MATHEMATICS

L. I. RONKIN

ON THE GENERAL FORM OF A FUNCTIONAL IN THE SPACE OF FUNCTIONS ANALYTIC IN A SEMICIRCULAR DOMAIN

(Presented by Academician S. N. Bernstein on 21 I 1961)

Let \(T\) be some domain in the space of the complex variables \(z,w\). Denote by \(A_T\) the space of functions \(f(z,w)\) analytic in the domain \(T\). Convergence of a sequence of elements of this space is defined as uniform convergence in every domain lying strictly inside \(T\). If the domain \(T\) is the bicylinder \(\{|z|<R_1,\ |w|<R_2\}\), then, as is known, any linear functional \(G\) in the space \(A_T\) is defined by the formula

\[ G(f)= \int_{|z|=R_1-\varepsilon} \int_{|w|=R_2-\varepsilon} f(z,w)g(z,w)\,dz\,dw \tag{1} \]

with a function \(g(z,w)\) analytic in the domain \(\{|z|>R_1-2\varepsilon,\ |w|>R_2-2\varepsilon\}\), where \(\varepsilon>0\) and depends on \(G\). This correspondence between the functionals \(G\) and the functions \(g(z,w)\) becomes one-to-one if one requires that \(g(\infty,\infty)=0\).

In the case when \(T\) is a complete \(n\)-circular domain, the general form of a functional in the space \(A_T\) was obtained by S. D. Okun and L. A. Aizenberg and B. S. Mityagin \((^1)\). In this note we consider the space of functions analytic in a semicircular domain*.

Let \(T\) be a complete semicircular domain with plane of symmetry \(w=0\). By definition of a semicircular domain,

\[ T=\{z\in H_T;\ |w|<R_T(z)\}, \]

where \(H_T\) is some domain in the \(z\)-plane, and \(R_T(z)\) is some nonnegative function defined in the domain \(H_T\). As is known, every function \(f(z,w)\in A_T\) expands into a Hartogs series

\[ f(z,w)=\sum_{k=0}^{\infty} w^k f_k(z). \]

Here the functions \(f_k(z)\), analytic in the domain \(H_T\), satisfy the condition

\[ \overline{\lim_{z'\to z}}\ \overline{\lim_{k\to\infty}}\sqrt[k]{|f_k(z')|} \le \frac{1}{R_T(z)}. \]

Considering the bicylinder \(\{|z|<R_1,\ |w|<R_2\}\) as a semicircular domain, we shall give formula (1) a somewhat different form, more convenient for us.

* The results of S. D. Okun were reported at the Fifth All-Union Conference on Function Theory in Yerevan.

** On semicircular domains see, for example, \((^2)\).

For this purpose, let us expand the functions \(f(z,w)\) and \(g(z,w)\) in Hartogs series

\[ f(z,w)=\sum_{k=0}^{\infty} w^{k} f_k(z), \qquad g(z,w)=\sum_{k=0}^{\infty} w^{-k-1} g_k(z). \]

Since the functions \(f(z,w)\) and \(g(z,w)\) are holomorphic respectively in the bicylinders \(\{|z|<R_1,\ |w|<R_2\}\) and \(\{|z|>R_1-2\varepsilon,\ |w|>R_2-2\varepsilon\}\), the functions \(f_k(z)\) and \(g_k(z)\) are analytic, respectively, in the domains \((|z|<R_1)\) and \((|z|>R_1-2\varepsilon)\), and for \(|z|=R_1-\varepsilon\) satisfy the conditions

\[ \overline{\lim}_{z'\to z}\ \overline{\lim}_{k\to\infty}\sqrt[k]{|f_k(z')|}<\frac{1}{R_2}, \qquad \overline{\lim}_{z'\to z}\ \overline{\lim}_{k\to\infty}\sqrt[k]{|g_k(z')|}<R_2-2\varepsilon . \tag{2} \]

Perform in formula (1) the integration with respect to \(w\):

\[ \begin{aligned} G(f) &= \int_{|z|=R_1-\varepsilon} \int_{|w|=R_2-\varepsilon} f(z,w)g(z,w)\,dz\,dw \\ &= \int_{|z|=R_1-\varepsilon} \int_{|w|=R_2-\varepsilon} \left(\sum_{k=0}^{\infty} w^k f_k(z)\right) \left(\sum_{k=0}^{\infty} w^{-k-1} g_k(z)\right)\,dz\,dw \\ &= \int_{|z|=R_1-\varepsilon} \sum_{k=0}^{\infty} f_k(z)g_k(z)\,dz = \sum_{k=0}^{\infty} \int_{|z|=R_1-\varepsilon} f_k(z)g_k(z)\,dz . \end{aligned} \]

Thus, a linear functional in the space under consideration can be specified by a sequence of functions \(g_k(z)\), analytic for \(|z|>R_1-2\varepsilon\) and satisfying, on some contour lying strictly inside \(H_T=\{|z|<R_1\}\), the condition

\[ \overline{\lim}_{z'\to z}\ \overline{\lim}_{k\to\infty}\sqrt[k]{|g_k(z')|}<R_T(z)-\varepsilon, \]

where \(R_T(z)=R_2\). Obviously, consideration of bicylinders whose center does not lie at the origin introduces nothing new in comparison with the case considered.

