Cauchy Integral Operator Research Articles

Thomson decompositions of measures in the disk

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

Existence of global solutions to the nonlocal Schrödinger equation on the line

AbstractWe address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation under the initial data with the small‐norm. The nonlocal NLS equation was first introduced by Ablowitz and Musslimani as a new nonlocal reduction of the well‐known Ablowitz–Kaup–Newell–Segur system. The main technical difficulty for proving its global well‐posedness on the line in is due to the fact that mass and energy conservation laws, being nonlocal, do not preserve any reasonable norm and may be negative. In this paper, we use the inverse scattering transform approach to prove the existence of global solutions in based on the representation of a Riemann–Hilbert (RH) problem associated with the Cauchy problem of the nonlocal NLS equation. A key of this approach is, by applying the Volttera integral operator and Cauchy integral operator, to establish a Lipschitz bijective map between the solution of the nonlocal NLS equation and reflection coefficients associated with the RH problem. By using the reconstruction formula and estimates on the solution of the time‐dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the nonlocal NLS equation.

Cauchy Integral Operator in Complex Structures with Degeneracies

Cauchy Integral Operator in Complex Structures with Degeneracies

A Derivative Formula for the Solid Cauchy Integral Operator and Its Applications

A Derivative Formula for the Solid Cauchy Integral Operator and Its Applications

On the asymptotic properties of a canonical diffraction integral.

We introduce and study a new canonical integral, denoted , depending on two complex parameters α 1 and α 2. It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in , and derive its rich asymptotic behaviour as |α 1 | and |α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +- arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener-Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math., in press). As a result, the integral can be used to mimic the unknown function G +- and to build an efficient 'educated' approximation to the quarter-plane problem.

A note on weak factorization of a Meyer-type Hardy space via a Cauchy integral operator

This paper provides a weak factorization for the Meyer-type Hardy space $H^1_b(\mathbb{R})$, and characterizations of its dual ${\rm BMO}_b(\mathbb{R})$ and its predual ${\rm VMO}_b(\mathbb{R})$ via boundedness and compactness of a suitable commutator with the Cauchy integral $\mathscr{C}_{\Gamma}$, respectively. Here $b(x)=1+iA'(x)$ where $A'\in L^{\infty}(\mathbb{R})$, and the Cauchy integral $\mathscr{C}_{\Gamma}$ is associated to the Lipschitz curve $\Gamma=\{x+iA(x)\, : \, x\in \mathbb{R}\}$.

The Cauchy integral, bounded and compact commutators

We study the commutator of the well-known Cauchy integral operator with a locally integrable function $b$ on $\mathbb R$, and establish the characterisation of the BMO space on $\mathbb R$ via the $L^p$ boundedness of this commutator. Moreover, we also establish the characterisation of the VMO space on $\mathbb R$ via the compactness of this commutator.

Two Weight Commutators on Spaces of Homogeneous Type and Applications

In this paper, we establish the two weight commutator theorem of Calderon–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $$A_2$$ weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderon–Zygmund operators: Cauchy integral operator on $${\mathbb {R}}$$ , Cauchy–Szego projection operator on Heisenberg groups, Szego projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

Hyponormality of singular Cauchy integral operators with matrix-valued symbols

In this paper, we study a class of hyponormal singular integral operators with matrix-valued symbols. First, we characterize hyponormal singular integral operators $S_{\Phi,\Psi}$ with trigonometric polynomial symbols $\Phi$ and $\Psi$. Next, we concentrate on the hyponormality of $S_{\Phi,\Psi}$ with some assumptions for the symbols $\Phi$ and $\Psi$.

The Boundedness of Cauchy Integral Operator on a Domain Having Closed Analytic Boundary

In this paper, we prove that the Cauchy integral operators (or Cauchy transforms) define continuous linear operators on the Smirnov classes for some certain domain with closed analytic boundary.

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let CΓ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,CΓ] is bounded (resp, compact) on the Morrey space for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if (resp, ). As an application, a factorization of the classical Hardy space in terms of CΓ and its adjoint operator is obtained.

