A general form of Green Formula and Cauchy Integral Theorem

Birth and life of the $L^{2}$ boundedness of the Cauchy Integral on Lipschitz graphs

We review various motives for considering the problem of estimating the Cauchy Singular Integral on Lipschitz graphs in the $L^{2}$ norm. We follow the thread that led to the solution and then describe a few of the innumerable applications and ramifications of this fundamental result. We concentrate on its influence in complex analysis, geometric measure theory and harmonic measure.

Some remarks about Cauchy integrals and totally real surfaces in C^m

There seems to be quite a bit of room for interesting things related to surfaces M in C^m with real dimension m which are totally real and aspects of several complex variables on C^m around M. A basic case occurs when m = 1, with Cauchy integral operators along curves, holomorphic functions on the complements of curves, and so on. In the present article we briefly indicate a few points concerning these matters.

Calderón commutators and the Cauchy integral on Lipschitz curves revisited I. First commutator and generalizations

This article is the first in a series of three papers, whose scope is to give new proofs to the well known theorems of Calderón, Coifman, McIntosh and Meyer. Here we treat the case of the first commutator and some of its generalizations.

Calderón commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions

This article is the last in a series of three papers, whose scope is to give new proofs to the well known theorems of Calderón, Coifman, McIntosh and Meyer. Here we extend the results of the previous two papers to the polydisc setting. In particular, we solve completely an open question of Coifman from the early eighties.

An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on $\mathbb R$

This paper concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights $w:=\exp(-Q)$ with finite moments and with smooth external fields $Q:\mathbb R\to [0,\infty)$, with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to $w^2$. We also study bounds for the first derivatives of a class of functions of the second kind for $w^2$.

关于通过柯西积分算子的梅耶型硬空间弱分解的注记

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

本文提供了梅耶型Hardy空间H^1_b(R)的弱分解，并通过与柯西积分C_Γ相关的有界性和紧性刻画其对偶空间BMO_b(R)和其预对偶空间VMO_b(R)

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogo…

Cauchy theorem for a surface integral in commutative algebras

We prove an analogue of the Cauchy integral theorem for hyperholomorphic functions given in three-dimensional domains with non piece-smooth boundaries and taking values in an arbitrary finite-dimensional commutative associative Banach algebra.

Bilipschitz maps, analytic capacity, and the Cauchy integral

Let vphi:C rightarrow C be a bilipschitz map. We prove that if E\subset\C is compact, and gamma(E), alpha(E) stand for its analytic and continuous analytic capacity respectively, then C^{-1}γ(E)\leq γ(\vphi(E)) \leq Cγ(E) and C^{-1}α(E)\leq α(\vphi(E)) \leq Cα(E), where C depends only on the bilipschitz constant of vphi. Further, we show that if mu is a Radon measure on C and the Cauchy transform is bounded on L^2(μ), then the Cauchy transform is also bounded on L^2(\vphi_\sharpμ), where vphi_\sharpμis the image measure of mu by vphi. To obtain these results, we estimate the curvature of vphi_…

Cauchy's formula on nonempty closed sets and a new notion of Riemann--Liouville fractional integral on time scales

We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.

Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent Abel integral operators, based on which we analyze the corresponding integral and differential equations; (ii) We prove that the sensitive dependence of the well-posedness of classical Riemann-Liouville fractional differential equations on the initial value could be resolved by adjusting the initial value of the variable exponent; (iii) We prove that the si…

The Cauchy-Schlomilch transformation

The Cauchy-Schlömilch transformation states that for a function $f$ and $a, \, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interval $[0, \infty)$ are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given.

