Abstract Generated abstract
This note extends properties of multiple power series to multiple Hartogs series in bounded complete multiple Hartogs domains. It establishes convergence criteria and coefficient estimates using the quantities defined by suprema of monomials over the domain, then proves isomorphisms between spaces of analytic functions on such domains and corresponding countably normed spaces of coefficient functions. The paper also constructs a continuous linear operator represented through derivatives and an integral kernel, and applies it to characterize when systems of analytic functions form bases or complete systems via associated systems on the projection domain.
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MATHEMATICS
A. F. SHADROV
SPACES OF FUNCTIONS ANALYTIC IN MULTIPLE HARTOGS DOMAINS
(Presented by Academician M. A. Lavrent’ev on 4 IV 1964)
In the present note, certain properties of multiple power series \((^1)\) are extended to multiple Hartogs series; with their aid an isomorphism is established for spaces of functions analytic in bounded multiple Hartogs domains. Then on these spaces a linear operator \(L(f)\) is constructed, analogous to a linear functional on \(E_R^{(n)}\) \((^2)\). The operator is applied to questions of basicity and completeness.
- Let \(C^n\) be the space of \(n\) complex variables \(z_1,\ldots,z_n\). A covering domain \(D\) over the space \(C^n\) having the property that, together with each point \(z_0=(z_1^{(0)},\ldots,z_n^{(0)})\), it also contains all points \(z=(z_1,\ldots,z_n)\) with coordinates
\[ z_1=(z_1^{(0)}-a_1)e^{i\theta}+a_1,\ldots,\ z_m=(z_m^{(0)}-a_m)e^{i\theta}+a_m;\ z_{m+1}=z_{m+1}^{(0)},\ldots,\ z_n=z_n^{(0)}, \]
where \(0\leq \theta \leq 2\pi\), will be called an \(m\)-fold Hartogs domain with planes of symmetry \(z_1=a_1,\ldots,z_m=a_m\). In the case \(m=1\) the domain \(D\) is called ((\(^3\), p. 109) a Hartogs domain. In the case of two variables a Hartogs domain is also called a semicircular domain ((\(^4\), p. 98; (\(^5\), p. 228).
If, together with each point \(z_0\) of the multiple Hartogs domain \(D\), it also contains all points of the closed polycylinder
\[ \{\,|z_j-a_j|\leq |z_j^{(0)}-a_j|,\ j=1,\ldots,m;\ z_{m+1}=z_{m+1}^{(0)},\ldots,\ z_n=z_n^{(0)}\,\}, \]
then the domain \(D\) is called complete.
To simplify notation we introduce the designations:
\[ w=(z_1,\ldots,z_m),\quad z=(z_{m+1},\ldots,z_n),\quad k=(k_1,\ldots,k_m),\quad \|k\|=k_1+\cdots+k_m, \]
\[ w^k=z_1^{k_1}\cdots z_m^{k_m},\quad (w-a)^k=(z_1-a_1)^{k_1}\cdots (z_m-a_m)^{k_m}. \]
Analogously to the case of a function of two variables ((\(^4\), p. 100), it is established that the domain of convergence of the \(m\)-fold Hartogs series
\[ f(w,z)=\sum_{k=0}^{\infty} f_k(z)(w-a)^k \tag{1} \]
is a complete \(m\)-fold Hartogs domain with planes of symmetry \(z_1=a_1,\ldots,z_m=a_m\).
In what follows we shall place the origin of coordinates in the planes of symmetry of the Hartogs domain \(D\), i.e., consider a complete \(m\)-fold Hartogs domain \(D\) with planes of symmetry \(z_1=0,\ldots,z_m=0\), which is the domain of convergence of the \(m\)-fold Hartogs series
\[ f(w,z)=\sum_{k=0}^{\infty} f_k(z)w^k. \tag{2} \]
The coordinates of all points of the domain \(D\) are determined by the inequalities
\(|z_1|<R_1(z),\ldots, |z_m|<R_m(z)\), where \(R_1(z),\ldots,R_m(z)\) are nonnegative functions defined in the projection \(H_D\) of the domain \(D\) onto the space of the variables \(z\). Therefore the complete multiple Hartogs domain \(D\) with planes of symmetry \(z_1=0,\ldots,z_m=0\) will be denoted as follows:
\[ D\{\, |z_j|<R_j(z),\ j=1,\ldots,m;\ z\in H_D\,\}. \]
The functions \(f_k(z)\) are regular in the domain \(H_D\) and satisfy the condition
\[ \varlimsup_{z'\to z}\ \varlimsup_{\|k\|\to\infty} \sqrt[\|k\|]{\,|f_k(z')|\, R_1^{k_1}(z)\cdots R_m^{k_m}(z)}\leqslant 1. \]
2. Analogously to (1), let us introduce into consideration the sequence
\[ d_k(D)=\sup_{(w,z)\in D}|w^k|,\qquad k_j=0,1,\ldots;\ j=1,\ldots,m. \]
Theorem 1. If the function \(f(w,z)\) is regular in the closed domain \(\overline D\) and
\[ f(w,z)=\sum_{k=0}^{\infty} f_k(z)\,w^k, \]
then
\[ |f_k(z)|\leqslant \frac{\max_{(w,z)\in \overline D}|f(w,z)|}{d_k(D)} . \tag{3} \]
The theorem is proved analogously to Theorem 3.8 from \((^{5})\), p. 62.
