UDC 517.53:517.55

MATHEMATICS

I. I. Bavrin

GENERAL INTEGRAL REPRESENTATIONS OF HOLOMORPHIC FUNCTIONS

(Presented by Academician M. A. Lavrent’ev, 22 IV 1966)

In the present note, in the case of several complex variables, a general integral formula is established for functions holomorphic in convex complete \(n\)-circular domains of the space \(C^n\), \(n \ge 2\) (Theorem 4), and a general integral formula for functions holomorphic in convex domains of the space \(C^n\), \(n \ge 2\)* (Theorem 5). Each of these two formulas, for every value of \(k\) from \(\{1,2,\ldots,\mu\}\), includes an infinite set of integral representations. In formulating Theorems 4 and 5 we adhere to the definitions and notation used in \((^1)\). One of the auxiliary formulas is the general integral formula obtained here in the case of one complex variable (Theorem 3) for functions holomorphic in star-shaped domains.

1. Let \(G\) be a star-shaped** domain with respect to the origin in the space \(C^n\) of complex variables \(z_1,\ldots,z_n\), \(n \ge 1\), and let the function \(F(z_1,\ldots,z_n)\) be holomorphic in \(G\). Further, let \(\gamma\) be an arbitrary positive number; \(p,q\) natural numbers with \(p \ge q\), and let \(\gamma_p,\gamma_{p-1},\ldots,\gamma_q,\gamma_0\) be arbitrary positive numbers. Introduce the notation (for brevity, instead of \(F(z_1,\ldots,z_n)\) we write \(F\))

\[ L_\gamma [F] = \gamma F + \sum_{\nu=1}^{n} z_\nu F'_{z_\nu}, \qquad L_{q,p}^{(p-q+1)}[F] = L_p[L_{p-1}\ldots[L_q[F]]\ldots], \]

\[ L_{\binom{\gamma_q}{\gamma_p}}^{(p-q+1)}[F] = L_{\gamma_p}[L_{\gamma_{p-1}}\ldots[L_{\gamma_q}[F]]\ldots] \]

and put

\[ L_{p,p-1}^{(0)}[F] = F, \qquad L_{\binom{\gamma_p}{\gamma_{p-1}}}^{(0)}[F] = F \quad (p \ge 1). \]

Theorem 1. If the function \(F(z_1,\ldots,z_n)\) \((n \ge 1)\) is holomorphic in the domain \(G\), then for every natural \(k\) in the domain \(G\) we have

\[ F(z_1,\ldots,z_n) = \int_0^1 d\varepsilon_1 \ldots \int_0^1 d\varepsilon_{k-1} \int_0^1 \varepsilon_1^{\gamma_1-1}\ldots \varepsilon_k^{\gamma_k-1} \times \]

\[ \times L_{\binom{\gamma_1}{\gamma_k}}^{(k)} \bigl[ F(\varepsilon_1\ldots \varepsilon_k z_1,\ldots,\varepsilon_1\ldots \varepsilon_k z_n) \bigr]\, d\varepsilon_k, \tag{1} \]

where \(\varepsilon_1,\ldots,\varepsilon_k\) are real and \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying the condition \(\gamma_j \ge 1\) \((j=1,\ldots,k)\).

For \(k=1\) formula (1) is proved in the same way as formula (1) in the author’s note \((^1)\), and then for any natural \(k\) by induction.

1. Let \(G_1\) be a star-shaped domain with respect to the origin in the space \(C^1\), bounded by a closed rectifiable Jordan curve \(\Gamma\), and

* The same is true in the case of star-shaped domains in \(C^n\), \(n \ge 2\).

** A domain star-shaped with respect to the origin is called a domain containing, together with each point, the entire segment connecting this point with the origin.

\(F(z)\) is a function holomorphic in the domain \(G_1\). On the basis of formula (1) (\(n=1\)) and Cauchy’s formula (\(n=1\)) one obtains

Theorem 2. If the function \(F(z)\), holomorphic in the domain \(G_1\), and all its derivatives up to order \(\mu\) \((\mu>0)\) inclusive are continuous in the closed domain \(\overline{G}_1\), then for \(k=1,\ldots,\mu\) and \(z \in G_1\) the formula holds
\[ F(z)=\frac{1}{2\pi i}\int_\Gamma K\binom{\gamma_1}{\gamma_k}(\xi,z)\, L^{(k)}\binom{\gamma_1}{\gamma_k}[F(\xi)]\,d\xi, \tag{2} \]
where
\[ K\binom{\gamma_1}{\gamma_k}(\xi,z) = \int_0^1 d\varepsilon_1\ldots \int_0^1 d\varepsilon_{k-1} \int_0^1 \frac{\varepsilon_1^{\gamma_1-1}\ldots \varepsilon_k^{\gamma_k-1}} {\xi-\varepsilon_1\ldots \varepsilon_k z}\,d\varepsilon_k, \tag{3} \]
\(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\), and integration is performed over the contour \(\Gamma\) in the positive direction.

