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_1-2\varepsilon,\ |w|>R_2-2\varepsilon}), the functions (f_k(z)) and (g_k(z)) are analytic, respectively, in the domains ((|z|R_1-2\varepsilon)), and for (|z|=R_1-\varepsilon) satisfy the conditions

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

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

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|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}) 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 A239)), it follows that the functional (G) is representable in the form}). Hence, by virtue of a proposition of V. P. Khavin ((({}^4), \text{p.

[
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)|}