UDC 517.55:513.881

MATHEMATICS

L. A. AIZENBERG

GENERAL FORM OF A LINEAR CONTINUOUS FUNCTIONAL IN SPACES OF FUNCTIONS HOLOMORPHIC IN CONVEX DOMAINS OF \(C^n\)

(Presented by Academician M. A. Lavrent'ev, 17-V 1965)

1. Let \(D\) be a domain in the complex space \(C^n\) of complex variables \((z_1, z_2,\ldots,z_n)=z\); \(A(D)\) a linear topological space of functions \(f(z)\) holomorphic in the domain \(D\); \(A(\overline D)\) a linear topological space of functions holomorphic in the closed domain \(\overline D\). The topology in \(A(D)\) and \(A(\overline D)\) is introduced in the generally accepted manner; here the space \(A(D)\) is a countably normed space (see \((^1)\), p. 29), while \(A(\overline D)\) is an inductive limit (union) of countably normed spaces (see \((^1)\), p. 89; \((^{2,3})\)). A sequence \(f_m(z)\), \(m=1,2,\ldots\), converges to \(f(z)\) in the topology of the space \(A(D)\) if this sequence converges to the function \(f(z)\) uniformly on every closed set \(M\subset D\). A sequence \(f_m(z)\), \(m=1,2,\ldots\), converges to \(f(z)\) in the topology of the space \(A(\overline D)\) if there exists a domain \(Q\supset \overline D\) in which all the functions \(f_m(z)\), \(m=1,2,\ldots\), are holomorphic, and \(\lim_{m\to\infty} f_m(z)=f(z)\) in the sense of the topology of the space \(A(Q)\).

One of the most important questions arising in the study of the spaces \(A(D)\) and \(A(\overline D)\) is the question of the general form of a linear continuous functional in these spaces. In the case \(n=1\) an exhaustive answer to this question has been given (see \((^{4-6})\)); moreover it turned out that the space \(A^*(D)\) is isomorphic to \(A(C^1\setminus D)\), and \(A^*(\overline D)\) is isomorphic to \(A(C^1\setminus \overline D)\), where \(A^*(D)\) (respectively \(A^*(\overline D)\)) is the space conjugate to \(A(D)\) (respectively to \(A(\overline D)\)). Similar results are valid \((^7)\) for polycylindrical domains in \(C^n\), \(n>1\) (i.e., for topological products of plane domains). For domains \(D\subset C^n\), \(n>1\), that are not polycylindrical, analogous assertions cannot be obtained.

The problem of describing the general form of a linear continuous functional in the space \(A(D)\) (in \(A(\overline D)\)) was solved (see \((^{8-14})\)) by other methods for certain classes of domains \(D\) (\(n\)-circular, \((p,q)\)-circular, semicircular, tubular). In the present note the indicated problem is solved for convex domains \(D\subset C^n\) by means of the integral representation obtained in \((^{15})\). For these domains \(D\) the concept of the conjugate set \(\widetilde D\) is introduced; in a certain sense it plays the same role as the exterior of a domain does when \(n=1\).

1. Let \(D\) be a convex domain, \((0,0,\ldots,0)\in D\). Put
   \[ \widetilde D=\{w:\; w_1z_1+\cdots+w_nz_n\ne 1,\ z\in D\},\qquad \widetilde{\overline D}=\{w:\; w_1z_1+\cdots+w_nz_n\ne 1,\ z\in \overline D\}. \]
   Then:

\(1^\circ\). \(\widetilde D\) is a closed star-shaped bounded set, and \(D\) is a star-shaped bounded domain.

\(2^\circ\). If \(D_1\subset D\), then \(\widetilde D_1\supset \widetilde D\), \(\widetilde{\widetilde D}_1\supset \widetilde{\widetilde D}\).

\(3^\circ\). Denote by \(D_r\), \(r>0\), the homothety \(D_r=r\cdot D\) of the domain \(D\). Then
\[ (\widetilde D_r)=(\widetilde D)_{1/r}. \]
Analogously,
\[ (\widetilde{\overline D}_r)=(\widetilde{\overline D})_{1/r}. \]

\(4^\circ\). If the domain \(D\) is circular, then \(\widetilde D\) and \(\widetilde{\widetilde D}\) are circular convex sets.*

