The Cauchy Integral Operator

Introduction

The Cauchy kernel $E(x-y)=\omega_m^{-1}(x-y)^{\natural}|x-y|^{-m-1}$ of Clifford Analysis defines an operator, and this article treats that operator at its own layer. The setting is the one of The Dirac Operator: a Clifford algebra $A=\mathrm{Cl}_{0,m}$ with frame $1,e_1,\dots,e_m$, a left Clifford module $\mathcal{S}$ of values, the vector variable $x\in\mathbb{R}^{m+1}$, and the Cauchy–Riemann operator $D=\sum_{\mu=0}^{m}e_\mu\partial_\mu$ with conjugate $\bar D$ and fundamental solution $E$.

Two operators are built from the kernel. The first is the boundary transform, the integral of a density on a surface against the kernel and the conormal element; it maps a datum on the boundary to a function away from it, it has one-sided traces, and its traces differ by the datum itself — the Plemelj–Sokhotski formulae — so that the odd part of the transform is an involution of the boundary density space, the singular Cauchy operator. The second is the volume transform, the integral of a function on a domain against the kernel, which inverts the operator $D$ and turns the Cauchy–Pompeiu formula into the statement that the boundary transform and the volume transform together reproduce every $C^1$ function. The article establishes the algebraic identities of these operators, their boundedness on the classical spaces, and the description of the monogenic Hardy space as the range of a projection.

The function theory is not repeated: the Cauchy–Pompeiu and Cauchy integral formulae and the kernel computation are Clifford Analysis's, the general integration theory of a hypercomplex system is Hypercomplex Integration's, and the boundary-value problems to which the operator is applied are Riemann Boundary Value Problems and Singular Integral Equations'. What belongs here is the operator: its definition, its traces, the involution property of the singular part, its mapping properties, and its relation to the projection onto boundary values. The Hermitian refinement of the operator is The Hermitian Cauchy Integral and the Boundary Values and The Hermitian Cauchy Kernel as an Adjoint; the measure-theoretic and fractal boundary theory is a section of Clifford Analysis and is cited, not developed; the Fourier and Calderón–Zygmund reading of the singular integral is Harmonic Analysis over Hypercomplex Systems.

The Boundary Transform

The Cauchy Transform of a Density

Definition. Let $\Omega\subseteq\mathbb{R}^{m+1}$ be a bounded domain with smooth boundary $\Gamma$, oriented as the boundary of $\Omega$, and let $h:\Gamma\to\mathcal{S}$ be a continuous density. The Cauchy transform of $h$ is

$$ (\mathcal{C}_\Gamma h)(x) = \int_\Gamma E(x-y)\,\nu_B(y)\,h(y)\,dS(y) , \qquad x\notin\Gamma , $$

where $\nu_B(y)=\sum_\mu\nu_\mu(y)e_\mu$ is the conormal element of The Dirac Operator. The transform is defined off the surface and is monogenic on each side: $D(\mathcal{C}_\Gamma h)=0$ on $\mathbb{R}^{m+1}\setminus\Gamma$, because $E(x-y)$ is monogenic in $x$ away from $y$, and the differentiation passes under the integral sign.

Definition. The one-sided extensions are $\mathcal{C}^+h(x)=\lim_{x'\to x,\,x'\in\Omega} \mathcal{C}_\Gamma h(x')$ and $\mathcal{C}^-h(x)=\lim_{x'\to x,\,x'\notin\overline\Omega} \mathcal{C}_\Gamma h(x')$, taken at points of $\Gamma$ at which the surface is smooth and the limits exist.

The Plemelj–Sokhotski Formulae

Theorem (Plemelj–Sokhotski; standard). For a $C^{0,\alpha}$ density $h$ on a smooth surface $\Gamma$ the one-sided extensions exist at every point of $\Gamma$, and

$$ \mathcal{C}^+h = \tfrac12 h + \mathcal{S}h , \qquad \mathcal{C}^-h = -\tfrac12 h + \mathcal{S}h , $$

where

$$ (\mathcal{S}h)(x) = 2\,\mathrm{p.v.}\!\int_\Gamma E(x-y)\,\nu_B(y)\,h(y)\,dS(y) $$

is the singular Cauchy operator, the principal value being taken over the intersection of $\Gamma$ with the complement of a small ball about $x$.

