Abstract
Full Text
L. A. AIZENBERG
INTEGRAL REPRESENTATION OF FUNCTIONS HOLOMORPHIC IN CONVEX DOMAINS OF THE SPACE (C^n)
(Presented by Academician V. I. Smirnov, 4 III 1963)
1. Cauchy’s integral formula
[
f(z)=\frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)}{\zeta-z}\,d\zeta
\tag{1}
]
(where the function (f(z)) is holomorphic in the domain (D) and continuous in the closed domain (\overline D); (\partial D) consists of a finite number of closed rectifiable Jordan curves) plays a very important role in the study of holomorphic functions of one complex variable, which, in our opinion, is explained mainly by two properties of this formula: 1) formula (1) is universal, i.e. it is valid and has one and the same form for any (D); 2) the Cauchy kernel (\dfrac{1}{\zeta-z}) is holomorphic in (z) (this ensures the holomorphy of the Cauchy-type integral, etc.).
For holomorphic functions of several complex variables it is not possible to obtain an integral formula possessing the two indicated properties of Cauchy’s formula: a) the Martinelli—Bochner integral representation (see (\left({}^{1}\right)), § 21) is universal, but has a nonholomorphic kernel; b) the integral representations of Weil (see (\left({}^{1}\right)), § 22), Bergman (\left({}^{2}\right)), Hua Loo-Keng (\left({}^{3}\right)), Temlyakov (\left({}^{4}\right)), and others (see (\left({}^{5,6}\right)); (\left({}^{1}\right)), § 23) are not universal, but do possess holomorphic kernels.
In the present paper an integral formula (2) with a holomorphic kernel is obtained for convex domains of the space (C^n) of (n) complex variables (z_1,z_2,\ldots,z_n). We know two proofs of this formula: A) formula (2) can be derived from the Cauchy—Fantappiè integral representation indicated by Leray (\left({}^{7}\right)); B) formula (2) can be obtained elementarily, without relying on the Cauchy—Fantappiè formula. We shall give the second proof. We note that, as Norguet (\left({}^{8}\right)) showed, the Martinelli—Bochner and Weil integral representations can also be obtained from the Cauchy—Fantappiè formula.
2.
Let the domain
[
D={(z_1,z_2,\ldots,z_n):\Phi(z_1,\bar z_1,z_2,\bar z_2,\ldots,\bar z_n,z_n)<0}
]
be convex and bounded, and let the function (\Phi) be twice continuously differentiable and all first-order derivatives of (\Phi) not vanish simultaneously at points of the boundary (\partial D) of the domain (D).
Theorem. If the function (f(z_1,z_2,\ldots,z_n)) is holomorphic in the domain (D) and continuous in the closed domain (\overline D), then for points ((z_1,z_2,\ldots,z_n)\in D)
[
f(z_1,\ldots,z_n)=
\frac{(n-1)!}{(2\pi i)^n}
\int_{\partial D}
\frac{
f(\zeta_1,\ldots,\zeta_n)
\left[
\sum_{k=1}^{n}\delta_k
\left(
\bigwedge_{1\le j\le n;\,j\ne k} d\bar\zeta_j
\right)
\right]
\bigwedge_{1\le j\le n} d\zeta_j
}{
\left[\Phi'{\zeta_1}\cdot(\zeta_1-z_1)+\cdots+\Phi'\cdot(\zeta_n-z_n)\right]^n
},
\tag{2}
]
where (\bigwedge) is the sign of exterior multiplication,
[
\delta_k=
\left|
\begin{array}{ccccc}
\Phi'{\zeta_1} & \Phi' & \ldots & \Phi'{\zeta_n}\
\Phi'' & \Phi''{\zeta_2\bar\zeta_1} & \ldots & \Phi''\
\ldots & \ldots & \ldots & \ldots\
\Phi''{\zeta_1\bar\zeta & \Phi''}{\zeta_2\bar\zeta & \ldots & \Phi''}{\zeta_n\bar\zeta\}
\Phi''{\zeta_1\bar\zeta & \Phi''}{\zeta_2\bar\zeta & \ldots & \Phi''}{\zeta_n\bar\zeta\}
\ldots & \ldots & \ldots & \ldots\
\Phi''{\zeta_1\bar\zeta_n} & \Phi''} & \ldots & \Phi''_{\zeta_n\bar\zeta_n
\end{array}
\right|.
