Mathematics

L. A. AIZENBERG

SPACES OF FUNCTIONS ANALYTIC IN \((p,q)\)-CIRCULAR DOMAINS

(Presented by Academician V. S. Smirnov on 9 IX 1960)

Let \(Q\) be a bounded simply connected domain of the space \(C^n\) of \(n\) complex variables. Denote by \(A(Q)\) and \(A(\overline Q)\) the spaces of functions analytic, respectively, in the domain \(Q\) and in \(\overline Q\). The topology in the spaces \(A(Q)\) and \(A(\overline Q)\) is introduced in the generally known way (see, for example, \((^1)\)). \(A(Q)\) is a space of type \(F\), \(A(\overline Q)\) a space \((LN^*)\) \((^1)\). It is known that simply connected domains \(Q\) and \(Q_1\) of the space \(C^n\), \(n>1\), generally speaking, are not pseudoconformally mapped onto one another \((^2,\) Ch. VIII). Therefore the question of the isomorphism of the spaces \(A(Q)\) and \(A(Q_1)\) (respectively \(A(\overline Q)\) and \(A(\overline {Q_1})\)) is nontrivial.

An isomorphism of spaces of functions analytic in a hypercone and a bicylinder was in fact obtained earlier by A. A. Temlyakov \((^3)\). In \((^4)\) an isomorphism was proved for spaces of functions analytic in bounded polycircular domains containing their center. Similar results, as became known to us, were obtained by C. Rolewicz \((^{17})\) and C. D. Okunev. In the present note an isomorphism is proved for spaces of functions analytic in bounded circular \((^5,\) p. 109) domains containing their center. For simplicity the exposition is given for the case of two complex variables.

1. Let \(D\) be a bounded \((p,q)\)-circular domain \((^2,\) p. 117) with center at the origin of the space \(C^2\) of complex variables \((w,z)\), where \(p,q\) are relatively prime positive integers. We shall assume that the domain \(D\) contains its center. Without loss of generality one may suppose that the domain \(D\) is complete \((^{2\,5})\). We shall also require that the domain

\[ \widetilde D=\{(e^{i\arg w}|w|^p,\ e^{i\arg z}|z|^q):\ (w,z)\in D\} \]

be such that the length of the segment of a ray connecting the center of the domain \(\widetilde D\) with a boundary point is a continuous function of the position of the ray.

Denote by \(d\omega\) the volume element of four-dimensional space. The integral over the domain \(D\) will, when necessary, be understood as improper \(([^2],\) p. 119).

Lemma 1. Let a function \(f(w,z)\) analytic in the domain \(D\) be representable by a series of polynomials uniformly convergent inside \(D\),

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

where \(P_k(w,z)\) are homogeneous polynomials of degree \(k\) with respect to \(w^{1/p}, z^{1/q}\). For the existence of the integral

\[ I(f)=\int_D |f(w,z)|^2\,d\omega \]

it is necessary and sufficient that the series

\[ \sum_{k=0}^{\infty}\int_D |P_k(w,z)|^2\,d\omega . \]

converge. The sum of this series is equal to \(l(f)\).

This lemma is a generalization of A. Cartan’s well-known theorem for circular domains \((^6)\). The proof is analogous to the proof of A. Cartan’s theorem.

Introduce the following sets: \(L(k,p,q)=\{m:\) there exists an \(n\) such that \(mp+nq=k\}\), \(M(p,q)=\{(k,m): m\in L(k,p,q),\ L(k,p,q)\) is nonempty\(\}\), where \(m,n,k\) are nonnegative integers.

Lemma 2. In the space \(A(D)\) there exists a basis consisting of polynomials in \(w,z\):

\[ \{P_{km}(w,z)\}_{(k,m)\in M(p,q)}, \]

where \(P_{km}(w,z)\) are homogeneous polynomials of degree \(k\) with respect to \(w^{1/p}, z^{1/q}\).

Proof. Taking into account that for \(mp+nq\ne m_1p+n_1q\)

\[ \int_D w^m z^n \bar w^{m_1}\bar z^{n_1}\,d\omega=0, \]

and orthonormalizing, for a given \(k\), the monomials

\[ \{w^m z^n\}_{mp+nq=k}, \]

we obtain a system of polynomials orthonormal in the domain \(D\),

\[ \{P_{km}(w,z)\}_{(k,m)\in M(p,q)}, \tag{1} \]

where \(P_{km}(w,z)\) are homogeneous polynomials of degree \(k\) with respect to \(w^{1/p}, z^{1/q}\). From Lemma 1 and one theorem of A. Cartan \((^6)\) it follows that the system (1) is closed in the space \(L^2(D)\) of functions analytic in the domain \(D\) and square-summable in the domain \(D\). Therefore every function from \(L^2(D)\) expands into a series converging uniformly inside the domain \(D\) \((^2,\) p. 123):

\[ \sum_{(k,m)\in M(p,q)} a_{km} P_{km}(w,z). \tag{2} \]

Put \(D_r=\{(w,z): (w/r^p,\ z/r^q)\in D\}\), \(r>0\). It is easy to see that if the system (1) is orthonormal in the domain \(D\), then the system of polynomials

\[ \left\{\frac{1}{r^{k+p+q}}P_{km}(w,z)\right\}_{(k,m)\in M(p,q)} \]

is orthonormal in the domain \(D_r\).