Proof. The kernel is written as the sum of its value at $y=x$ plus the difference; the difference is integrable by the homogeneity of $E$, and the excision of a small ball about $x$ isolates the principal value. Inside the ball the kernel is that of the Laplacian's fundamental solution, and its flux over the half-sphere gives the terms $\pm\tfrac12 h$; the outside contribution is the principal value. The computation is the classical one, carried out in the Clifford setting in Clifford Analysis and Hypercomplex Integration; the two displays are quoted there and used here. $\square$

Corollary (the jump and the sum). The two formulae give

$$ \mathcal{C}^+h-\mathcal{C}^-h = h , \qquad \mathcal{C}^+h+\mathcal{C}^-h = 2\,\mathcal{S}h , $$

so the jump of the transform across the surface is the density, and the singular operator is the mean of the two traces.

Corollary (reality of the surface for a monogenic function). If $F$ is monogenic on $\Omega$ with continuous extension to $\bar\Omega$, its Cauchy integral is $F$ itself, so the first formula reads $F^+=\tfrac12 F|_\Gamma+\mathcal{S}(F|_\Gamma)$, the boundary version of the Cauchy integral formula.

The Involution

Theorem (the singular operator is an involution). On a smooth surface $\Gamma$ the singular Cauchy operator satisfies

$$ \mathcal{S}^2 = I $$

on the space of densities for which it is defined as a bounded operator.

Proof. Apply the Plemelj formulae to the two functions $\mathcal{C}^+h$ and $\mathcal{C}^-h$; both are monogenic off $\Gamma$ and are themselves represented by the Cauchy transform of their boundary values. The double-layer potentials of a monogenic function reproduce it, which expresses $\mathcal{C}^\pm$ as the projections of the trace space onto the two complementary subspaces; since $\mathcal{C}^\pm=\tfrac12(\mathcal{S}\pm I)$, idempotency of either projection is equivalent to $\mathcal{S}^2=I$. The Clifford proof is the one of the classical complex case, with the kernel and the conormal element in place of $dz/(z-\zeta)$. $\square$

Corollary (the projections). The operators

$$ P^+ = \tfrac12(I+\mathcal{S}) = \mathcal{C}^+ , \qquad P^- = \tfrac12(I-\mathcal{S}) = -\mathcal{C}^- , \qquad P^+ + P^- = I , \qquad P^+P^- = P^-P^+ = 0 $$

are the complementary projections of the trace space, and they are orthogonal when the surface is smooth and the space is $L^2(\Gamma)$. The image of $P^+$ is the space of boundary values of monogenic functions on $\Omega$, and the image of $P^-$ is the space of boundary values of monogenic functions vanishing at infinity on the exterior.

Remark (the complex case). For $m=1$ and $A\cong\mathbb{C}$ the surface is a contour, the kernel is $1/(2\pi i(\zeta-z))$, the singular operator $\mathcal{S}$ is the classical Hilbert transform on the contour, and the projections $P^\pm$ are the Szegő projections onto the boundary values of holomorphic functions inside and outside. The whole of the classical boundary theory of holomorphic functions is the case $m=1$ of the statements above.

The Volume Transform

The Teodorescu Transform

Definition. On a bounded domain $\Omega$ the Teodorescu transform of a continuous function $f:\Omega\to\mathcal{S}$ is

$$ (\mathcal{T}_\Omega f)(x) = \int_\Omega E(x-y)\,f(y)\,dy , \qquad x\in\Omega . $$

Theorem (the volume transform inverts the operator). On a domain $\Omega$ with sufficiently regular boundary,

$$ D\,\mathcal{T}_\Omega = I , \qquad \mathcal{T}_\Omega\,D = I - \mathcal{C}_{\partial\Omega} $$

on the classes for which the operators are defined; the second identity is the operator form of the Cauchy–Pompeiu formula.

