MATHEMATICS

V. P. KHAVIN

ON THE SPACE OF BOUNDED REGULAR FUNCTIONS

(Presented by Academician V. I. Smirnov, November 17, 1959)

1. Let \(R\) be the extended complex plane; \(F\) a closed set of its points; \(G=R\setminus F\ne \Lambda\); \(A(G)\) the totality of all functions regular in \(G\); \(A(F)\) the totality of all functions regular on \(F\) (two elements \(\varphi_1,\varphi_2\in A(F)\) such that \(\varphi_1(z)=\varphi_2(z)\) for all \(z\) in some neighborhood of \(F\) are identified).

We shall consider only those regular functions which are equal to zero at \(z=\infty\), if this point belongs to their set of definition.

An open set \(g\supset F\), whose boundary \(\partial g\) consists of a finite number of closed rectifiable pairwise nonintersecting Jordan curves, will be called a canonical neighborhood of \(F\). We introduce in \(A(F)\) the natural algebraic operations and topology \((^{1,2})\). Then every element \(\psi\in A(G)\) can be identified with a certain additive functional \(\Phi_\psi\), defined in \(A(F)\) by formula (1):

\[ \Phi_\psi(x)=\frac{1}{2\pi i}\int_{\partial g_x} x(\zeta)\psi(\zeta)\,d\zeta . \tag{1} \]

Here \(\partial g_x\) is the boundary of a canonical neighborhood \(g_x\) of the set \(F\), in the closure of which the function \(x\) is regular.

Many classes of functions regular in \(G\) can be completely characterized as additive functionals, defined on \(A(F)\) and continuous in some topology weaker than the topology of \(A(F)\) \((^3)\). In this note we shall indicate such a characterization of the class \(B(G)\) of all functions regular and bounded in \(G\).

2. Let \(\mu\) be a complex measure defined on the Borel subsets of \(G\); if there exists a set \(P\), \(\overline P=P\subset G\), such that for every Borel \(e\subset F\setminus P\) \(\mu(e)=0\), then we shall say that \(\mu\) is concentrated on \(P\) and that \(P\) is a support of \(\mu\). To every element \(\varphi\in A(F)\) we assign the totality \(M(\varphi)\) of all such measures \(\mu\) with supports in \(G\) for which

\[ \varphi(z)=\int_G \frac{d\mu_\zeta}{\zeta-z} \]

in some neighborhood of \(F\). We now define on \(A(F)\) a seminorm \(|||\cdot|||\), putting

\[ |||\varphi|||=\inf_{\mu\in M(\varphi)}\left\{\int_G |d\mu|\right\}. \tag{2} \]

The linear set \(A(F)\), endowed with the seminorm (2), is transformed in the usual way into a linear topological space (generally speaking, non-Hausdorff), which we shall denote by \(\mathcal L(F)\). The space \(\mathcal L(F)\) is certainly not complete.

Theorem 1. Let \(f\in B(G)\); then the functional \(\Phi_f\) is continuous in \(\mathcal L(F)\), and

\[ \|\Phi_f\|=\sup_{z\in G}|f(z)|. \]

If \(\Phi\) is an additive functional continuous in \(\mathcal L(F)\), then there exists \(f\in B(G)\) such that \(\Phi=\Phi_f\). This function \(f\) is unique.

Let the norm \(\|f\|_{B(G)}=\max_{z\in G}|f(z)|\) be introduced in \(B(G)\). Thus the normed space \(B(G)\) turns out to be conjugate to \(\mathcal L(F)\).

Corollary 1. Let \(S_1(G)\) be the unit ball in \(B(G)\), \(\varphi\in A(F)\). Then

\[ \sup_{f\in S_1(R)}\frac1{2\pi}\left|\int_{\partial g_\varphi}\varphi(\zeta)f(\zeta)\,d\zeta\right| = \inf_{\mu\in M(\varphi)}\int_G |d\mu|. \]

Here \(g_\varphi\) is a canonical neighborhood of \(F\), in whose closure \(\varphi\) is regular. The supremum on the left is always attained.

This corollary is close to the well-known “duality relations” of S. Ya. Khavinson \((^4)\).