1. The domain \(D\) can be represented in the form \(D=\lim\limits_{m\to\infty} D_m\), where \(\overline D_m \subset D_{m+1}\), \((0,0,\ldots,0)\in D_m\),

\[ D_m=\{z:\Phi_m(z,\bar z)<0\},\qquad m=1,2,\ldots, \tag{1} \]

the domains \(D_m\) are bounded, the functions \(\Phi_m\) are twice continuously differentiable and convex in some neighborhoods of \(\overline D_m\). Denote by \(\partial D_m\) the boundary of the domain \(D_m\), and by \(\partial\widetilde D_m\) the boundary of \(\widetilde D_m\). Note that \(\widetilde D=\lim\limits_{m\to\infty}\widetilde D\), \(\widetilde D_{m+1}\subset(\widetilde D_m\setminus \partial\widetilde D_m)\), \(m=1,2,\ldots\). Let

\[ \omega(\Phi)=(n-1)!\left(\sum_{k=1}^n \delta_k\, d\bar\xi[k]\right)\wedge d\xi /(2\pi i)^n (\xi_1\Phi'_{\xi_1}+\cdots+\xi_n\Phi'_{\xi_n})^n, \]

where

\[ d\bar\xi[k]=d\bar\xi_1\wedge\cdots\wedge d\bar\xi_{k-1}\wedge d\bar\xi_{k+1}\wedge\cdots\wedge d\bar\xi_n;\qquad d\xi=d\xi_1\wedge\cdots\wedge d\xi_n; \]

\(\wedge\) denotes exterior multiplication; \(\Phi=\Phi(\xi,\bar\xi)\) is a twice continuously differentiable function,

\[ \delta_k= \begin{vmatrix} \Phi'_{\xi_1} & \Phi'_{\xi_2} & \cdots & \Phi'_{\xi_n}\\ \Phi''_{\xi_1\bar\xi_1} & \Phi''_{\xi_2\bar\xi_1} & \cdots & \Phi''_{\xi_n\bar\xi_1}\\ \vdots & \vdots & & \vdots\\ \Phi''_{\xi_1\bar\xi_{k-1}} & \Phi''_{\xi_2\bar\xi_{k-1}} & \cdots & \Phi''_{\xi_n\bar\xi_{k-1}}\\ \Phi''_{\xi_1\bar\xi_{k+1}} & \Phi''_{\xi_2\bar\xi_{k+1}} & \cdots & \Phi''_{\xi_n\bar\xi_{k+1}}\\ \vdots & \vdots & & \vdots\\ \Phi''_{\xi_1\bar\xi_n} & \Phi''_{\xi_2\bar\xi_n} & \cdots & \Phi''_{\xi_n\bar\xi_n} \end{vmatrix}. \]

Further, let \(\tau(\Phi)=(\tau_1(\Phi),\ldots,\tau_n(\Phi))\),

\[ \tau_i(\Phi)=\Phi'_{\xi_i}(\xi_1\Phi'_{\xi_1}+\cdots+\xi_n\Phi'_{\xi_n})^{-1},\qquad i=1,2,\ldots,n. \]

Theorem 1. Every continuous linear functional \(F(f)\) in the space \(A(D)\) has the form

\[ F(f)=F_\varphi(f)=\int_{\partial\widetilde D_m} f(\xi)\,\varphi(\tau(\Phi_m))\,\omega(\Phi_m), \tag{2} \]

where the function \(\varphi\) is holomorphic on the set \(\widetilde D\); \(m\) depends only on \(\varphi\) and is such that \(\varphi\) is holomorphic on the set \(\widetilde D_m\supset \widetilde D\). This correspondence between continuous linear functionals \(F(f)\) in the space \(A(D)\) and functions \(\varphi\in A(\widetilde D)\) is an isomorphism of the linear topological spaces \(A^*(D)\) and \(A(\widetilde D)\).

Proof. 1) First of all, note that, by the convexity of \(\Phi_m\), all first-order derivatives of \(\Phi_m\) do not vanish simultaneously at points of the boundary \(\partial D_m\) of the domain \(D_m\). At each point \(\xi\in\partial D_m\) there exists a tangent complex \((n-1)\)-dimensional analytic plane

\[ \{z:(\xi_1-z_1)\Phi'_{m\xi_1}+\cdots+(\xi_n-z_n)\Phi'_{m\xi_n}= \]

\[ =0\}=\{z:z_1\tau_1(\Phi_m)+\cdots+z_n\tau_n(\Phi_m)=1\} \]

\[ (\xi_1\Phi'_{m\xi_1}+\cdots+\xi_n\Phi'_{m\xi_n}\ne0,\ \text{since }(0,0,\ldots,0)\in D_m), \]

which does not intersect the convex domain \(D_m\); therefore the point \(\tau(\Phi_m)\in\widetilde D_m\).

2) Let the functional \(F(f)\in A^*(D)\); then there exists an \(m_0\) such that

* If the domain \(D\) is not circular, then one cannot guarantee the convexity of \(\widetilde D\) or \(\widetilde{\widetilde D}\). For example, if \(D=\{z:|\operatorname{Re} z|+|\operatorname{Im} z|<1\}\subset C^1\), then the set \(\widetilde D=\{w:(|\operatorname{Re} w|-1/2)^2+(|\operatorname{Im} w|-1/2)^2\le 1/2\}\) is star-shaped, but not convex.