Proof. The kernel satisfies $D_xE(x-y)=\delta_0(x-y)$ by the theorem on the fundamental solution of The Dirac Operator; differentiating under the integral sign gives the first identity. The second is the Cauchy–Pompeiu formula of Clifford Analysis, $f=\mathcal{C}_{\partial\Omega}f-\mathcal{T}_\Omega(Df)$, read as an operator identity on $f$. $\square$

Corollary (a right inverse and the monogenic class). The Teodorescu transform is a right inverse of $D$ on $\Omega$, and its kernel is exactly the boundary transform's: a function $f$ on $\Omega$ is monogenic precisely when $f=\mathcal{C}_{\partial\Omega}f$, that is, when $\mathcal{T}_\Omega(Df)=0$.

Remark (the volume transform on all of $\mathbb{R}^{m+1}$). On the whole space the Teodorescu transform is the convolution with $E$, the distributional inverse of $D$; it is the operator whose symbol is $\sigma(\xi)^{-1}$ in the sense of The Dirac Operator, and its study on $\mathbb{R}^{m+1}$ is the study of the elliptic operator $D$ by its parametrix, as in Harmonic Analysis over Hypercomplex Systems.

Mapping Properties

Boundedness of the Singular Operator

Theorem (Coifman–McIntosh–Meyer, quoted). Let $\Gamma$ be a Lipschitz surface with a small Lipschitz constant. Then the singular Cauchy operator $\mathcal{S}$ extends to a bounded operator on $L^2(\Gamma;\mathcal{S})$ and, with the kernel $E(x-y)\nu_B(y)$ homogeneous of degree $-(m-1)$ on $\Gamma$ and odd, to a bounded operator on $L^p(\Gamma;\mathcal{S})$ for a range of $p$ around $2$.

Proof. Quoted from the Clifford form of the Calderón–Zygmund theory; the kernel of $\mathcal{S}$ is a singular integral kernel of the standard type, its restriction to the surface is homogeneous of degree $-(m-1)$ and odd, and the theorem of Coifman–McIntosh–Meyer supplies the $L^2$ boundedness for Lipschitz surfaces with small constant; the $L^p$ statement follows from the $L^2$ result by interpolation and duality in the range in which the operator is of weak type. The Clifford case with operator-valued kernels is the one developed in Harmonic Analysis over Hypercomplex Systems and Clifford Analysis. $\square$

Remark (the measure-theoretic refinements). When the boundary carries only a Federer normal, or is a fractal of Hausdorff dimension $d$ strictly between $m$ and $m+1$, there is no boundary measure to integrate against, and the boundary transform is replaced by the Teodorescu transform together with a Whitney extension of the datum; the trace of the new transform is the fractal Hilbert transform, and the Plemelj calculus holds with an approximate dimension in place of the metric one. This is the measure-theoretic boundary theory of Clifford Analysis, and it is not repeated here.

The Range and the Hardy Space

Definition. For a domain $\Omega$ with smooth boundary $\Gamma$, the monogenic Hardy space is

$$ H^2(\Omega) = \{F:\Omega\to\mathcal{S} \text{ monogenic}\ :\ \sup_{t>0}\textstyle\int_{\Gamma_t}|F|^2\,dS<\infty\} , $$

where $\Gamma_t$ are the level surfaces receding from $\Gamma$ inside $\Omega$.

Theorem (the boundary values and the projection). The space $H^2(\Omega)$ is a Hilbert space, the boundary trace $F\mapsto F|_\Gamma$ is an isometry onto a closed subspace of $L^2(\Gamma)$, and that subspace is the range of $P^+$; the orthogonal projection of $L^2(\Gamma)$ onto it is $P^+=\tfrac12(I+\mathcal{S})$.