General Clark model for finite-rank perturbations

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$. For a purely contractive $\Gamma$ the resulting perturbed operator $T_\Gamma$ is a contraction (a completely non-unitary contraction under the natural assumption about cyclicity of the range), so they admit the functional model. In this paper we investigate the Clark operator, i.e. a unitary operator that intertwines $T_\Gamma$ (presented in the spectral representation of the non-perturbed operator $U$) and its model. We make no assumptions on the spectral type of the unitary operator $U$; absolutely continuous spectrum may be present. We find a representation of the adjoint Clark operator in the coordinate free Nikolski--Vasyunin functional model. This representation features a special version of the vector-valued Cauchy integral operator. Regularization of this singular integral operator yield representations of the adjoint Clark operator in the Sz.-Nagy--Foias transcription. In the special case of inner characteristic functions (purely singular spectral measure of $U$) this representation gives what can be considered as a natural generalization of the normalized Cauchy transform (which is a prominent object in the Clark theory for rank one case) to the vector-valued settings.

Cauchy independent measures and almost-additivity of analytic capacity

We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. However, we need the discs in question to be separated, and it is not clear whether the separation condition is essential. We apply this result to find families {μj} of measures in ℂ with the following property. If the Cauchy integral operators $${C_{{\mu _j}}}$$ from L2(μj) to itself are bounded uniformly in j, then Cμ, μ = Σμj, is also bounded from L2(μ) to itself.

Solvability of singular integro-differential equations via Riemann–Hilbert problem

The article is to study singular integro-differential equations involving convolutional operators and Cauchy integral operators via Riemann–Hilbert problem. To do this, we adopt a new approach through Fourier transform on L2 subspace which is Hölder-continuous with a certain decay at infinity. The Fourier transform converts the equations into a Riemann–Hilbert problem with Hölder-continuous coefficients and with nodal points, which allows us to construct the general solutions.

A class of planar vector fields with homogeneous singular points: Solvability and boundary value problems

This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered.

Cauchy Integral Operators Involving Higher Order Lipschitz Classes in the Poly-Analytic Function Theory

In this note, we consider poly-analytic Cauchy integral operators introduced in a rather natural way on the higher order Lipschitz classes. We prove some boundary properties of the function represented by such an integral operator in the general context of rectifiable Jordan curves.

Normal singular Cauchy integral operators with operator-valued symbols

In this paper we characterize a class of normal (isometric, coisometric, unitary, hyponormal) singular Cauchy integral operators with operator-valued (or matrix-valued) symbols on vector-valued (or Cn-valued) L2 spaces.

Cauchy integral and singular integral operator over closed Jordan curves

This paper is mostly a review paper. It contains a description of old and recent results concerning the regularity conditions on a Jordan curve in the plane that imply the boundedness of the singular integral operator as well as the boundary behavior of the Cauchy type integral. These results are of significance for boundary value problems in domains with non-smooth and non-rectifiable boundaries.

The integral equation method for electromagnetic scattering problem at oblique incidence

We consider the scattering of electromagnetic waves scattered by an infinitely long impedance cylinder at oblique incidence, which is modeled as a system of a pair of the two-dimensional Helmholtz equations with coupled oblique boundary conditions. The solvability of such a scattering problem is proven by using the boundary integral equation method. By expressing the scattered fields in the form of single-layer potentials, our oblique scattering problem is transformed to a system of two integral equations. It is not a usual Fredholm system of the second kind as that in the case of normal incidence, since the system involves the tangential derivatives of the single-layer potential. By relating it to the Cauchy integral operator, we show that this system of operators is of Fredholm type with index 0. Therefore, the solvability of the integral system follows from the uniqueness of its solutions due to the Fredholm theory. A numerical scheme for solving the integral equations is also presented with some numerics. The numerical results illustrate the validity and efficiency of the proposed method.

The Cauchy integral operator on weighted Hardy space

The authors show that the Cauchy integral operator is bounded from Hωp(R1) to hwp(R1) (the weighted local Hardy space). To prove the results, a kind of generalized atoms is introduced and a variant of weighted “Tb theorem” is considered.