Positive Definite Kernels in Clifford Analysis

Introduction

A reproducing kernel Hilbert space is a Hilbert space of functions on which the point evaluations are continuous; equivalently, it is the space $H_K$ attached by the Aronszajn construction to a positive definite kernel $K$ on the product $X\times X$. The general theory — the positivity condition, the Aronszajn theorem, the Hermitian kernels and their integral operators — belongs to Analysis on Linear Spaces, in Reproducing Kernel Hilbert Spaces, Positive Definite Functions, Hermitian Kernels and the Integral Operator and Hermitian Integral Kernels; this article develops the instances that occur in Clifford analysis, in which the kernel takes values in the Clifford algebra or in its modules and the functions take values in a Clifford module, and in which the two classical instances are the Bergman kernel of the monogenic $L^2$ space and the Szegő kernel of the monogenic Hardy space.

The setting is that of Hermitian Hilbert Modules over a Clifford Algebra: the Clifford algebra $A=\mathrm{Cl}_{0,m}$ with conjugation $*$, a definite Clifford module $\mathcal{S}$ of values with a compatible Hermitian form, and the Hilbert modules of sections $L^2(\Omega;\mathcal{S})$ and their closed monogenic subspaces $\mathcal{A}^2=\ker D\cap L^2$ and $H^2$. The positivity is the positivity of the module form, the Hermitian symmetry is the symmetry of that form combined with the involution $*$, and the reproducing property is the continuity of the point evaluation, which the elliptic theory supplies. The operator-theoretic background is Hermitian Hilbert Modules over a Clifford Algebra; the Cauchy kernel and the boundary projections are The Cauchy Integral Operator; the Hermitian refinement of the kernels is The Hermitian Cauchy Integral and the Boundary Values and The Hermitian Cauchy Kernel as an Adjoint. Nothing of the general theory is repeated: what is specific here is the Clifford-valued kernel, its Hermitian symmetry under $*$, and the two kernels of the monogenic spaces.

Positive Definite Kernels with Clifford Values

Definition and Positivity

Definition. Let $X$ be a set and let $\mathcal{S}$ be a definite Clifford module with a compatible Hermitian form $(\cdot,\cdot)$. A kernel is a map $K:X\times X\to\mathrm{End}_\mathbb{R}(\mathcal{S})$ (or, in the scalar case, $K:X\times X\to A$); it is positive definite when

$$ \sum_{i,j=1}^{n}\bigl(K(x_i,x_j)\,s_j,\,s_i\bigr)\ \ge\ 0 $$

for every finite family $x_1,\dots,x_n\in X$ and every $s_1,\dots,s_n\in\mathcal{S}$.

Definition. The kernel is Hermitian when

$$ K(x,y) = K(y,x)^{*} , $$

the adjoint being taken with respect to the module form; for a scalar kernel with values in $A$ this reads $K(x,y)=K(y,x)^{*}$, the Clifford conjugation of the value.

Theorem (Aronszajn; quoted from the general theory). A Hermitian positive definite kernel $K$ on $X\times X$ determines a Hilbert space $H_K$ of $\mathcal{S}$-valued functions on $X$ on which the point evaluations are continuous, with the reproducing property

$$ (K(x,\cdot)s,\,f)_{H_K} = (s,\,f(x)) \qquad x\in X,\ s\in\mathcal{S},\ f\in H_K , $$

and every such space arises from exactly one kernel. The theorem is the general Aronszajn theorem of Reproducing Kernel Hilbert Spaces; the Clifford-valued case is its instance over the module $\mathcal{S}$.

Proof. Quoted. The proof is the general one: the span of the functions $K(x,\cdot)s$ with the inner product induced by positivity is a pre-Hilbert space, its completion is $H_K$, and the reproducing property makes the evaluation continuous and recovers the kernel from the space. $\square$

Remark (what the Clifford structure adds). Three features distinguish the case at hand from the scalar theory. The kernel takes values in a non-commutative algebra, so the order of the factors in the reproducing property matters and the Hermitian symmetry carries the conjugation $*$; the module form is the one of Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint, so the positivity of the kernel is the positivity of a family of module forms; and the functions of the classical spaces are monogenic, so the kernel inherits monogenicity in each variable — it is monogenic in the first and anti-monogenic in the second — and the kernel is determined by its diagonal. These are the properties the next two sections use.