In the case of a semicircular domain the following holds:

Theorem. Let \(G\) be a continuous linear functional in the space \(A_T\), where \(T\) is a complete semicircular domain. Then there exists a contour \(C\), lying strictly inside \(H_T\), and a sequence of functions \(g_k(z)\), analytic outside the contour \(C\) and on the contour \(C\) itself, such that for every function \(f(z,w)\in A_T\)

\[ G(f)=\sum_{k=0}^{\infty}\int_C f_k(z)g_k(z)\,dz . \tag{3} \]

Moreover, a linear functional in \(A_T\) is determined by the sequence \(\{g_k(z)\}\) if and only if the sequence under consideration can be represented in the form

\[ g_k(z)=\sum_{i=1}^{N} g_{k,i}(z), \]

where \(N\) depends on \(G\) and does not depend on \(k\), and the functions \(g_{k,i}(z)\) satisfy the conditions: 1) for every \(k\), the function \(g_{k,i}(z)\) must be holomorphic outside some contour \(C_i\), lying inside \(H_T\), and also on the contour itself; 2) for every \(z\in C_i\) the condition must hold

\[ \overline{\lim}_{z'\to z}\ \overline{\lim}_{k\to\infty}\sqrt[k]{|g_{k,i}(z')|}<R_T(z)-\varepsilon . \]

The representation is unique under the additional requirement \(g_k(\infty)=0\), \(k=1,2,\ldots\).

Proof. Take a sequence of finite semicircular domains \(\{T_p\}\) such that \(T_p \subset T_{p+1}\) for every \(p\) and \(T_p \to T\) as \(p \to \infty\). Denote by \(\dot T\) the boundary of the domain \(T\). Also put

\[ \varepsilon_p = \inf_{\substack{(z,w)\in \dot T;\ (z',w')\in T_p}} \sqrt{|z'-z|^2+|w-w'|^2}, \]

and require that \(\varepsilon_p>\varepsilon_{p+1}>0\) for every \(p\). It is obvious that such a choice of the sequence \(\{T_p\}\) is possible for any semicircular domain \(T\). With the aid of the domains \(T_p\), introduce in the space \(A_T\) a countable sequence of norms

\[ \|f\|_p=\max_{(z,w)\in T_p}|f(z,w)|. \]

The topology defined by this set of norms is, clearly, equivalent to the topology of the space \(A_T\) introduced earlier.

Let \(G\) be a linear continuous functional in the space \(A_T\). As is known \((^3)\), a functional continuous in a countably normed space is continuous with respect to one of the norms of this space. Consequently, the functional \(G\) will be continuous with respect to some norm \(\|\ \|_q\). Take in the domain \(T_q\) an arbitrary disk \(\{|w|<R_T(z_0);\ z=z_0\in H_{T_q}\}\) and cover it by the bicylinder

\[ K(z_0)=\left\{|z-z_0|<\frac{\varepsilon_q}{2\sqrt2};\ |w|<R_{T_q}(z_0)+\frac{\varepsilon_q}{2\sqrt2}\right\}. \]

Note that

\[ \begin{aligned} &\inf_{(z,w)\in \dot T}\ \inf_{(z',w')\in K(z_0)} \sqrt{|z-z'|^2+|w-w'|^2} \ge \\ &\ge \inf_{(z,w)\in \dot T}\ \inf_{(z_0,w'')\in T_q} \sqrt{|z-z_0|^2+|w-w''|^2} - \\ &\qquad -\sup_{\substack{(z',w')\in K(z_0)\\ |w''|<R_{T_q}(z_0)}} \inf \sqrt{|z'-z_0|^2+|w'-w''|^2} \ge \varepsilon_q-\frac12\varepsilon_q=\frac12\varepsilon_q, \end{aligned} \]

i.e., for \(z_0\in H_{T_q}\) each bicylinder \(K(z_0)\) lies strictly inside \(T\). The totality of all bicylinders \(K(z)\), \(z\in H_{T_q}\), covers not only the domain \(T_q\), but also its closure \(\overline T_q\). Consequently, one can choose a finite number \(N\) of bicylinders \(K_i=K(z_i)\), \(i=1,2,\ldots,N\), so that

\[ \overline T_q\subset K=\bigcup K_i\subset T. \]