Let us point out the distinctive features of the integral representation (2): a) it expresses the values of the function \(F(z)\) in the domain \(G_1\) through the values of the operator
\[ L^{(k)}\binom{\gamma_1}{\gamma_k}[F] \]
on the boundary of the domain \(G_1\); b) since \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\), formula (2), for each value of \(k\) from \(\{1,2,\ldots,\mu\}\), includes an infinite set of integral representations.

Putting
\[ K\binom{\gamma_1}{\gamma_0}(\xi,z)=\frac{1}{\xi-z}, \]
one can, in the case of the domain \(G_1\), combine Cauchy’s formula and formula (2) into a single formula (2). Thus we obtain the following general theorem 3.

Theorem 3. If the function \(F(z)\), holomorphic in the domain \(G_1\), and all its derivatives up to order \(\mu\) \((\mu \ge 0;\ F^{(0)}=F)\) inclusive are continuous in the closed domain \(\overline{G}_1\), then for \(k=0,1,\ldots,\mu\) and \(z \in G_1\) formula (2) holds, where \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\).

1. Let us pass to the case \(n \ge 2\). Let \(B\) and \(H\) be bounded star-shaped domains with respect to the origin in \(C^n\), respectively with piecewise-smooth and smooth boundaries.

Theorem 4. Let \(D \in (T)\), and let the function \(f(z)\) \((n \ge 2)\) be holomorphic in \(D\), and let \(\alpha\) be a number equal to 0 or 1. Then, if the functions
\[ f_{z^{(\alpha)}_\nu}(z),\qquad \nu=1,\ldots,n^{**}, \]
and all their partial derivatives up to order \(\mu\) \((\mu \ge 0)\) inclusive are continuous

* We note that if the domain \(G_1\) is the disk \(|z|<1\), then under the conditions of theorem 3, for \(k=0,1,\ldots,\mu\) and \(|z|<1\) the formulas
\[ F(z)=\frac{1}{2\pi}\int_0^{2\pi} P\binom{\gamma_1}{\gamma_k}(\rho,\varphi-\psi)\, L^{(k)}\binom{\gamma_1}{\gamma_k}[F(e^{i\varphi})]\,d\varphi \quad (z=\rho e^{i\psi}), \]
\[ F(z)=i\,\operatorname{Im}F(0)+\frac{1}{2\pi} \int_0^{2\pi} S\binom{\gamma_1}{\gamma_k}(e^{i\varphi},z)\, \operatorname{Re}L^{(k)}\binom{\gamma_1}{\gamma_k}[F(e^{i\varphi})]\,d\varphi, \]
hold, where the kernels
\[ P\binom{\gamma_1}{\gamma_k}(\rho,\varphi-\psi) \quad\text{and}\quad S\binom{\gamma_1}{\gamma_k}(e^{i\varphi},z) \]
for \(k=1,\ldots,\mu\) have the form obtained from (3) by replacing in (3)
\[ \frac{1}{\xi-\varepsilon_1\ldots\varepsilon_k z} \]
respectively by
\[ \frac{1-(\varepsilon_1\ldots\varepsilon_k\rho)^2} {1+(\varepsilon_1\ldots\varepsilon_k\rho)^2 -2\varepsilon_1\ldots\varepsilon_k\rho\cos(\varphi-\psi)} \]
and
\[ \frac{e^{i\varphi}+\varepsilon_1\ldots\varepsilon_k z} {e^{i\varphi}-\varepsilon_1\ldots\varepsilon_k z}, \]
and for \(k=0\) these kernels are, respectively, the Poisson and Schwarz kernels.

