Integral representations in multi-dimensional complex analysis

The representation of a holomorphic function in terms of its boundary values by means of integral formulas is one of the most important tools in classical complex analysis (cf. also Boundary value problems of analytic function theory; Analytic continuation into a domain of a function given on part of the boundary). In the case of one complex variable, the familiar Cauchy integral formula plays a dominant and unique role in the theory of functions. In contrast, in higher dimensions there are numerous generalizations which have been discovered gradually over a period of many decades, each having its special properties and applications. Moreover, these integral formulas typically depend on the domain under consideration, and they intimately reflect complex-analytic/geometric properties of the boundaries of such domains.

The simplest and oldest such formula involves iteration of the one-variable formula on product domains. For example, on a poly-disc $P = \{ ( z _ { 1 } , \dots , z _ { n } ) : | z _ { j } - a _ { j } | < r _ { j } , j = 1 , \dots , n \}$ in $\mathbf{C} ^ { n }$ (the product of $n$ discs), one obtains

\begin{equation*} f ( z ) = \frac { 1 } { ( 2 \pi i ) ^ { n} } \int _ { b _ { 0 } P } \frac { f ( \zeta ) d \zeta _ { 1 } \ldots d \zeta _ { n } } { ( \zeta _ { 1 } - z _ { 1 } ) \ldots ( \zeta _ { n } - z _ { n } ) } , z \in P, \end{equation*}

for a continuous function $f : \overline{P} \rightarrow \mathbf{C}$ which is holomorphic in each variable separately (and thus, in particular, for $f$ holomorphic). Here, integration is over the distinguished boundary $b _ { 0 } P = \{ ( \zeta _ { 1 } , \dots , \zeta _ { n } ) : | \zeta _ { j } - a _ { j } | = r _ { j } , j = 1 , \dots , n \}$, the product of $n$ circles and thus a strictly smaller subset of the topological boundary $\partial P$ when $n > 1$. As in dimension one, this formula implies the standard local properties of holomorphic functions, for example, the local power series representation. Under suitable hypothesis, an analogous formula (the Bergman–Weil integral formula, cf. also Bergman–Weil representation) holds on analytic polyhedra i.e., regions $A$ described by $A = \{ | h _ { 1 } ( z ) | < 1 , \dots , | h _ { \text{l} } ( z ) | < 1 \}$ for some holomorphic functions $h _ { 1 } , \dots , h _ { \operatorname {l} }$ in a neighbourhood of $\bar{A}$. This formula explicitly involves the functions $h _ { 1 } , \dots , h _ { \operatorname {l} }$, and integration is over the corresponding distinguished boundary of $A$ as above. An important feature of the Bergman–Weil formula is the holomorphic dependence of the integrand on the free variable $z$, as is obvious in the poly-disc formula above.

In contrast, the Bochner–Martinelli integral formula (cf. also Bochner–Martinelli representation formula)

\begin{equation*} f ( z ) = \int _ { \partial D } f ( \zeta ) K _ { \text{BM} } ( \zeta , z ), \end{equation*}

\begin{equation*} K _ { \text{BM} } ( \zeta , z ) = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \frac { \omega _ { \zeta } ^ { \prime } ( \overline { \zeta } - \overline {z} ) \wedge \omega ( \zeta ) } { | \zeta - z | ^ { 2 n } } ,\; \omega _ { \zeta } ^ { \prime } ( \overline { \zeta } - \overline {z} )= \end{equation*}

\begin{equation*} = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } ( \overline { \zeta _ { j } } - \overline { z _ { j } } ) d \overline { \zeta _ { 1 } } \bigwedge \ldots \bigwedge [ d \overline { \zeta _ { j } } ] \bigwedge \ldots \bigwedge d \overline { \zeta _ { n } } , \omega ( \zeta ) = d \zeta _ { 1 } \bigwedge \cdots \bigwedge d \zeta _ { n }, \end{equation*}

which is valid for holomorphic functions $f$ on arbitrary regions, involves integration over the full topological boundary (assumed differentiable here), but the integrand is no longer holomorphic in $z$, except for $n = 1$; here, $[ d \overline { \zeta _ { j } } ]$ means that one has to "leave out dzj" , so that a bidegree-$( n , n - 1 )$-form is integrated. This formula is an easy consequence of the Green formulas in potential theory: the kernel $K_{\text{BM}} (\zeta , z )$ above is equal to $- 2 * \partial _ { \zeta } N ( \zeta , z )$, where $*$ is the Hodge operator (cf. Laplace operator) and $N$ is the Newton potential on $\mathbf{R} ^ { 2 n }$ ($= {\bf C}^ { n }$). As in dimension one, there is a more general representation formula valid for $C ^ { 1 }$-functions:

\begin{equation*} f ( z ) = \int _ { \partial D } f ( \zeta ) K _ { \text{BM} } ( \zeta , z ) - \int _ { D } \overline { \partial } f ( \zeta ) \bigwedge K _ { \text{BM} } ( \zeta , z ), \end{equation*}

\begin{equation*} z \in D. \end{equation*}

In applications involving the construction of global holomorphic functions satisfying special properties, and in order to solve explicitly the inhomogeneous Cauchy–Riemann equation $\overline { \partial } u = f$ (cf. also Cauchy-Riemann equations) for a given $\overline { \partial }$-closed $( 0,1 )$-form $f = \sum _ { j = 1 } ^ { n } f _ { j } d \overline { z _ { j } }$ (i.e., for solving the system $\partial u / \partial \overline { z _ j } = f_j $, $j = 1 , \ldots , n$), it is important to replace the Bochner–Martinelli kernel $K _ { \operatorname{BM} } $ by kernels which are holomorphic in $z$. The existence of such kernels can be proved abstractly by functional-analytic methods (for example, the Szegö kernel, or the kernel of A.M. Gleason [a8]), but in applications one needs much more explicit information. A concrete general method to construct a class of integral representation formulas for holomorphic functions, the so-called Cauchy–Fantappiè integral formulas, was introduced in 1956 by J. Leray [a4] (cf. also Leray formula). (See [a6] for the origins of the terminology.) Together with its generalizations to differential forms (see below), this method has had numerous important applications. The ingredient for this construction is a so-called Leray mapping (or Leray section) for $D$, that is, a (differentiable) mapping $s = ( s _ { 1 } , \dots , s _ { n } ) : \partial D \times D \rightarrow \mathbf{C} ^ { n }$ with the property that $\langle s ( \zeta , z ) , \zeta - z \rangle = \sum _ { j = 1 } ^ { n } s _ { j } ( \zeta , z ) ( \zeta _ { j } - z _ { j } ) \neq 0$ on $\partial D \times D$. To any such $s$, Leray associates the following explicit $( n , n - 1 )$-form in $\zeta$:

\begin{equation*} K ( s ) = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \frac { 1 } { \langle s , \zeta - z \rangle ^ { n } } \times \end{equation*}

\begin{equation*} \times \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } s _ { j } d s _ { 1 } \bigwedge \ldots \bigwedge [ d s _ { j } ] \bigwedge \ldots \bigwedge d s _ { n } \bigwedge \omega ( \zeta ), \end{equation*}

and he obtains the representation $f ( z ) = \int \partial_{Df} ( \zeta ) K ( s )$ for holomorphic $f$. Notice that for $n = 1$, $K ( s )$ is independent of $s$ and equals the Cauchy kernel, while for $n > 1$ one has many different possibilities. For example, $s = ( \overline { \zeta } - \overline{z} )$ gives the Bochner–Martinelli kernel. Another important case arises for Euclidean-convex domains $D$ with $C ^ { 2 }$ boundary. If $D$ is described as $\{ z : r ( z ) < 0 \}$, where $r \in C ^ { 2 }$ and $d r \neq 0$ on $\partial D$, convexity implies that

\begin{equation*} \sum _ { j = 1 } ^ { n } \frac { \partial r } { \partial \zeta _ { j } } ( \zeta _ { j } ) ( \zeta _ { j } - z _ { j } ) \neq 0 \end{equation*}

for $\zeta \in \partial D$, $z \in D$, so $s _ { r } ( \zeta , z ) = ( \partial r / \partial \zeta _ { 1 } ( \zeta ) , \ldots , \partial r / \partial \zeta _ { n } ( \zeta ) )$ defines a Leray mapping that is holomorphic in $z$. The associated kernel $K ( s _ { r } )$ is then also holomorphic in $z$. Its pullback $C _ { D }$ to the boundary $\partial D$, which only depends on the geometry of $\partial D$ and not on the particular function $r$ chosen to describe $D$, is known as the Cauchy–Leray kernel for the (convex) domain $D$.