Theorem 2. If the series (2) converges in the bounded complete multiple Hartogs domain \(D\), then for every subdomain \(D_0\) of it, \(\overline D_0\subset D\), the series
\[ \sum_{k=0}^{\infty}\sup_{D_0}|f_k(z)|\,d_k(D_0) \tag{4} \]
converges.
Proof. Consider the domain \(D_r\{(w,z):(\frac{w}{r},z)\in D\}\), where \(0<r<\infty\). Choose numbers \(r_1\) and \(r_2\), \(0<r_1<r_2<1\), so that \(\overline D_0\subset D_{r_1}\subset D_{r_2}\subset D\). Since \(d_k(D_{r_1})=r_1^{\|k\|}d_k(D)\), \(d_k(D_{r_2})=r_2^{\|k\|}d_k(D)\), we have
\[ \sum_{k=0}^{\infty}\sup_{D_0}|f_k(z)|\,d_k(D_0) \leqslant \sum_{k=0}^{\infty}\sup_{D_{r_1}}|f_k(z)|\,d_k(D_{r_1}) \leqslant \]
\[ \leqslant M\sum_{k=0}^{\infty}\frac{d_k(D_{r_1})}{d_k(D_{r_2})} = M\sum_{k=0}^{\infty}\left(\frac{r_1}{r_2}\right)^{\|k\|} = M\sum_{j=0}^{\infty} \frac{(n+j-1)!}{j!(n-1)!} \left(\frac{r_1}{r_2}\right)^j <\infty, \]
where \(M=\max_{\overline D_{r_2}}|f(w,z)|\).
Corollary. If the series (2) converges in the bounded complete multiple Hartogs domain \(D\), then in every subdomain \(D_0\) of it, \(\overline D_0\subset D\), the series
\[ \sum_{k=0}^{\infty} f_k(z)d_k(D_0) \]
converges uniformly.
Theorem 3. For convergence of the series (2) in the bounded complete multiple Hartogs domain
\[ D\{\, |z_1|<R_1(z),\ldots, |z_m|<R_m(z);\ z\in H_D\,\} \]
it is necessary and sufficient that the series
\[ \sum_{k=0}^{\infty}\sup_{z\in H_D}|f_k(z)|\,d_k(D)\,w^k \tag{5} \]
converge in the domain \(D^{(1)}\{\, |z_1|<1,\ldots, |z_m|<1;\ z\in H_D\,\}\).
Proof. Consider the domain \(D_r,\; 0<r<1\). Suppose that the series (5) converges in the domain \(D^{(1)}\). This means that, for every \(r<1\), the series
\[ \sum_{k=0}^{\infty}\sup_{D_r}|f_k(z)|\,d_k(D)\,r^{\|k\|} \]
converges, or the series
\[ \sum_{k=0}^{\infty}\sup_{D_r}|f_k(z)|\,d_k(D_r). \]
converges. But
\[ |f_k(z)w^k|\leq \sup_{D_r}|f_k(z)|\,d_k(D_r), \]
and therefore the series (2) converges in the domain \(D_r\) for every \(r<1\), and consequently it also converges in the domain \(D\).
The converse assertion follows from Theorem 2.
Analogously to Theorem 3 one proves
Theorem 4. For the convergence of the series (2) in the domain \(D\) it is necessary and sufficient that the series
\[ \sum_{k=0}^{\infty} f_k(z)d_k(D)w^k \]
converge in the domain \(D^{(1)}\).
3. Consider the space \(A(D)\) of functions analytic in the complete multiple Hartogs domain \(D\), with the topology determined by uniform convergence of the functions \(f(w,z)\in A(D)\) in each domain \(D_0,\; D_0\Subset D\).
Introduce the countably normed space \(B(D)\) of sequences of functions \(a=\{f_k(z)\}\) with the system of norms
\[ \|a\|_r=\sum_{k=0}^{\infty}\sup_{z\in H_D}|f_k(z)|\,d_k(D)\,r^{\|k\|},\qquad 0<r<1. \]
Next consider the space \(A(\overline D)\) of functions analytic in the closed domain \(\overline D\), and the countably normed space \(B(\overline D)\) of sequences of functions \(a=\{f_k(z)\}\) such that, for some \(r>1\), depending, generally speaking, on the sequence,
\[ \|a\|_r=\sum_{k=0}^{\infty}\sup_{z\in H_D}|f_k(z)|\,d_k(D)\,r^{\|k\|}<\infty. \]
The topologies of \(A(\overline D)\) and \(B(\overline D)\) are defined analogously to \((^1)\).