致敬 Richard Askey，及其钟爱的若干 Beta 积分的广阔视角

A Tribute to Richard Askey, and an Expansive View of Some of his Favorite Beta Integrals

本文致敬 Richard Askey，并重新审视其钟爱的 Beta 积分，提出 Cauchy beta 积分在实对称矩阵空间上的多维推广，进而推导出 Cauchy--Selberg 积分的特例。

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

We study AAK as well as Padé approximants to functions f, where f is a sum of a Cauchy transform of a complex measure μsupported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of μ. It is shown that the approximants converge to f locally uniformly in the domain of holomorphy of f, intersected with the unit disk in the case of AAK approximants. In the case of Padé approximants we need …

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to rel…

Cauchy non-integral formulas

We study certain generalized Cauchy integral formulas for gradients of solutions to second order divergence form elliptic systems, which appeared in recent work by P. Auscher and A. Rosén. These are constructed through functional calculus and are in general beyond the scope of singular integrals. More precisely, we establish such Cauchy formulas for solutions $u$ with gradient in weighted $L_2(\R^{1+n}_+,t^αdtdx)$ also in the case $|α|<1$. In the end point cases $α= \pm 1$, we show how to apply Carleson duality results by T. Hytönen and A. Rosén to establish such Cauchy formulas.

Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

Cauchy Type Integrals of Algebraic Functions

We consider Cauchy type integrals $I(t)={1\over 2πi}\int_γ {g(z)dz\over z-t}$ with $g(z)$ an algebraic function. The main goal is to give constructive (at least, in principle) conditions for $I(t)$ to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function $g$, the geometry of the integration curve $γ$, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functi…

An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind

In the paper, the author establishes an integral representation for Cauchy numbers of the second kind, finds the complete monotonicity, minimality, and logarithmic convexity of Cauchy numbers of the second kind, and presents some inequalities for determinants and products of Cauchy numbers of the second kind.

A family of singular integral operators which control the Cauchy transform

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-bo…

Properties of fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function

This paper aims to investigate properties associated with fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function. We prove that the Cauchy problem and the Volterra integral equation are equivalent. We find a closed-form to the solution of the Cauchy problem using successive approximations method and $ψ$-Caputo fractional derivative.

Cauchy Fluxes and Gauss-Green Formulas for Divergence-Measure Fields over General Open Sets

We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in $L^p$ over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth a…

The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces

Let $p:\R\to(1,\infty)$ be a globally log-Hölder continuous variable exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space $L^{p(\cdot)}(\R,w)=\{f:fw\in L^{p(\cdot)}(\R)\}$ if and only if the weight $w$ satisfies \[ \sup_{-\infty<a<b<\infty} \frac{1}{b-a}\|wχ_{(a,b)}\|_{p(\cdot)}\|w^{-1}χ_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1). \]

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let $C_Γ$ be the Cauchy integral operator on a Lipschitz curve $Γ$. In this article, the authors show that the commutator $[b,C_Γ]$ is bounded (resp., compact) on the Morrey space $L^{p,\,λ}(\mathbb R)$ for any (or some) $p\in(1, \infty)$ and $λ\in(0, 1)$ if and only if $b\in {\rm BMO}(\mathbb R)$ (resp., ${\rm CMO}(\mathbb R)$). As an application, a factorization of the classical Hardy space $H^1(\mathbb R)$ in terms of $C_Γ$ and its adjoint operator is obtained.

广义散度度量场、Gauss-Green公式与Cauchy通量

Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes

本文针对开集上的广义散度度量场建立了Gauss-Green公式，将法向迹表示为逼近集测度值法向迹的极限，并借此将经典连续介质物理的平衡律推广至由Radon测度度量产出和Cauchy通量的一般框架，最终建立了多维散度形式非线性偏微分方程熵解与物理平衡律数学表述的等价性。

关于一类柯西积分方程的解

On the solution of a class of Cauchy integral equations

本文详细讨论了在仅有有限且含噪声数据时，如何数值实现之前提出的求解一类柯西积分方程的新方法，并重点分析了数值稳定性问题。

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1<p finite. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L^2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

Commutators of Cauchy--Szegő type integrals for domains in $\mathbb C^n$ with minimal smoothness

In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ as in the recent result of Lanzani--Stein. We point out that in this setting the Cauchy type integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calderón--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calderón--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{blac…

A Gronwall inequality and the Cauchy-type problem by means of $ψ$-Hilfer operator

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $ψ$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.