]
Proof will be carried out, for simplicity, in the case of two complex variables; then formula (2) takes the form
[
f(z_1,z_2)=\frac{1}{(2\pi i)^2}\int_{\partial D}
\frac{
f(\zeta_1,\zeta_2)\left(
\left|
\begin{array}{cc}
\Phi'{\zeta_1} & \Phi'\
\Phi'{\bar\zeta_1} & \Phi'
\end{array}
\right|\,d\bar\zeta_1+
\left|
\begin{array}{cc}
\Phi'{\zeta_1} & \Phi'\
\Phi'{\bar\zeta_2} & \Phi'
\end{array}
\right|\,d\bar\zeta_2
\right)\wedge d\zeta_1\wedge d\zeta_2
}{
\left[\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)\right]^2
}.
\tag{3}
]
Let us first note that, by virtue of the conditions imposed by us on the function (\Phi), at every point ((\zeta_1,\zeta_2)\in \partial D) there exists a tangent analytic plane
[
{(z_1,z_2):\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)=0}.
]
Since the domain (D) is convex, it follows that
[
\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)\ne 0
]
for ((z_1,z_2)\in D).
I. First suppose that the function (f(z_1,z_2)) is holomorphic in the closed domain (\overline D). Represent the boundary (\partial D) of the domain (D) in the form (\partial D=\Gamma_1\cup\Gamma_2), where
[
\Gamma_1={(\zeta_1,\zeta_2):(\zeta_1,\zeta_2)\in\partial D,\ |\zeta_1-z_1-\zeta_2+z_2|\ge \sigma},
]
[
\Gamma_2={(\zeta_1,\zeta_2):(\zeta_1,\zeta_2)\in\partial D,\ |\zeta_1-z_1-\zeta_2+z_2|<\sigma},\quad \sigma>0.
]
Denote the right-hand side of formula (3) by (I). Then (I=I_1+I_2), where the integrals (I_1) and (I_2) are obtained from (I) by replacing the set of integration (\partial D) respectively by (\Gamma_1) and (\Gamma_2). Obviously, for sufficiently small (\sigma) the inequality (|I_2|<\varepsilon/2) holds, where (\varepsilon) is any fixed positive number.
Since the integrand in formula (3) is the differential of the exterior differential form
[
\mu=
\frac{
f(\zeta_1,\zeta_2)(\Phi'{\zeta_1}+\Phi')\,d\zeta_1\wedge d\zeta_2
}{
(\zeta_1-z_1-\zeta_2+z_2)\left[\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)\right]
},
]
then, with the aid of Stokes’ formula (see (9), § 6), we obtain
[
I_1=\int_{\partial\Gamma_1}\mu
=
\frac{1}{4\pi^2 i}\int_0^{2\pi}dt
\int_{C_{t,\sigma}}
\frac{
f(\zeta_1,\zeta_1-z_1+z_2-\sigma e^{it})(\Phi'{\zeta_1}+\Phi')\,d\zeta_1
}{
\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)
},
]
where
[
C_{t,\sigma}={(\zeta_1,\zeta_2):(\zeta_1,\zeta_2)\in\partial D,\
\zeta_1-z_1-\zeta_2+z_2=\sigma e^{it}},
]
[
\partial\Gamma_1=\bigcup_{0\le t\le 2\pi} C_{t,\sigma}.
]
Hence we find
[
I_1=
\frac{1}{4\pi^2 i}\int_0^{2\pi}dt
\int_{C_{t,\sigma}}
\frac{
f(\zeta_1,\zeta_1-z_1+z_2-\sigma e^{it})\,d\zeta_1
}{
\zeta_1-z_1
}
+
]
[
+
\frac{\sigma}{4\pi^2 i}\int_0^{2\pi}dt
\int_{C_{t,\sigma}}
\frac{
f(\zeta_1,\zeta_1-z_1+z_2-\sigma e^{it})\Phi'{\zeta_2}e^{it}\,d\zeta_1
}{
(\zeta_1-z_1)\left[\Phi'(\zeta_2-z_2)\right]}(\zeta_1-z_1)+\Phi'_{\zeta_2
}
=I_3+\sigma I_4.