Corollary 2. Let \(G\) be a domain, \(\infty\in G\). In order that \(B(G)\) contain a function \(f\) different from identically zero, it is necessary and sufficient that for some \(z_0\in G\)

\[ \|\|\psi_{z_0}\|\|>0 \qquad \left(\psi_{z_0}(z)=\frac1{z-z_0}\right). \]

Remark. Let \(\mathcal L_1(F)\) be the totality of all measures with supports lying in \(G\). For \(\mu\in\mathcal L_1(F)\) put

\[ \|\mu\|^1=\int_G |d\mu|. \]

Let \(\mathfrak K(F)\) be the subset of \(\mathcal L_1(F)\) consisting of all measures \(\mu\) such that

\[ \int_G \frac{d\mu_\zeta}{\zeta-z}\equiv 0 \]

in some neighborhood of \(F\). Then the quotient space \(\mathcal L_1(F)/\mathfrak K(F)\) is isomorphic to \(\mathcal L(F)\). In some simple cases one can construct a complete space whose conjugate is isomorphic to \(B(G)\). Thus, for example, if \(G\) is the unit disk, then such a space will be \(L/H_1\). Here \(L\) is the space of all functions defined on the unit circle and summable (with the usual norm). For the definition of the set \(H_1\), see \((^5)\). Hence it is not difficult to derive the following fact.

Let \(a_1,a_2,\ldots,a_n,\ldots\) be a sequence of complex numbers,

\[ |a_1|\le |a_2|\le \cdots \le |a_n|\le \cdots <1. \]

The following assertions are equivalent:

\[ 1)\ \sum_1^\infty (1-|a_n|)=\infty; \]

2) whatever \(\varphi\in L\) and \(\varepsilon>0\) may be, there is a fraction

\[ \sum \frac{\lambda_k}{z-a_k} \]

and \(\psi\in H_1\) such that

\[ \int_0^{2\pi}\left|\varphi(e^{i\theta})-\psi(e^{i\theta})-\sum\frac{\lambda_k}{e^{i\theta}-a_k}\right|\,d\theta<\varepsilon. \]

Here \(\lambda_1,\lambda_2,\ldots\) is a sequence of complex numbers, only finitely many of which are different from zero.

1. Let \(G\) be a domain, \(\infty\in G\). The number

\[ \Omega(F)=\sup_{f\in S_1(G)}\left|\int_{\partial g} f(\zeta)\,d\zeta\right| \]

is called the analytic capacity of \(F\). The importance of this concept in approximation theory was discovered by A. G. Vitushkin (see, for example, \((^6)\)).

Theorem 2.

\[ \Omega(F)=\inf_{\mu\in M(1)}\int_G |d\mu|. \]

Here \(1(z)\equiv 1\).

We shall say that \(F\) has finite enclosure if there exists a number \(l(F)\) such that every neighborhood of \(F\) contains a canonical neighborhood whose boundary length does not exceed \(l(F)\).

If \(F\) has finite enclosure, then any \(f\in B(G)\) has the form

\[ f(z)=\int_F \frac{d\nu_t}{t-z}\quad (z\in G), \]

where \(\nu\) is a measure defined on the Borel subsets of \(F\).

The proof follows easily from the theorems of our paper \((^7)\).

It follows from this remark that the analytic capacity is the exact upper bound of the “masses” \(\left|\int_F d\nu\right|\) that can be placed on \(F\) so that the “potential” \(\int_F \frac{d\nu}{t-z}\) at any point \(z\in G\) does not exceed one in modulus.

Theorem 2 shows that, on the other hand, \(\Omega(F)\) is the exact lower bound of the “masses” that can be placed outside \(F\) so that the “potential” they create in a neighborhood of \(F\) is equal to one. In particular, \(\Omega(F)=0\) if and only if an arbitrarily small “mass” situated outside \(F\) is capable of producing on \(F\) a uniformly large “potential.”