Theorem 5. If \(D\) is a bounded complete multiple Hartogs domain, then the spaces \(A(D)\) and \(B(D)\), and also \(A(\overline D)\) and \(B(\overline D)\), are isomorphic.
Proof. Theorems 3 and 4 put each function \(f(w,z)\in A(D)\) into correspondence with a sequence of functions \(\{f_k(z)\}\), and conversely. This correspondence is continuous in both directions with respect to the topologies of the spaces \(A(D)\) and \(B(D)\), since for any numbers \(r\) and \(r_1,\; 0<r<r_1<1\), the inequalities
\[ \max_{D_r}|f(w,z)| \leq \sum_{k=0}^{\infty}\sup_{D_r}|f_k(z)|\,d_k(D_r) = \sum_{k=0}^{\infty}\sup_{D_r}|f_k(z)|\,d_k(D)\,r^{\|k\|} = \|a\|_r, \]
\[ \|a\|_r \leq \sum_{k=0}^{\infty} \frac{d_k(D)r^{\|k\|}}{d_k(D_{r_1})} \max_{D_{r_1}}|f(w,z)| = \left(\max_{D_{r_1}}|f(w,z)|\right) \sum_{k=0}^{\infty}\left(\frac{r}{r_1}\right)^{\|k\|}. \]
are valid.
The isomorphism of the spaces \(A(\overline D)\) and \(B(\overline D)\) is proved analogously.
Theorem 6. If \(D\{|z_j|<R_j(z),\; j=1,\ldots,m;\; z\in H_D\}\) and \(D_1\{|z_j|<R_j^{(1)}(z),\; j=1,\ldots,m;\; z\in H_{D_1}\}\) are arbitrary bounded complete multiple Hartogs domains having the same projections \(H_D=H_{D_1}\), then the spaces \(A(D)\) and \(A(D_1)\) are isomorphic. The spaces \(A(\overline D)\) and \(A(\overline{D_1})\) are also isomorphic.
Proof. By Theorem 3, \(B(D)\) is isomorphic to \(B(D^{(1)})\). Hence, from Theorem 5, it follows that \(A(D)\) is isomorphic to \(A(D^{(1)})\). Similarly, \(A(D_1)\) is isomorphic to \(A(D^{(1)})\). Consequently, \(A(D)\) and \(A(D_1)\) are isomorphic.
The spaces \(A(\overline D)\) and \(A(\overline{D_1})\) are considered analogously.
Remark. If the domain \(D\) is univalent, then the space \(A(D)\) is isomorphic to the space \(A(E_1)\), where \(E_1\{|z_1|<1,\ldots,|z_m|<1\}\), which follows as a special case also from Theorem 2 \((^6)\).
4. Theorem 7. In order that an operator \(L(f)\), defined on \(A(D)\) and taking values in \(A(H_D)\), be linear and continuous, it is necessary
and it is sufficient that
\[ L(f)=\sum_{k=0}^{\infty} l_k \frac{1}{k!}\frac{\partial^k f(0,z)}{\partial w^k}, \tag{6} \]
where
\[ \varlimsup_{|k|\to\infty}\sqrt[|k|]{|l_k|} = l < \varlimsup_{|k|\to\infty}\sqrt[|k|]{a_k(D)} \left( k! = k_1!\cdots k_m!,\quad \frac{\partial^k}{\partial w^k} = \frac{\partial^{k_1+\cdots+k_m}}{\partial z_1^{k_1}\cdots \partial z_m^{k_m}} \right). \]
Consider the function
\[ \varphi(w)=\sum_{k=0}^{\infty}\frac{l_k}{w^{k+1}}. \]
It is analytic in the domain \(\{|z_j|>l,\ j=1,\ldots,m\}\), and moreover \(\varphi(w)=0\) if, for some \(j\), \(1\le j\le m\), \(z_j=\infty\). With the aid of this function the operator \(L(f)\) can be represented in the form
\[ L(f)=\frac{1}{(2\pi i)^m} \int_{|z_1|=r_1}\cdots \int_{|z_m|=r_m} f(w,z)\varphi(w)\,dw, \]
where \(l<r_j<R_j(z)\), \(j=1,\ldots,m\).
Theorem 8. In order that the system
\[ \left\{ f_l(w,z)=\sum_{k=0}^{\infty} f_{k,l}(z)w^k \right\} \tag{7} \]
be a basis in \(A(D)\), it is necessary and sufficient that the system
\[ \{\varphi_l(z)=L(f_l(w,z))\} \tag{8} \]
be a basis in \(A(H_D)\).
Theorem 9. In order that the system (7) be complete in \(A(D)\), it is necessary and sufficient that the system (8) be complete in \(A(H_D)\).
With the aid of Theorems 8 and 9, theorems analogous to Theorems 4—11, established in [7] for the case \(m=1\), are proved; moreover, the conditions of these theorems can now be formulated as necessary and sufficient.
Kuibyshev Civil Engineering Institute
named after A. I. Mikoyan
Received
2 IV 1964
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