\(F(f)\) is bounded with respect to the norm \(\|f\|_{m_0}=\max_{\overline{D}_{m_0}} |f|\) (see (1), p. 49). Using the integral representation

\[ f(z)=\int_{\partial D_m}\frac{f(\zeta)\,\omega(\Phi_m)} {[1-z_1\tau_1(\Phi_m)-\cdots-z_n\tau_n(\Phi_m)]^n}, \tag{3} \]

\(z\in D_m,\ \zeta\in \partial D_m\), we obtain, for \(m>m_0\), formula (2), where
\(\varphi(\tau(\Phi_m))= F([1-z_1\tau_1(\Phi_m)-\cdots-z_n\tau_n(\Phi_m)]^{-n})\), the function \(\varphi\) is holomorphic on the set \(\widetilde{D}_m\), and the integral on the right-hand side of (2) does not depend on the choice of \(m>m_0\).

Conversely, let \(\varphi\in A(\widetilde{D})\); then there is an \(m\) such that \(\varphi\in A(\widetilde{D}_m)\). Then, obviously, formula (2) defines a linear continuous functional \(F(f)\) in the space \(A(D)\).

3) From the boundedness of the domain \(D_m\) it follows that there exists an \(r>0\) such that \(D_m\) is contained in the hypersphere \(I_r=\{z:\ |z_1|^2+\cdots+|z_n|^2<r^2\}\). It can be shown that equality (2), considered only for \(f(z)\in A(\overline{I}_r)\), is transformed into the form

\[ F_\varphi(f)= \frac{(n-1)!}{(2\pi i)^n r^{2n}} \int_{\partial I_r} f(\zeta)\varphi\left(\frac{\overline{\zeta}}{r^2}\right) \left(\sum_{k=1}^n (-1)^{k-1}\overline{\zeta}_k\,d\overline{\zeta}\,[k]\right)\wedge d\zeta = \]

\[ = \frac{(n-1)!\,r^{2n}}{(2\pi i)^n} \int_{\partial I_{1/r}} f(\overline{\eta}r^2)\varphi(\eta) \sum_{k=1}^n (-1)^{k-1}\overline{\eta}_k\,d\overline{\eta}\,[k]\wedge d\eta, \tag{4} \]

where \(\zeta\in \partial I_r,\ \eta\in \partial I_{1/r},\ \overline{I}_{1/r}=\widetilde{I}_r\). Put
\(f_w(z)=(1-w_1z_1-\cdots-w_nz_n)^{-n}\), where \(w\) is a fixed point of \(I_{1/r}\). Applying (3) to the domain \(I_{1/r}\) and to the function \(\varphi\) (instead of the function \(f\)) and (4), we obtain

\[ F_\varphi(f_w)=\varphi(w) \tag{5} \]

for all \(w\in I_{1/r}\). On the other hand, it follows from (2) that the function
\(\psi(w)=F_\varphi(f_w)\) is holomorphic in \(w\) in the domain \(\widetilde{D}_m\); therefore \(\psi(w)=\varphi(w)\) for \(w\in \widetilde{D}_m\), and hence formula (5) is also valid for \(w\in \widetilde{D}_m\).

4) In 2) a correspondence was established between functionals \(F(f)\in A^*(D)\) and functions \(\varphi\in A(\widetilde{D})\). We shall show that this correspondence is one-to-one. Let \(\varphi_1,\varphi_2\in A(\widetilde{D})\); choose \(m\) so that \(\varphi_1,\varphi_2\in A(\widetilde{D}_m)\). If \(\varphi_1=\varphi_2\), then from (2) we find \(F_{\varphi_1}(f)=F_{\varphi_2}(f)\). Conversely, if \(F_{\varphi_1}(f)=F_{\varphi_2}(f)\), then from (5) it follows that \(\varphi_1=\varphi_2\) in the domain \(\widetilde{D}_m\subset \widetilde{D}_m\), i.e. \(\varphi_1=\varphi_2\) also on the set \(\widetilde{D}_m\).

From the one-to-one character of the correspondence under consideration and formulas (2) and (5), we obtain that the spaces \(A^*(D)\) and \(A(\widetilde{D})\) are algebraically isomorphic. It remains to establish that this isomorphism is also topological. The spaces \(A^*(D)\) and \(A(\widetilde{D})\) are, as can be shown, spaces \((LN^*)\) (see \((2,3)\)); therefore it suffices to prove that a sequence \(\varphi_k,\ k=1,2,\ldots\), converges in the topology of the space \(A(\widetilde{D})\) to a function \(\varphi\) if and only if the sequence of functionals \(F_{\varphi_k},\ k=1,2,\ldots\), converges in the sense of the topology of the space \(A^*(D)\) to the functional \(F_\varphi\).