Theorem 3. If \(\Omega(F)=0\), then, whatever \(\varphi\in A(F)\) and \(\varepsilon>0\) may be, there exists a measure \(\mu\in \mathcal L_1(F)\) such that
\[ \varphi(z)=\int_G \frac{d\mu}{\xi-z} \]
for all \(z\) in some neighborhood of \(F\), and
\[ \int_G |d\mu|<\varepsilon . \]

Theorem 4. 1) If \(\Omega(F)=0\), then, whatever the continuous function \(\lambda\) on \(F\) and the number \(\varepsilon>0\) may be, there exists a fraction
\[ R(z)=\sum_{k=1}^{n}\frac{\lambda_k}{z-a_k} \]
\((a_1,a_2,\ldots,a_n\in G)\) such that
\[ \max_{z\in F}|R(z)-\lambda(z)|<\varepsilon,\qquad \sum_{k=1}^{n}|\lambda_k|<\varepsilon . \]

2) If \(F\) has a finite cover and if, for every \(\varepsilon>0\), one can find a fraction
\[ R(z)=\sum_{k=1}^{n}\frac{\lambda_k}{z-a_k}\quad (a_1,a_2,\ldots,a_n\in G) \]
such that
\[ \max_{z\in F}|1-R(z)|<\varepsilon,\qquad \sum_{k=1}^{n}|\lambda_k|<\varepsilon, \]
then \(\Omega(F)=0\).*

1. Theorem 5. Let \(F\) contain a nondegenerate continuum or, being a discontinuum, have a finite cover and positive analytic capacity. Then for every \(\varphi\in A(F)\), \(\varphi\ne0\), one can find an element \(f\in B(G)\) such that \(\Phi_f(\varphi)\ne0\).

Corollary 1. Let \(F\) have a finite cover. Then the following alternative is valid: either \(B(G)\) contains no elements different from zero, or \(B(G)\) is dense in \(A(G)\) (in the sense of uniform convergence on compact sets lying in \(G\)).

Corollary 2. Under the conditions of Corollary 1, either (2) is a norm on \(A(F)\), or \(\|\varphi\|=0\) for every \(\varphi\in A(F)\).

Corollary 3. Under the conditions of Corollary 1, either \(\mathscr K(F)\) is dense in \(\mathcal L_1(F)\), or \(\mathscr K(F)\) is closed in \(\mathcal L_1(F)\).

The proof of Theorem 3 is based on the results of §§ 2 and 3 and on the following obvious lemma. Let \(F\) be a bounded discontinuum; let \(\lambda\) be a function regular on \(F\), \(\lambda\ne0\), and vanishing on \(F\) only at the points \(a_1,a_2,\ldots,a_s\); let the multiplicity of the zero \(a_j\) be \(k_j\) \((j=1,2,\ldots,s)\). If \(\xi\) is a function regular on \(F\) having at the point \(a_j\) a zero of multiplicity not less than \(R_j\) \((j=1,2,\ldots,s)\), then for every \(\varepsilon>0\) one can find closed sets \(F_1,F_2,\ldots,F_N\) and numbers \(c_1,c_2,\ldots,c_N\) such that
\[ \bigcup_{k=1}^{n} F_k=F,\qquad F_{k'}\cap F_{k''}=\Lambda\ (k'\ne k''),\qquad \max_{z\in F}\left|\sum_{k=1}^{n} c_k[\lambda]^k(z)-\xi(z)\right|<\varepsilon . \]
Here
\[ [\lambda]^k(z)= \begin{cases} 0, & z\in F\setminus F_k,\\ \lambda(z), & z\in F_k. \end{cases} \]

* S. Ya. Khavinson kindly informed me that all the facts stated in § 3 had been found by him earlier from somewhat different considerations.

In conclusion, I express my sincere gratitude to S. Ya. Khavinson, whose advice guided me in writing this note.

Leningrad State University
named after A. A. Zhdanov

Received
16 X 1959

References

1. G. Köthe, J. reine u. angew. Math., 191, 30 (1953).
2. J. Sebastião e Silva, Matematika, 1, 60 (1957).
3. V. P. Khavin, Applications of functional analysis to certain problems in the theory of analytic functions, Dissertation, LSU, 1958.
4. S. Ya. Khavinson, Matem. sborn., 36 (78), 3, 445 (1955).
5. I. I. Privalov, Boundary Properties of Analytic Functions, Moscow–Leningrad, 1950.
6. A. G. Vitushkin, DAN, 123, No. 5, 778 (1958); 123, No. 6, 959 (1958); 128, No. 1 (1959).
7. V. P. Khavin, Vestnik LGU, 1, No. 1, 66 (1958).