Proof. The hard part is that $P^+$ is a projection, which is the involution theorem above; given it, the reproducing kernel of the range is the Cauchy kernel, and the Cauchy integral formula identifies the monogenic extensions with the elements of the range. The statement is the Clifford form of the classical Hardy-space theorem; its development is Clifford Analysis's and the general theory of the boundary value map is Hypercomplex Integration's. $\square$

Remark (the two realisations of one operator). The Cauchy integral operator has two faces. As a boundary operator, $\mathcal{S}=2\,\mathcal{C}^+-I$ is the odd part of the traces, and it is the Hilbert transform of the surface; as a volume operator, $\mathcal{T}_\Omega$ inverts $D$ and reconstructs the function from its data. The link between them is the Cauchy–Pompeiu formula, which reads the failure of $\mathcal{T}_\Omega D$ to be the identity as the boundary projection. Without the surface — on all of $\mathbb{R}^{m+1}$ — only the second face survives, and the operator is the parametrix of $D$.

Summary

The Cauchy kernel $E$ of Clifford Analysis defines two operators. The boundary Cauchy transform $\mathcal{C}_\Gamma h(x)=\int_\Gamma E(x-y)\nu_B(y)h(y)dS(y)$ maps a density on a surface $\Gamma$ to a monogenic function off it, with one-sided traces $\mathcal{C}^\pm h=\pm\tfrac12h+\mathcal{S}h$, whose difference is the density and whose mean is the singular Cauchy operator $\mathcal{S}h=2\,\mathrm{p.v.}\!\int_\Gamma E\nu_Bh\,dS$. On a smooth surface $\mathcal{S}^2=I$, so $P^\pm=\tfrac12(I\pm\mathcal{S})=\pm\mathcal{C}^\pm$ are complementary projections of the trace space, the joint statement of the Plemelj–Sokhotski formulae and the classical continuation principle; the image of $P^+$ is the monogenic Hardy space $H^2(\Omega)$, and the orthogonal projection onto it is the Cauchy transform. On a Lipschitz surface with small constant, $\mathcal{S}$ is bounded on $L^2$ and on an interval of $L^p$ spaces by the Clifford form of the Coifman–McIntosh–Meyer theorem, the kernel being homogeneous of degree $-(m-1)$ and odd. The volume Teodorescu transform $\mathcal{T}_\Omega f(x)=\int_\Omega E(x-y)f(y)dy$ is a right inverse of $D$, and $\mathcal{T}_\Omega D=I-\mathcal{C}_{\partial\Omega}$ is the Cauchy–Pompeiu formula read as an operator identity. For $m=1$ the article is the classical theory of the Hilbert transform and the Szegő projection. The function theory is Clifford Analysis's, the general integration theory is Hypercomplex Integration's, the boundary-value problems are Riemann Boundary Value Problems and Singular Integral Equations', the Fourier and Calderón–Zygmund reading is Harmonic Analysis over Hypercomplex Systems', and the Hermitian refinement is The Hermitian Cauchy Integral and the Boundary Values.

Summary of Notation

Symbol	Meaning
$E(x-y)=\omega_m^{-1}(x-y)^{\natural}\|x-y\|^{-m-1}$	Cauchy kernel
$\nu_B=\sum_\mu\nu_\mu e_\mu$	Conormal element
$\mathcal{C}_\Gamma h$	Boundary Cauchy transform of a density
$\mathcal{C}^\pm h=\pm\tfrac12 h+\mathcal{S}h$	One-sided traces; Plemelj–Sokhotski
$\mathcal{S}=2\,\mathrm{p.v.}\!\int_\Gamma E\nu_B(\cdot)dS$	Singular Cauchy operator, the Hilbert transform of $\Gamma$
$P^\pm=\tfrac12(I\pm\mathcal{S})$	Complementary projections; $\mathcal{S}^2=I$
$\mathcal{T}_\Omega f=\int_\Omega E(x-\cdot)f$	Teodorescu transform; $D\mathcal{T}_\Omega=I$
$H^2(\Omega)$	Monogenic Hardy space; range of $P^+$
$\Gamma,\ dS$	Boundary surface and surface measure