Since the functional \(G\) is continuous with respect to the norm \(\|\ \|_q\), and \(T_q\subset K\), the functional under consideration will also be continuous in the topology of the space \(A_K\). Since \(K=\bigcup K_i\), we have \(A_K=\bigcap A_{K_i}\). Hence, by virtue of a proposition of V. P. Khavin \((({}^4), \text{p. }239)\), it follows that the functional \(G\) is representable in the form

\[ G=\sum_{i=1}^{N}G_i, \]

where each functional \(G_i\) is defined in the space \(A_{K_i}\) and is continuous in the topology of this space. From the remarks made at the beginning of the note it follows that to the functional \(G_i\) there corresponds a sequence of functions \(\{g_{k,i}(z)\}_K\), analytic in the domain

\[ |z-z_i|>\frac{\varepsilon_q}{2\sqrt2}-2\varepsilon>0, \]

such that, for \(f(z,w)\in A_{K_i}\) and \(C_i=|z-z_i|=\varepsilon_q/2\sqrt2-\varepsilon\),

\[ G_i(f)=\sum_{k=0}^{\infty}\int_{C_i} f_k(z)g_k(z)\,dz. \]

Since the distance between any boundary points of the domains \(T\) and \(A_{K_i}\) is greater than \(\frac12 \varepsilon_q\), we have \(R_{K_i}(z)<R_T(z)-\varepsilon_q/2\) for every \(z\in H_{h_i}\). Hence, from (2) it follows that for \(z\in C_i\) the sequence \(g_{k,i}(z)\) satisfies the condition

\[ \varlimsup_{z'\to z}\,\varlimsup_{k\to\infty}\sqrt[k]{|g_{k,i}(z')|} < R_T(z)-\frac12\varepsilon_q . \]

Now let \(f(z,w)\in A_T\). Then the corresponding functions \(f_k(z)\) will be analytic in the domain \(H_T\), containing all the disks \(H_{K_i}\). Take inside the domain \(H_T\) a contour \(C\) such that every disk \(|z-z_i|<\varepsilon_q/2\sqrt{2}\) lies strictly inside \(C\). Then, for all \(k\) and \(i\),

\[ \int_{C_i} f_k(z) g_k(z)\,dz = \int_C f_k(z) g_k(z)\,dz, \]

and, consequently,

\[ G(f)=\sum_{i=1}^{N}G_i(f) = \sum_{i=1}^{N}\sum_{k=0}^{\infty}\int_{C_i} f_k(z)g_{k,i}(z)\,dz = \sum_{k=0}^{\infty}\int_C f_k(z)g_k(z)\,dz . \]

The necessity is proved.

The sufficiency is almost obvious. Indeed, if \(f(z,w)\in A_T\) and the functions \(g_{k,i}(z)\) satisfy the conditions of the theorem, then, as follows from the properties of functions analytic in a semicircular domain, for every \(\eta>0\), beginning with some \(k=k_0(\eta)\), the inequalities

\[ \sqrt[k]{|f_k(z)|}<\frac{1+\eta}{R_T(z)}; \qquad \sqrt[k]{|g_{k,i}(z)|}<R_T(z)-\varepsilon+\eta \quad (z\in C_i) \]

will hold. But then the series

\[ \sum_{k=0}^{\infty} f_k(z)g_{k,i}(z) \]

converge absolutely and uniformly on the corresponding contours \(C_i\), and, consequently, the functional defined by the formula

\[ G(f)=\sum_{k=0}^{\infty}\int_C f_k(z)g_k(z)\,dz = \sum_{i=1}^{N}\sum_{k=0}^{\infty}\int_{C_i} f_k(z)g_{k,i}(z)\,dz \]

is defined on all functions \(f(z,w)\in A_T\) and is continuous.

Finally, to prove the uniqueness of the representation (3), it is enough to consider \(G(w^k f_k(z))\), using the uniqueness of the corresponding representation of a functional in the case of one variable.

In conclusion we note that the result obtained is transferred without difficulty to the case of a number of variables greater than two. In this case, instead of semicircular domains, one considers domains that are domains of uniform convergence of series of the form

\[ \sum_{k=0}^{\infty} w^k f_k(z_1,\ldots,z_n). \]

Received
14 I 1961

References

\(^{1}\) L. A. Aizenberg, B. S. Mityagin, Siberian Mathematical Journal, 1, No. 2 (1960).
\(^{2}\) B. A. Fuks, Theory of Analytic Functions of Several Complex Variables, 1948.
\(^{3}\) I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 2, 1958.
\(^{4}\) V. P. Khavin, DAN, 121, No. 2 (1958).