5) Let \(\varphi=\lim_{k\to\infty}\varphi_k\) in the topology of the space \(A(\widetilde{D})\); then there exists an \(m\) such that the indicated convergence is uniform on the closed set \(\widetilde{D}_m\), and, by virtue of (2), \(\lim_{k\to\infty}F_{\varphi_k}(f)=F_\varphi(f)\) for all \(f\in A(D)\), i.e. the sequence of functionals \(F_{\varphi_k}\) converges weakly to the functional \(F_\varphi\). By Montel’s theorem (see \((16)\), p. 195) it follows that the space \(A(D)\) is perfect ((1), p. 73); therefore \(\lim_{k\to\infty}F_{\varphi_k}=F_\varphi\) with respect to the topology of the space \(A^*(D)\) ((1), p. 77).

Conversely, let \(\lim_{k\to\infty}F_{\varphi_k}=F_\varphi\) in the topology of the space \(A^*(D)\). There exists an \(m\) such that all \(F_{\varphi_k}\in B^*(D_m)\) and \(\lim_{k\to\infty}F_{\varphi_k}=F_\varphi\) in the norm of the space

spaces \(B^*(D_m)\), where \(B^*(D_m)\) is the space conjugate to the Banach space \(B(D_m)\) of functions holomorphic in the domain \(D_m\), continuous in the closed domain \(\overline{D}_m\), with norm \(\|f\|_m=\max_{\overline{D}_m}|f|\) (see (1), p. 78). Repeating the arguments of 2), we obtain that all \(\varphi_k,\ k=1,2,\ldots,\) are holomorphic on the set \(\widetilde D_{m+1}\). Moreover,
\[ \lim_{k\to\infty}F_{\varphi_k}(f_w)=F_\varphi(f_w) \]
uniformly for \(w\in\widetilde D_{m+1}\). Consequently, by virtue of (5),
\[ \lim_{k\to\infty}\varphi_k(w)=\varphi(w), \]
and this convergence is uniform on the closed set \(\widetilde D_{m+1}\supset\widetilde D\).

1. A bounded domain \(D\) can be represented in the form
   \[ D=\lim_{m\to\infty}D_m, \]
   \(\overline{D}_{m+1}\subset D_m\), where the domains \(D_m\) have the form (1) under the same restrictions as in item 3. Then, analogously to Theorem 1, the following can be established.*

Theorem 2. Let the domain \(D\) be bounded. Every linear continuous functional \(F(f)\) on the space \(A(\overline D)\) has the form (2), where the function \(\varphi\) is holomorphic in the domain \(\widetilde D\), \(m\) depends on \(f\) and is such that \(f\) is holomorphic on the set \(\overline{D}_m\supset\overline D\). This correspondence between linear continuous functionals on the space \(A(\overline D)\) and functions \(\varphi\in A(\widetilde D)\) is an isomorphism of linear topological spaces \(A^*(\overline D)\) and \(A(\widetilde D)\).

1. If the domain \(D=\{z:\Phi(z,\bar z)<0\}\) is bounded, the function \(\Phi\) is twice continuously differentiable and convex in some neighborhood of \(\overline D\), then instead of the domains \(D_m\) in Theorem 1 (in Theorem 2) one may consider homothetic domains \(D_r,\ r<1\) (\(r>1\)) or domains
   \[ D(\rho)=\{z:\Phi(z,\bar z)<\rho\},\quad \rho<0\quad(\rho>0). \]
   Then we obtain

Corollary 1. If, for any nonnegative integers \(k_1,k_2,\ldots,k_n\),
\[ \int_{\partial D}\zeta_1^{k_1}\cdots \zeta_n^{k_n}\varphi(\tau(\Phi))\,\omega(\Phi)=0, \]
where \(\zeta\in\partial D\), and the function \(\varphi\) is holomorphic on the set \(\widetilde D\), then \(\varphi=0\).

Corollary 2. In order that, for every function \(f(z)\) holomorphic in the closed domain \(\overline D\), the formula
\[ f(z)=\int_{\partial D} f(\zeta)\Psi(z,\bar z,\tau(\Phi))\,\omega(\Phi) \]
hold, where \(\zeta\in\partial D,\ z\in D\), and \(\Psi(z,\bar z,\tau(\Phi))\) is holomorphic in \(\tau\) on the set \(\widetilde D\) for each \(z\in D\), it is necessary and sufficient that
\[ \Psi(z,\bar z,\tau(\Phi))= \frac{1}{[1-z_1\tau_1(\Phi)-\cdots-z_n\tau_n(\Phi)]^n}. \]

Institute of Physics
Siberian Branch of the Academy of Sciences of the USSR

Received
17 V 1965

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* Theorem 2 is obtained also as a consequence of Martineau’s results \((^{17})\).