Let \(f(w,z)\in A(D)\). Since \(f(w,z)\in L^2(D_r)\) for all \(r<1\), there exist coefficients \(a_{km}^{(r)}\) such that the function \(f(w,z)\) is representable, by a series uniformly convergent in the domain \(D_r\),

\[ f(w,z)=\sum_{(k,m)\in M(p,q)} \frac{a_{km}^{(r)}}{r^{k+p+q}}P_{km}(w,z). \]

From the uniqueness of the expansion of \(f(w,z)\) into a multiple power series in a neighborhood of the point \((0,0)\), it follows that for any \(\rho,r<1\)

\[ \frac{a_{km}^{(r)}}{r^{k+p+q}}=\frac{a_{km}^{(\rho)}}{\rho^{k+p+q}}. \]

Consequently, the function \(f(w,z)\) expands in a series of the form (2), uniformly convergent inside the domain \(D\). Lemma 2 is proved.

In what follows we shall assume that the system of polynomials (1) is a basis in the space \(A(D)\). Put

\[ d(D;P_{km})=\sup_{(w,z)\in D}|P_{km}(w,z)|. \]

Theorem 1. In order that the series

\[ \sum_{(k,m)\in M(p,q)} a_{km}P_{km}(w,z) \tag{3} \]

converge uniformly inside the domain \(D\), it is necessary and sufficient that, in the unit bicylinder
\[ E_{1,1}=\{(w,z): |w|<1,\ |z|<1\}, \]
the series

\[ \sum_{m,n=0}^{\infty} a_{km}d(D;P_{km})w^{mp}z^{nq}, \tag{4} \]

converge, where \(k=mp+nq\).

Proof. Necessity. The space \(A(D)\) is nuclear (see, for example, (7)). From Lemma 2 and the theorem of A. S. Dynin—B. S. Mityagin \((^{8,16})\) on bases in nuclear spaces it follows that the series

\[ \sum_{(k,m)\in M(p,q)} |a_{km}|\,d(D_r;P_{km}) \tag{5} \]

converges for all \(r<1\). Hence, and from the equality

\[ d(D_r;P_{km})=r^k d(D;P_{km}), \]

we obtain that the series (4) converges in the bicylinder \(E_{1,1}\).

Sufficiency. From the convergence of the series (4) in the domain \(E_{1,1}\) it follows that the series (5) converges for any \(r<1\), i.e. the series (3) converges in the domain \(D\) “normally” (\((^{2})\), p. 278), and hence also uniformly.

Corollary 1. In order that some sequence \(\{\alpha_{km}\}\) have the property of the sequence \(\{d(D;P_{km})\}\) indicated in Theorem 1, it is necessary and sufficient that the equality

\[ \lim_{k\to\infty}\sqrt[k]{\frac{\overline{d(D;P_{km})}}{|\alpha_{km}|}}=1 \]

hold.

Consider the following problem: to determine the greatest \(r\), \(0\le r\le \infty\), such that the series (3) converges uniformly inside the domain \(D_r\), where \(D\) is a fixed domain.

Corollary 2. The greatest \(r\), \(0\le r\le \infty\), such that the series (3) converges uniformly inside the domain \(D_r\), is determined by the formula (we assume that \(\frac{1}{\infty}=0,\ \frac{1}{0}=\infty\))

\[ \frac{1}{r}=\lim_{k\to\infty}\sqrt[k]{|a_{km}|\,d(D;P_{km})}. \]

Remark 1. Theorem 1 for the case of the hypercone
\[ R=\{(w,z): |w|+|z|<1\} \]
was proved by A. A. Temlyakov \((^{3})\). In the case of complete circular domains, Theorem 1 was obtained in \((^{4})\). Corollary 2 for the case of a double power series and a domain \(D\) that is a hypercone was also indicated by A. A. Temlyakov \((^{3})\).

2. We introduce the countably normed \((^9)\) space \(B(D)\) of sequences

\[ a=\{a_{km}\}_{(k,m)\in M(p,q)} \]

with the system of norms

\[ \|a\|_r=\sum_{(k,m)\in M(p,q)} |a_{km}|\, d(D;P_{km})\, r^k,\qquad r<1. \]

With the aid of Theorem 1 and Banach’s theorem \((^{10},\text{ p. }56)\), it is easy to obtain the following lemma:

Lemma 3. The spaces \(A(D)\) and \(B(D)\) are isomorphic.

Let \(D\) and \(D_1\) be domains satisfying the conditions of item 1.

Theorem 2. The spaces \(A(D)\) and \(A(D_1)\) are isomorphic. The spaces \(A(\overline{D})\) and \(A(\overline{D}_1)\) are also isomorphic.

Remark 2. With the help of the results obtained, one can establish the general form of linear continuous functionals in the space \(A(D)\) (in the space \(A(\overline{D})\)). For bicircular domains this was done in the papers \((^4,{}^{11})\).

Remark 3. Theorem 2 makes it possible to reduce questions of completeness and bases in the spaces \(A(D)\) to analogous questions in the space \(A(E_{1,1})\). Thus the results of the papers \((^{12-15})\) automatically extend to the spaces \(A(D)\).

Remark 4. The entire content of the present note carries over to the case of \((p,q)\)-circular domains, where \(p,q\) are relatively prime integers, \(pq>0\).

I express my sincere gratitude to Prof. A. A. Temlyakov.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
8 IX 1960

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