A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula

Let $a=(a_1,a_2,...c,a_n)$ for $n\in\mathbb{N}$ be a given sequence of positive numbers. In the paper, the authors establish, by using Cauchy's integral formula in the theory of complex functions, an integral representation of the principal branch of the geometric mean {equation*} G_n(a+z)=\Biggl[\prod_{k=1}^n(a_k+z)\Biggr]^{1/n} {equation*} for $z\in\mathbb{C}\setminus(-\infty,-\min\{a_k,1\le k\le n\}]$, and then provide a new proof of the well known GA mean inequality.

Holomorphic extension on product Lipschitz surfaces in two complex variables

In this work we prove a new $L^p$ holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journé-Semmes and Christ's $Tb$ theorem to the Cauchy integral operator, we prove a biparameter $Tb$ theorem and apply it to prove $L^p$ space bounds for the biparameter Cauchy integral operator. We also prove some new biparameter Li…

Some remarks concerning integrals of curvature on curves and surfaces

In these notes we discuss some relations between complex analysis (derivatives of Cauchy integrals) and curvatures of curves and surfaces. In higher dimensions the Cauchy integrals are based on generalizations of complex analysis using quarternions or Clifford algebras. These topics came up in "Analysis of and on Uniformly Rectifiable Sets" (with Guy David, American Mathematical Society, 1993), and here we describe some of the basic points involved. A presentation based on this paper was made at the AMS Special Session "Surface Geometry and Shape Perception" (Hoboken, 2001).

Generalized substantial fractional operators and well-posedness of Cauchy problem

In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.

通过柯西积分公式计算转折点问题的渐近展开：贝塞尔函数

Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions

本文提出利用柯西积分公式高精度计算具有大实参数的二阶微分方程转折点问题的渐近展开，重点在于贝塞尔函数的Airy型展开。

超复分析中正则函数的边界问题

On Boundary Problems for Regular Functions in Hypercomplex Analysis

本文综述了 Clifford 分析中正则函数边值问题的研究，涵盖广义 Cauchy 定理等理论基础、积分表示，以及 Clifford 代数下的边值问题与奇异积分方程。

Cauchy Means of Dirichlet polynomials

We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q} \frac{\dd t}{π( t^2+1)}.$$ These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\ge 1$, $\s\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.

On the Green function and Poisson integrals of the Dunkl Laplacian

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $Δ_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $Δ_k$-subharmonic functions and Hardy-Stein identities for the Poisson integrals of $Δ_k$. We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in $\mathbb{R}^d$. These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.

A note on the discrete Cauchy-Kovalevskaya extension

In this paper we exploit the umbral calculus framework to reformulate the so-called discrete Cauchy-Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space-time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving th…

Kontsevich-Zagier Integrals for Automorphic Green's Functions. I

In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-$k$ automorphic Green's functions invariant under modular transformations in $\varGamma_0(N)$ ($N\in\mathbb Z_{\geq1} $), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight $k\geq4$. These Kontsevich-Zagier integral representations for automorphic Green's functions give explicit formulae for certain Eichler-Shimura maps connecting Eichler cohomology to Maaß cusp forms. We construct integral representations for weight-4 Gross-Zagier renormalized G…

关于 Cauchy $\ m{P.V.}$ 积分的逐点存在性

On the pointwise existence of Cauchy $\rm{P.V.}$ integrals

本文扩展了标量 Riemann-Hilbert 理论中的 Sokhotski-Plemelj 公式，在比 Dini 连续性更弱的逐点连续性与 $L^1$ 条件下证明了该公式，并给出了 Cauchy $\ m{P.V.}$ 积分存在的充要条件。