Cauchy Integral Research Articles

Endpoint compactness of singular integrals and perturbations of the Cauchy integral

We prove sufficient and necessary conditions for the compactness of Calderón–Zygmund operators on the endpoint from L∞(R) into CMO(R). We use this result to prove the compactness on Lp(R) with 1<p<∞ of a certain perturbation of the Cauchy integral on curves with normal derivatives satisfying a CMO-condition.

Exponentially Convergent Method For An Abstract Nonlocal Problem With Integral Nonlinearity

We consider a nonlocal problem for the first-order differential equation with unbounded operator coefficient in a Banach space and a nonlinear integral nonlocal condition. We propose an exponentially convergent method for the numerical solution of this problem and justified this method under the assumptions that the indicated operator coefficient A is sectorial and that certain conditions for the existence and uniqueness of the solution are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein-type equation, discretization of this equation by the method of collocation, and its subsequent solution by the method of simple iterations. Each iteration of the method is based on the Sinc-quadrature approximation of the exponential operator function represented by the Dunford–Cauchy integral over the hyperbola enveloping the spectrum of A. The integral part of the nonlocal condition is approximated by using the Clenshaw–Curtis quadrature formula.

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

A direct approach to the $S$-functional calculus for closed operators

The $S$-functional calculus for slice hyperholomorphic functions generalizes the Riesz-Dunford-functional calculus for holomorphic functions to quaternionic linear operators and to $n$-tuples of noncommuting operators. For an unbounded closed operator, it is defined, as in the classical case, using an appropriate transformation and the $S$-functional calculus for bounded operators. This is however only possible if the $S$-resolvent set of the operator contains a real point. In this paper, we define the $S$-functional calculus directly via a Cauchy integral, which allows us to consider also operators whose resolvent sets do not contain real points. We show that the product rule and the spectral mapping theorem also hold true with this definition and that the $S$-functional calculus is compatible with polynomials, although polynomials are not included in the set of admissible functions if the operator is unbounded. We also prove that the $S$-functional calculus is able to create spectral projections. In order to do this, we remove another assumption usually made: we do not assume that admissible functions are defined on slice domains. Instead, we also consider functions that are defined on not necessarily connected sets. This leads to an unexpected phenomenon: the $S$-functional calculi for left and right slice hyperholomorphic functions become inconsistent and give different operators for functions that are both left and right slice hyperholomorphic. We show that any such function is the sum of a locally constant and an intrinsic function. For intrinsic functions both functional calculi agree, but for locally constant functions they must give different operators unless any spectral projection commutes with arbitrary scalars.

Investigation of an arbitrarily shaped inclusion embedded in a two-dimensional finite domain

As one of the fundamental problems in contemporary composite mechanics and micromechanics, the disturbance induced by an inclusion embedded in an elastic matrix has been garnering increasing attention in a wide variety of engineering applications. The focus of this study is particularly placed on constructing a general methodology to seek for the analytical solutions of the disturbed elastic field within a two-dimensional (2D) finite domain containing an arbitrarily shaped inclusion. In the theoretical derivation, the complex potential method is adopted and Laurent series expansion is employed to express the governing potential functions within the finite elastic domain. To overcome the mathematical challenges imposed by the complex geometries of inclusion-matrix interface and matrix boundary, the inclusion is replaced with an arbitrarily-sided polygon, which enables 1) the shape of inclusion to be closely approximated and 2) the gap function on the inclusion-matrix interface to be evaluated by Cauchy integrals performed on the straight edges. Then conformal mapping is utilized to map the real domain to a virtual unit circle. It is found that after the boundary condition is enforced, the coefficient of every term in the Laurent series expansions for the potential functions can be explicitly determined by evaluating the Fourier transform of the boundary effect function and its conjugate on the exterior boundary. To further delineate the mathematical framework, several examples are presented, and the obtained results are verified by numerical analysis and some analytical benchmark solutions available in the literature.

Riemann boundary value problems on the positive real axis

In this paper, some Riemann boundary value problems on the positive real axis are presented. Firstly, we introduce the concepts of principle part and order at infinity and zero point for the holomorphic function on the complex plane cut along the positive real axis, which are more extensive than those in the classic sense. Then, the behaviors of Cauchy-type integral and Cauchy principal value integral on the positive real axis at infinity and zero point are discussed, respectively. Based on those, Riemann boundary value problems for sectionally holomorphic functions with the positive real axis as their jump curve are established explicitly, which are different from Riemann problems on the finite curves and more complicated than those on the whole real axis. Finally, some boundary value problems for matrix-valued functions are also constructed, which play a very important role in the asymptotic analysis for orthogonal polynomials on the positive real axis.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

Harmonic reflection in quasicircles and well-posedness of a Riemann–Hilbert problem on quasidisks

A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H. Osborn. We call the set of such boundary values the Douglas–Osborn space. One may then attempt to solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasicircles are precisely those Jordan curves for which this reflection is defined and bounded. We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj–Sokhotski jump formula holds on quasicircles with boundary data in the Douglas–Osborn space. This enables us to prove the well-posedness of a Riemann–Hilbert problem with boundary data in the Douglas–Osborn space on quasicircles.

Difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces

ABSTRACTIn this paper, we characterize boundedness and compactness of difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces. Our characterizations are free from pseudo-hyperbolic metric, which is a common feature of all the characterizations of difference of composition operators acting between different spaces of holomorphic functions. Exact value of operator norm of difference of composition operators acting between these spaces is also computed.

The Riemann boundary value problem in the class of Cauchy type integrals with densities of grand variable exponent Lebesgue spaces

Abstract The present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

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.

Criterion for connections of solutions of the general convolution type paired integral equations in banach algebras with weight

Existence and connection between the solutions of the abstract general convolution type paired integral equations with arbitrary right-hand side and equal to Dirac’s delta function are considered. The theorem - criterion for such connection is formulated and proved. Procedure is a free from the theory of Cauchy integral and Hölder requirement

The Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue spaces

The present paper deals with a solution of the Riemann–Hilbert problem in the class of Cauchy type integrals with densities of certain new nonstandard Banach function spaces. The solvability conditions are explored and the solutions (if any) are constructed explicitly.

Estimating the Norm of the Holomorphic Component of a Meromorphic Function in a Finitely Connected Domain

In this paper, we extend a Gonchar–Grigorjan type estimate for the norm of the holomorphic parts of meromorphic functions in finitely connected Jordan domains with C2-smooth boundary in the case where the poles are in a compact set. A uniform estimate for Cauchy type integrals is also given.

On Secure Wireless Communications for IoT Under Eavesdropper Collusion

Wireless communication is one of the key technologies that actualize the Internet of Things (IoT) concept into the real world. Understanding the security performance of wireless communications lays the foundation for the security management of IoT. Eavesdropper collusion represents a significant threat to wireless communication security, while physical-layer security serves as a promising approach to providing a strong form of security guarantee. This paper studies the important secrecy outage performance of wireless communications under eavesdropper collusion, where the physical layer security is adopted to counteract such attack. Based on the classical Probability Theory, we first conduct analysis on the secrecy outage of the simple noncolluding case in which eavesdroppers do not collude and operate independently. For the secrecy outage analysis of the more hazardous M-colluding scenario, where any M eavesdroppers can combine their observations to decode the message, the techniques of Laplace transform, keyhole contour integral, and Cauchy Integral Theorem are jointly adopted to work around the highly cumbersome multifold convolution problem involved in such analysis, such that the related signal-to-interference ratio modeling for all colluding eavesdroppers can be conducted and thus the corresponding secrecy outage probability can be analytically determined. Finally, simulation and numerical results are provided to illustrate our theoretical achievements. An interesting observation suggests that the SOP increases first superlinearly and then sublinearly with M.

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.

An isotropic circular inhomogeneity partially bonded to an anisotropic medium

We study the generalized plane strain deformations of an elastically isotropic circular inhomogeneity partially bonded to an unbounded generally anisotropic elastic matrix. The two-phase composite is subjected to a uniform loading at infinity, and meanwhile a line force and a line dislocation are applied both in the inhomogeneity and in the matrix. An elegant closed-form solution is obtained by reducing the original boundary value problem to a non-homogeneous Riemann–Hilbert problem of vector form which can be analytically solved by using a decoupling method and evaluating the Cauchy integrals. Surface traction on the bonded part of the interface, displacement jump across the debonded part of the interface, and the complex and real stress intensity factors at the crack tips are explicitly derived when the composite is only subjected to a remote uniform loading.

Computing Partial Spectra with Least-Squares Rational Filters

We present a method for computing partial spectra of Hermitian matrices, based on a combination of subspace iteration with rational filtering. In contrast with classical rational filters derived from Cauchy integrals or from uniform approximations to a step function, we adopt a least-squares (LS) viewpoint for designing filters. One of the goals of the proposed approach is to build a filter that will lead to linear systems that are easier to solve by iterative methods. Among the main advantages of the proposed LS filters is their flexibility. Thus, we can place poles in more general locations than with the formulations mentioned above, and we can also repeat these poles a few times for better efficiency. This leads to a smaller number of required poles than in existing methods. As a consequence, factorization costs are reduced when direct solvers are used and the scheme is also beneficial for iterative solvers. The paper discusses iterative schemes to solve the linear systems resulting from the filtered s...

Cauchy representation formulas for Maxwell equations in 3-dimensional domains with fractal boundaries

The aim of this paper is to offer a new definition of Cauchy integral associated with Maxwell equations on 3-dimensional domains with fractal boundaries. This new Cauchy integral leads to several types of integral representation formulas, including the Cauchy representation formula for time-harmonic electromagnetic fields. Our approach is based on the framework of exploiting the close and well-defined connection between the quaternionic analysis and Maxwell equations.

Chord-arc curves and the Beurling transform

We study the relation between the geometric properties of a quasicircle~$\Gamma$ and the complex dilatation~$\mu$ of a quasiconformal mapping that maps the real line onto~$\Gamma$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-\mu S)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves.