View source for Integral representations in multi-dimensional complex analysis

The representation of a [[Holomorphic function|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|Boundary value problems of analytic function theory]]; [[Analytic continuation into a domain of a function given on part of the boundary|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200401.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200402.png" /> (the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200403.png" /> discs), one obtains

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200404.png" /></td> </tr></table>

for a [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200405.png" /> which is holomorphic in each variable separately (and thus, in particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200406.png" /> holomorphic). Here, integration is over the distinguished boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200407.png" />, the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200408.png" /> circles and thus a strictly smaller subset of the topological boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i1200409.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004010.png" />. 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|Bergman–Weil representation]]) holds on analytic polyhedra i.e., regions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004011.png" /> described by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004012.png" /> for some holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004013.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004014.png" />. This formula explicitly involves the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004015.png" />, and integration is over the corresponding distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004016.png" /> as above. An important feature of the Bergman–Weil formula is the holomorphic dependence of the integrand on the free variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004017.png" />, as is obvious in the poly-disc formula above.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004018.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004019.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004020.png" /></td> </tr></table>

which is valid for holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004021.png" /> on arbitrary regions, involves integration over the full topological boundary (assumed differentiable here), but the integrand is no longer holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004022.png" />, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004023.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004024.png" /> means that one has to  "leave out dzj" , so that a bidegree-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004025.png" />-form is integrated. This formula is an easy consequence of the [[Green formulas|Green formulas]] in [[Potential theory|potential theory]]: the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004026.png" /> above is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004028.png" /> is the Hodge operator (cf. [[Laplace operator|Laplace operator]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004029.png" /> is the [[Newton potential|Newton potential]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004031.png" />). As in dimension one, there is a more general representation formula valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004032.png" />-functions:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004034.png" /></td> </tr></table>

In applications involving the construction of global holomorphic functions satisfying special properties, and in order to solve explicitly the inhomogeneous Cauchy–Riemann equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004035.png" /> (cf. also [[Cauchy–Riemann equations|Cauchy–Riemann equations]]) for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004036.png" />-closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004037.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004038.png" /> (i.e., for solving the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004040.png" />), it is important to replace the Bochner–Martinelli kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004041.png" /> by kernels which are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004042.png" />. 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 [[#References|[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 [[#References|[a4]]] (cf. also [[Leray formula|Leray formula]]). (See [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004043.png" />, that is, a (differentiable) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004044.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004046.png" />. To any such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004047.png" />, Leray associates the following explicit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004048.png" />-form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004049.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004051.png" /></td> </tr></table>

and he obtains the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004052.png" /> for holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004053.png" />. Notice that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004055.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004056.png" /> and equals the Cauchy kernel, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004057.png" /> one has many different possibilities. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004058.png" /> gives the Bochner–Martinelli kernel. Another important case arises for Euclidean-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004059.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004060.png" /> boundary. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004061.png" /> is described as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004065.png" />, convexity implies that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004066.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004068.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004069.png" /> defines a Leray mapping that is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004070.png" />. The associated kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004071.png" /> is then also holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004072.png" />. Its pullback <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004073.png" /> to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004074.png" />, which only depends on the geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004075.png" /> and not on the particular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004076.png" /> chosen to describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004077.png" />, is known as the Cauchy–Leray kernel for the (convex) domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004078.png" />.

To construct Leray mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004079.png" /> which are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004080.png" /> for more general domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004081.png" />, is much more complicated. Such domains must necessarily be domains of holomorphy, and hence pseudo-convex, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004082.png" /> is a holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004083.png" /> which is singular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004084.png" /> (cf. also [[Domain of holomorphy|Domain of holomorphy]]; [[Pseudo-convex and pseudo-concave|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) [[#References|[a1]]] and, independently, by E. Ramirez [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004086.png" />-equation with supremum-norm estimates on strictly pseudo-convex domains (cf. also [[Neumann d-bar problem|Neumann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004087.png" />-problem]]).

These methods generalize to yield integral representation formulas for differential forms. In 1967, W. Koppelman [[#References|[a3]]] introduced (double) differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004089.png" />, of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004090.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004091.png" /> and type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004092.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004093.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004094.png" />, and proved a representation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004095.png" />-forms on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004096.png" /> with piecewise-differentiable boundary, as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004097.png" /></td> </tr></table>

Koppelman also introduced the analogous forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004098.png" /> of Cauchy–Fantappiè type in dependence of a given Leray mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i12004099.png" />, and proved corresponding integral representation formulas. By applying the Leray mapping of Khenkin and Ramirez, these methods lead to integral solution operators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i120040100.png" /> on forms of arbitrary degree.

Standard reference texts for these topics are [[#References|[a2]]] and [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i120040101.png" />-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, [[#References|[a9]]] and [[Singular integral|Singular integral]]).

Another fundamental integral representation formula involves the non-explicit [[Bergman kernel function|Bergman kernel function]], which is defined abstractly in the context of Hilbert spaces of square-integrable holomorphic functions on a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120040/i120040102.png" />. 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|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 [[#References|[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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. Henkin,   "Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications"  ''Math. USSR Sb.'' , '''7'''  (1969)  pp. 597–616  ''Mat. Sb.'' , '''78'''  (1969)  pp. 611–632</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. Henkin,   J. Leiterer,   "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Koppelman,   "The Cauchy integral for differential forms"  ''Bull. Amer. Math. Soc.'' , '''73'''  (1967)  pp. 554–556</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Leray,   "Le calcul différentiel et intégral sur une variété analytique complexe: Problème de Cauchy III"  ''Bull. Soc. Math. France'' , '''87'''  (1959)  pp. 81–180</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Ramirez,   "Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis"  ''Math. Ann.'' , '''184'''  (1970)  pp. 172–187</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.M. Range,   "Cauchy–Fantappié formulas in multidimensional complex analysis" , ''Geometry and Complex Variables, Univ. Bologna 1989'' , M. Dekker  (1991)  pp. 307–321</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representations in several complex variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Gleason,   "The abstract theorem of Cauchy–Weil"  ''Pac. J. Math.'' , '''12'''  (1962)  pp. 511–525</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.M. Stein,   "Hilbert Integrals, singular integrals, and Radon transforms II"  ''Invent. Math.'' , '''86'''  (1986)  pp. 75–113</TD></TR></table>