To construct Leray mappings $s$ which are holomorphic in $z$ for more general domains $D$, is much more complicated. Such domains must necessarily be domains of holomorphy, and hence pseudo-convex, since $h _ { \zeta } ( z ) = \langle s , \zeta - z \rangle ^ { - 1 }$ is a holomorphic function on $D$ which is singular at $\zeta \in \partial D$ (cf. also Domain of holomorphy; Pseudo-convex and pseudo-concave). The most complete results are known for strictly pseudo-convex domains. Locally near each boundary point, such a domain is biholomorphically equivalent to a (strictly) convex domain. Thus, a local holomorphic Leray mapping is easily obtained from the convex case by applying an appropriate change of coordinates. The major obstacle then involves passing from local to global. This was achieved in 1968–1969 by G.M. Khenkin (also spelled G.M. Henkin) [a1] and, independently, by E. Ramirez [a5], by using deep global results in multi-dimensional complex analysis. The corresponding kernel is known as the Khenkin–Ramirez kernel (also written as Henkin–Ramirez kernel). Shortly thereafter, Khenkin and, independently, H. Grauert and I. Lieb used the new kernels to construct quite explicit integral operators to solve the $\overline { \partial }$-equation with supremum-norm estimates on strictly pseudo-convex domains (cf. also Neumann $\overline { \partial }$-problem).

These methods generalize to yield integral representation formulas for differential forms. In 1967, W. Koppelman [a3] introduced (double) differential forms $K _ { q }$, $0 \leq q \leq n$, of type $( n , n - q - 1 )$ in $\zeta$ and type $( 0 , q )$ in $z$, with $K _ { 0 } = K _ { \text{BM} }$, and proved a representation for $( 0 , q )$-forms on a domain $D$ with piecewise-differentiable boundary, as follows:

\begin{equation*} f = \int _ { \partial D } f \bigwedge K _ { q } - \overline { \partial _ { z } } \int f \bigwedge K _ { q- 1 } + \int _ { D } \overline { \partial } f \bigwedge K _ { q }. \end{equation*}

Koppelman also introduced the analogous forms $K _ { q } ( s )$ of Cauchy–Fantappiè type in dependence of a given Leray mapping $s$, and proved corresponding integral representation formulas. By applying the Leray mapping of Khenkin and Ramirez, these methods lead to integral solution operators for $\overline { \partial }$ on forms of arbitrary degree.

Standard reference texts for these topics are [a2] and [a7].

The rather explicit form of these integral operators on strictly pseudo-convex domains makes it possible to study refined regularity properties and estimates for solutions of the $\overline { \partial }$-problem in many classical and newer function spaces. In particular, the non-isotropic nature of the singularities of the kernels has led to new classes of singular integral operators which have been thoroughly investigated by E.M. Stein and his collaborators (see, for example, [a9] and Singular integral).

Another fundamental integral representation formula involves the non-explicit Bergman kernel function, which is defined abstractly in the context of Hilbert spaces of square-integrable holomorphic functions on a region $D$. In particular, the Bergman kernel is used to define the Bergman metric (i.e., the Poincaré metric on the unit disc, cf. also Poincaré model). Biholomorphic mappings between two domains are isometries in the respective Bergman metrics. This leads to important applications of the Bergman kernel to the study of such mappings. On strictly pseudo-convex domains, the principal part of the Bergman kernel can be expressed explicitly by kernels closely related to the Khenkin–Ramirez kernel (see [a7]).

On more general weakly pseudo-convex domains no comparable precise results are known, except under additional quite restrictive assumptions. One major difficulty is the fact that such domains in general are not locally biholomorphic to a convex domain. Furthermore, the local complex-analytic geometry is considerably more complicated, and not yet fully understood. Much research work is continuing in this area.

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