]
The point ((z_1,z_2)) is a fixed point of the domain (D); therefore there exists an (h_1>0) such that
[
|\Phi'{\zeta_1}(\zeta_1-z_1)+\Phi'(\zeta_2-z_2)|>h_1
]
for ((\zeta_1,\zeta_2)\in\partial\Gamma_1\subset\partial D). We further note that
[
\partial\Gamma_1={(\zeta_1,\zeta_2):(\zeta_1,\zeta_2)\in\partial D,\
|\zeta_1-z_1-\zeta_2+z_2|=\sigma},
]
for sufficiently small (\sigma), does not intersect the plane
[
{(\zeta_1,\zeta_2):\zeta_1-z_1=0}.
]
Indeed, otherwise the intersection
[
\partial D\cap{(\zeta_1,\zeta_2):\zeta_1-z_1-\zeta_2+z_2=0}
\cap{(\zeta_1,\zeta_2):\zeta_1-z_1=0}
]
would not be empty, and we arrive at a contradiction, since the only common point of the two analytic planes
[
{(\zeta_1,\zeta_2):\zeta_1-z_1-\zeta_2+z_2=0}
\quad\text{and}\quad
{(\zeta_1,\zeta_2):\zeta_1-z_1=0}
]
is the point ((z_1,z_2)\notin\partial D). From the compactness of the set (\partial\Gamma_1) it follows that there exists an (h_2>0) such that
[
|\zeta_1-z_1|>h_2
]
for ((\zeta_1,\zeta_2)\in\partial\Gamma_1). Therefore the quantity (I_4) is bounded in modulus. But then, for sufficiently small (\sigma),
[
\sigma |I_4|<\varepsilon/2.
]
Further,
[
I_3=
\frac{1}{4\pi^2 i}\int_0^{2\pi}dt
\int_{C_{t,\sigma}^{+}}
\frac{
f(\zeta_1,\zeta_1-z_1+z_2-\sigma e^{it})\,d\zeta_1
}{
\zeta_1-z_1
}
=
]
[
\frac{1}{2\pi}\int_0^{2\pi} f(z_1,z_2-\sigma e^{it})\,dt
f(z_1,z_2).
]
Thus,
[
I=I_1+I_2=I_3+\sigma I_4+I_2=f(z_1,z_2)+(\sigma I_4+I_2),
]
where
[
|\sigma I_4+I_2|\le \sigma |I_4|+|I_2|<\varepsilon/2+\varepsilon/2=\varepsilon.
]
Hence, by the arbitrariness of (\varepsilon), it follows that (I=f(z_1,z_2)).
II. Now suppose that the function (f(z_1,z_2)) is holomorphic in the domain (D) and continuous in the closed domain (\overline D). Then (f(z_1,z_2)) can be represented as the limit of the sequence
[
{f_m(z_1,z_2)\equiv f(z_1(1-1/m),\,z_2(1-1/m)),\quad m=2,3,\ldots},
]
consisting of functions holomorphic in the closed domain (\overline D) and converging uniformly in this closed domain (we assume that the point ((0,0)\in D); otherwise one can make a parallel translation). Writing formula (3) for each of the functions (f_m(z_1,z_2)) and passing to the limit as (m\to\infty) in both parts of the equality obtained, we arrive at formula (3) for the function (f(z_1,z_2)).
- We note some consequences of the theorem proved.