The Bergman Kernel of the Monogenic Space

Definition. For a bounded domain $\Omega$ the monogenic Bergman space is $\mathcal{A}^2(\Omega;\mathcal{S})=\ker D\cap L^2(\Omega;\mathcal{S})$ and the Bergman kernel is the reproducing kernel $B:\Omega\times\Omega\to\mathrm{End}(\mathcal{S})$ of the space, so that

$$ f(w) = \int_\Omega B(w,z)\,f(z)\,dz , \qquad f\in\mathcal{A}^2(\Omega;\mathcal{S}) . $$

Theorem (properties of the Bergman kernel). The Bergman kernel of the monogenic space exists and is unique; it is Hermitian, $B(w,z)=B(z,w)^{*}$; it is monogenic in the variable $w$ for the operator $D$ and anti-monogenic in $z$; it is determined by the diagonal through the identity $\sum_i B(w,z_i)B(z_i,z)\to B(w,z)$; and it is the kernel of the orthogonal projection of $L^2(\Omega;\mathcal{S})$ onto $\mathcal{A}^2(\Omega;\mathcal{S})$.

Proof. The point evaluation is continuous on $\mathcal{A}^2$ because $D$ is elliptic and the domain is bounded, so the Aronszajn construction applies; the monogenicity in each variable follows by applying $D$ in $w$ and $\bar D$ in $z$ to the reproducing identity, and the projection statement is the reproducing property read as an integral operator. $\square$

Remark (the relation to the Cauchy kernel). The Bergman kernel of the monogenic space is not the Cauchy kernel: the Cauchy kernel is the reproducing kernel of the space of boundary values, i.e. the Szegő kernel of the following section, while the Bergman kernel reproduces the space of $L^2$ monogenic functions on the whole domain. The two agree only on the boundary in the limiting sense. For the half-space and the ball the Bergman kernel of the monogenic space has the same shape as the Bergman kernel of the holomorphic space, with the Clifford kernel in place of the complex one; the explicit formula is the Clifford refinement of the classical Bergman kernel, cited to the literature. The general Bergman operator of a domain is The Bergman Operator in a later category, and the metric it defines is Hermitian Symmetric Spaces and the Bergman Metric.

The Szegő Kernel and the Boundary

Definition. For a domain $\Omega$ with smooth boundary the monogenic Szegő kernel is the reproducing kernel $S$ of the monogenic Hardy space $H^2(\Omega;\mathcal{S})$ of The Cauchy Integral Operator, so that

$$ f(w) = \int_{\partial\Omega} S(w,z)\,f(z)\,dS(z) , \qquad f\in H^2(\Omega;\mathcal{S}) . $$

Theorem (the Szegő kernel is the Cauchy kernel read on the boundary). The Szegő kernel is determined by the Cauchy kernel and the conormal element:

$$ S(w,z) = E(w-z)\,\nu_B(z) $$

up to the normalisation of the Cauchy–Pompeiu formula of Clifford Analysis, and it is Hermitian in the sense of the module form, monogenic in $w$ and anti-monogenic in $z$; the projection it defines is the Szegő projection $P^+=\tfrac12(I+\mathcal{S})$ of The Cauchy Integral Operator.

Proof. The boundary values of a monogenic function are reproduced by the Cauchy integral formula against the Cauchy kernel and the conormal element, which is exactly the reproducing property for the kernel $E(w-z)\nu_B(z)$; the Hermitian symmetry follows from the Hermitian symmetry of the kernel $E$ and the reality of the conormal element, and the projection statement is the Cauchy integral formula as the orthogonal projection onto the space of boundary values. $\square$

Remark (why the boundary kernel is the Cauchy one). In the monogenic theory the Hardy space is the range of the boundary Cauchy transform, so its reproducing kernel is the Cauchy kernel itself; this is the sense in which the Cauchy integral formula is the reproducing property of the monogenic Hardy space, a statement that the general theory of The Cauchy Integral Operator records as the identification of the Szegő projection with the Cauchy transform. The Bergman and the Szegő kernels are therefore the two ends of the same structure: the second is the Cauchy kernel, and the first is its interior analogue.

