DMS 0604561.     

Global L 2-Boundedness Theorems for a Class of Fourier Integral Operators

We prove sharp interpolatory estimates between directional Haar projections and Riesz Transforms. We apply those to prove a conjecture of L. Tartar that arose within the theory of compensated compactness. To provide a frame of reference for the analytical estimates addressed in this paper we briefly review the core ideas of compensated compactness as developed by Murat and Tartar.     

Fourier Integral Operators Defined by Classical Symbols with Exit Behaviour

We present the detailed construction of the classical version of the calculus for Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour ( SG symbols). In particular, we analyse what happens when one restricts the choice of amplitude and phase functions to the subset of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a “classical” counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1), refining the known results about the analogous problem for general SG hyperbolic operators. The theory developed here will be used in forthcoming papers to study the propagation of singularities and the Weyl formula for suitableclasses of operators defined on manifolds with cylindrical ends.     

On the Algebra of Pair Integral Operators with Homogeneous Kernels

In this paper, we study the Banach algebra generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra to be Fredholm and present a formula for calculating the index.     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

傅里叶变换中一个含参变量广义积分的计算

本文利用含参变量广义积分计算中常用的求导和变量替换等技巧等给出了傅里叶变换中一个重要的含参变量广义积分的计算方法.     

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.     

BMO与弱核条件下的Calderón-Zygmund算子交换子的有界性

本文证明了BMO与弱核条件下Calderón-Zygmund奇异积分算子的交换子 [B,T]f=BTf-T(Bf)的L2有界性,这里B ∈BMO,T对增生的b1,b2,满足 T(b1)∈BMO,T*(b2)∈BMO,b2Tb1∈WBP．     

BMO与弱核条件下的Calderón-Zygmund算子交换子的有界性

本文证明了BMO与弱核条件下Calderón-Zygmund奇异积分算子的交换子 [B,T]f=BTf-T(Bf)的L2有界性,这里B ∈BMO,T对增生的b1,b2,满足 T(b1)∈BMO,T*(b2)∈BMO,b2Tb1∈WBP．     

Wave Front Set at Infinity and Hyperbolic Linear Operators with Multiple Characteristics

We discuss a notion of wave front set which allows us tocontrol the behaviour `at infinity' of temperate distributions. Weobtain the microlocality and microellipticity properties with respect toa class of global pseudodifferential operators and a propagation theoremfor the corresponding class of Fourier Integral Operators. Through theseresults, we prove an adapted global version of the classical theoremconcerning the singularities of solutions of hyperbolic Cauchy problemsfor linear operators with multiple characteristics of constantmultiplicities. Finally, we make a comparison with the scattering wavefront set introduced by Melrose.     

一类粗糙核奇异积分算子的端点估计

设T_Ω是带粗糙核的Calderón-Zygmund奇异积分算子,I为任意真包含在单位圆周S~1上的闭圆弧.本文证明,若Ω支在I上并在I上单调,那么T_Ω是从Hardy空间H~1(R~2)到L~1(R~2)的有界算子当且仅当‖Ω‖_(LlogL(S~1))∞.     

Approximating the Approximant: A Numerical Code for Polynomial Compression of Discrete Integral Operators

The action of various one-dimensional integral operators, discretized by a suitable quadrature method, can be compressed and accelerated by means of Chebyshev series approximation. Our approach has a different conception with respect to other well-known fast methods: its effectiveness rests on the smoothing effect of integration, and it works in linear as well as nonlinear instances, with both smooth and nonsmooth kernels. We describe a Matlab toolbox which implements Chebyshev-like compression of discrete integral operators, and we present several numerical tests, where the basic O(n ²) complexity is shown to be reduced to O(mn), with mn.     

Regularization and Fourier Series for Fredholm Integral Equations of the Second Kind with a Weakly Singular Kernel

To compute an approximate solution, of a weakly singular integral equation, we first regularize the kernel, and then truncate the associated Fourier series. An application to an astrophysical transfer equation is presented.     

和一类积分算子相关的单叶函数的子类

在Ruscheweyh定义了解析函数的Ruscheweyh导数[1]之后,许多学者相继研究了与Ruscheweyh导数有关的单叶或者多叶解析函数类.近来,Jung,Ki m和Srivastava[5]引入了下面的单参数积分算子类:Iσf(z)=zΓ2(σσ)∫0zlogtzσ-1f(t)dt,σ0,f∈Α.算子Iσ和Flett[6]研究的乘数变换密切相关.本文利用算子Iσ定义了两个函数类.首先研究在单位圆内解析的单叶函数类Rσ(A,B),给出函数类的包含关系Rσ(A,B)Rσ+1(A,B),同时也考虑了在积分算子Fλ的作用下的函数类的包含关系以及当λ取特殊值1时的特殊情况.其次研究了函数类Rσ(A,B)中系数为正实数的函数类Sσ(A,B),给出函数f(z)属于类Sσ(A,B)的充分必要条件.     

Singular Integral Operators,Morrey Spaces and Fine Regularity of Solutions to PDE's

Boundedness in Morrey spaces is studied for singular integral operators with kernels of mixed homogeneity and their commutators with multiplication by a BMO-function. The results are applied in obtaining fine (Morrey and Hölder) regularity of strong solutions to higher-order elliptic and parabolic equations with VMO coefficients.     

新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 and introduced by Duong,Yan and Tang, respectively. Assuming that T is a bounded linear operator on L~2(X), we find the sufficient condition on the kernel of T so that T is bounded from BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.     

多复变Cauchy积分和被动算子与多重色散关系

本文阐述多重色散关系联系着矩阵z(p)的实部与虚部:,式中z(p)是阻抗z(ζ)的边值.     

核为调和方程基本解的积分算子在L2上的紧性

本文给出了以调和方程基本解１ｒ和ｌｎ１ｒ为核的积分算子在Ｌ２（Ω）上的紧性证明     

Infinite type homeomorphisms of the circle and convergence of Fourier series

We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.