A. In the case (n=1), formula (2) yields the Cauchy formula (1).
B. Let the bicircular domain
[
D={(z_1,z_2): |z_2|<\Phi(|z_1|),\ 0\le |z_1|\le r}
]
be convex and bounded, and let the function (\Phi) be twice continuously differentiable. Then for ((z_1,z_2)\in D) the formula
[
f(z_1,z_2)=\frac{-1}{(2\pi i)^2}\int_{\partial D}
\frac{
f(\zeta_1,\zeta_2)\,d\varphi_1(|\zeta_1|)\wedge \dfrac{d\zeta_1}{\zeta_1}\wedge \dfrac{d\zeta_2}{\zeta_2}
}{
\left[1-z_1\overline{\zeta}_1\,\dfrac{\varphi_1(|\zeta_1|)}{|\zeta_1|^2}
-z_2\overline{\zeta}_2\,\dfrac{\varphi_2(|\zeta_2|)}{|\zeta_2|^2}\right]^2
},
\tag{4}
]
is valid, where
[
\varphi_1(|\zeta_1|)=
\frac{|\zeta_1|\Phi'(|\zeta_1|)}
{|\zeta_1|\Phi'(|\zeta_1|)-\Phi(|\zeta_1|)},
\quad
\varphi_2(|\zeta_2|)=\varphi_2(\Phi(|\zeta_1|))=1-\varphi_1(|\zeta_1|).
]
(4) is Temlyakov’s integral representation, written in another form (see ((^4,^6)); ((^4)), § 23). Formula (4) can also be obtained under more general assumptions regarding the smoothness of the function (\Phi) (see ((^6,^10))).
C. For the hyperellipsoid
[
D=\left{(z_1,z_2): z_1=x_1+iy_1,\ z_2=x_2+iy_2,\right.
]
[
\left.
\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}+\frac{x_2^2}{c^2}+\frac{y_2^2}{d^2}<1
\right}
]
the integral representation (2) has the form
[
f(z_1,z_2)=\frac{1}{(2\pi i)^2}\int_{\partial D}
\frac{
f(\zeta_1,\zeta_2)(-b_1\beta\,d\overline{\zeta}_1+b_2\alpha\,d\overline{\zeta}_2)\wedge d\zeta_1\wedge d\zeta_2
}{
[\alpha(\zeta_1-z_1)+\beta(\zeta_2-z_2)]^2
},
]
where
[
\alpha=a_1\zeta_1+b_1\overline{\zeta}_1,\quad
\beta=a_2\zeta_2+b_2\overline{\zeta}_2,\quad
a_1=\frac12\left(\frac1{a^2}-\frac1{b^2}\right),\quad
b_1=\frac12\left(\frac1{a^2}+\frac1{b^2}\right),
]
[
a_2=\frac12\left(\frac1{c^2}-\frac1{d^2}\right),\quad
b_2=\frac12\left(\frac1{c^2}+\frac1{d^2}\right).
]
- One may consider an “integral of type (2),” i.e., the integral from the right-hand side of formula (2) of an arbitrary function (f(\zeta_1,\zeta_2)), summable on the boundary (\partial D) of the domain (D). This integral defines a function holomorphic in the domain (D). In particular, integrals of Temlyakov type were studied in detail by us earlier ((^{11},^{12})). It follows from these studies that the integral (2) for (n>1), generally speaking, is not equal to zero outside the domain (D), while an integral of type (2) for (n>1), generally speaking, defines a function that is not holomorphic outside the domain (D).
Shuya State
Pedagogical Institute
Received
2 III 1963
CITED LITERATURE
- B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Moscow, 1962.
- S. Bergmann, Matem. sborn., 1 (43), No. 6 (1936).
- Hua Lo-keng, Harmonic Analysis of Functions of Several Complex Variables in Classical Domains, Moscow, 1959.
- A. A. Temlyakov, DAN, 120, No. 5 (1958).
- S. G. Gindikin, DAN, 145, No. 6 (1962).
- L. A. Aizenberg, DAN, 138, No. 1 (1961).
- J. Leray, Differential and Integral Calculus on a Complex Analytic Manifold, Moscow, 1960.
- F. Norguet, C. R., 250, No. 10 (1960).
- J. de Rham, Differentiable Manifolds, Moscow, 1956.
- L. A. Aizenberg, Uch. zap. Moscow Regional Pedagogical Institute, 110, 11 (1962).
- L. A. Aizenberg, DAN, 120, No. 5 (1958).
- L. A. Aizenberg, DAN, 125, No. 5 (1959).