The Kernel as an Operator

Theorem (the integral operator of a Hermitian kernel). To a Hermitian kernel $K$ on $\Omega\times\Omega$ associate the integral operator

$$ (T_Kf)(x) = \int_\Omega K(x,y)\,f(y)\,dy . $$

The kernel is Hermitian exactly when the operator is self-adjoint on $L^2(\Omega;\mathcal{S})$, and positive definite exactly when the operator is positive, $\langle T_Kf,f\rangle\ge0$; the two properties are the operator form of the two conditions defining the reproducing kernel.

Proof. The kernel of the adjoint of $T_K$ is the conjugate-reverse $K(y,x)^{*}$ by the general theorem of Hermitian Kernels and the Integral Operator, so $T_K$ is self-adjoint exactly when $K(x,y)=K(y,x)^{*}$; the positivity of the quadratic form is the definition of positive definiteness read as the integral $\int\!\!\int(K(x,y)f(y),f(x))\,dy\,dx\ge0$. The statement is the Clifford-module instance of the general theory of Banach and Hilbert Spaces. $\square$

Remark (the spectral consequences). A compact self-adjoint integral operator with a positive definite kernel therefore has a nonnegative spectrum, an orthonormal basis of eigenspinors and a Mercer-type expansion of the kernel in that basis; the reproducing kernel Hilbert space of the kernel is the range of the square root of the operator, and the Cauchy and Bergman kernels of the preceding sections are the entries of the projection operators onto the corresponding monogenic spaces. The spectral theory of the integral operator, and the compactness on which the expansion rests, are the subjects of Hermitian Integral Kernels, Compact Operators and Self-Adjoint Operators and the Spectral Theorem; the Clifford case differs only in that the kernel and the eigenspinors take values in the algebra and the module.

Summary

A Clifford-valued kernel $K:X\times X\to\mathrm{End}(\mathcal{S})$ is positive definite when $\sum_{i,j}(K(x_i,x_j)s_j,s_i)\ge0$ and Hermitian when $K(x,y)=K(y,x)^{*}$, the adjoint being taken in the compatible module form; by the Aronszajn theorem of Reproducing Kernel Hilbert Spaces it defines a Hilbert space $H_K$ with the reproducing property $(K(x,\cdot)s,f)_{H_K}=(s,f(x))$. The Clifford case adds the non-commutativity of the algebra, the module form of Part II and the monogenicity of the functions, so that the kernel is monogenic in the first and anti-monogenic in the second variable. The Bergman kernel $B$ of the monogenic $L^2$ space $\mathcal{A}^2=\ker D\cap L^2$ is Hermitian, monogenic in each variable and the kernel of the orthogonal projection onto $\mathcal{A}^2$; the Szegő kernel $S$ of the monogenic Hardy space is the Cauchy kernel read on the boundary, $S(w,z)=E(w-z)\nu_B(z)$, and its projection is the boundary Cauchy transform $P^+=\tfrac12(I+\mathcal{S})$ of The Cauchy Integral Operator. The two kernels are the interior and the boundary instances of the same structure, whose general theory is Analysis on Linear Spaces' and whose module background is Hermitian Hilbert Modules over a Clifford Algebra.

Summary of Notation

Symbol	Meaning
$K(x,y)$, $K(x,y)=K(y,x)^{*}$	Clifford-valued kernel and its Hermitian symmetry
$\sum_{i,j}(K(x_i,x_j)s_j,s_i)\ge0$	Positive definiteness
$H_K$, $(K(x,\cdot)s,f)=(s,f(x))$	The kernel's Hilbert space and the reproducing property
$B(w,z)$	Bergman kernel of $\mathcal{A}^2(\Omega;\mathcal{S})$
$S(w,z)=E(w-z)\nu_B(z)$	Szegő kernel of $H^2(\Omega;\mathcal{S})$; the Cauchy kernel on the boundary
$\mathcal{A}^2=\ker D\cap L^2(\Omega;\mathcal{S})$	Monogenic Bergman space
$H^2(\Omega;\mathcal{S})$, $P^+$	Monogenic Hardy space and Szegő projection