** If \(\alpha=0\), then these functions are one and the same function \(f(z)\).

are continuous in \(D \cup S\), then for \(k=0,1,\ldots,\mu\) and \(z \in D\)

\[ f(z)=\alpha f(0)+\frac{1}{n+\alpha(1-n)} \sum_{\nu=1}^{n}\frac{z_\nu^\alpha}{(2\pi)^{n_i}} \int d\omega_* \int d\omega_\theta \int_{|\xi|=1} L_{\alpha+1,n-1}^{(n-1-\alpha)} \left[ K\binom{\gamma_1}{\gamma_k}(\xi,\mu) \right]\times \]

\[ \times L\binom{\gamma_1}{\gamma_k}^{(k)} \left[ F_{0z_\nu^\alpha}^{(\alpha)}(\xi,r,\theta) \right]\,d\xi, \tag{4} \]

where \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\).*

Theorem 5. Let the function \(F(z_1,\ldots,z_n)\) \((n \ge 2)\) be holomorphic in the domain \(D^{**}\), and let \(\alpha\) be a number equal to \(0\) or \(1\). Then, if the functions \(F_{z_\nu^\alpha}^{(\alpha)}(z_1,\ldots,z_n)\), \(\nu=1,\ldots,n\), and all their partial derivatives up to order \(\mu\) \((\mu \ge 0)\) inclusive are continuous in the closed domain \(\overline D\), then for \(k=0,1,\ldots,\mu\) and points \((z_1,\ldots,z_n)\in D\) there holds a formula (in view of the brevity of the presentation we do not write it out here), analogous in character to formula (4), expressing the values of the function \(F(z_1,\ldots,z_n)\) in the domain \(D\) through the values

\[ L\binom{\gamma_1}{\gamma_k}^{(k)} \left[ F_{z_\nu^\alpha}^{(\alpha)} \right], \]

where \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\), on the boundary of the domain \(D\), up to the summand \(\alpha F(0,\ldots,0)\). Similarly in the case of the domains \(B\) and \(H^{***}\).

For the proof of Theorem 4 one should use Theorem 3 of this note and the integral formula (for \(k=0\)) in Theorem 2 from \((^1)\); for the proof of Theorem 5 in the case of the domain \(D\), formula (1) of the present note, formula (2) from \((^1)\), and formula (2) from \((^2)\); while in the case of the domains \(B\) and \(H\) the only difference is that instead of formula (2) from \((^2)\) one uses, respectively, the Martinelli–Bochner integral representation (see, for example, \((^3)\)) and integral formula (2) from \((^4)\).

Remark 1. Each of the four integral formulas in Theorems 4 and 5 (in Theorem 5 three formulas are meant: one in the case of the domain \(D\), another for the domain \(B\), and a third for \(H\)) for any value of \(k\) from \(\{1,2,\ldots,\mu\}\) includes an infinite set of integral representations, since \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers satisfying \(\gamma_j \ge 1\) \((j=1,\ldots,k)\).

1. Formula (1) \((k=1)\) remains valid also in the case when \(\gamma_1\) is any positive number. But for \(0<\gamma_1<1\) the integral entering this formula should be understood as an improper one. Taking into account the same remark concerning analogous integrals, Theorem 1 and all the other results of this note connected with \(\gamma_1,\ldots,\gamma_k\) remain valid also in the case when \(\gamma_1,\ldots,\gamma_k\) are arbitrary positive numbers.

2. With the aid of his integral representations, A. A. Temlyakov \((^{5,6})\) obtained integral representations for a certain class of meromorphic functions in the case of two complex variables. In an analogous way, with the aid of Theorem 4 of the present note, integral representations are established for the corresponding class of meromorphic functions of \(n\) \((n \ge 2)\) complex variables.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
27 III 1966

REFERENCES

\(^1\) I. I. Bavrin, DAN, 169, No. 3 (1966).
\(^2\) L. A. Aizenberg, DAN, 151, No. 6 (1963).
\(^3\) V. S. Vladimirov, Methods of the Theory of Functions of Several Complex Variables, Moscow, 1964.
\(^4\) L. A. Aizenberg, DAN, 155, No. 1 (1964).
\(^5\) A. A. Temlyakov, DAN, 143, No. 5 (1962).
\(^6\) A. A. Temlyakov, Scientific Notes of the Moscow Regional Pedagogical Institute named after N. K. Krupskaya, 110, 3 (1962).

* Under the conditions of Theorem 4 there are also two more general formulas corresponding to the formulas indicated in the footnote on p. 1252.

** \(D\) here is the same convex domain in \(C^n\) that was considered in \((^{1,2})\).

*** The integral formula in the case of the domain \(H\) is more general in construction than in the case of the domain \(B\).