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MATHEMATICS
M. SHIRINBEKOV
CONSTRUCTION OF ENVELOPES OF HOLOMORPHY FOR SEMITUBE DOMAINS
(Presented by Academician N. N. Bogolyubov, June 26, 1964)
1. A (univalent) domain \(T\) over the space of complex variables \(z=(z_1,\ldots,z_n)\), \(w=u+iv\) is called a semitube domain if it can be written in the form
\[ T=[(z,w):(z,u)\in B,\ |v|<\infty], \]
where \(B\) is a domain in the space of variables \(z,u\).
A semitube domain \(T\) is called normal if every line \(z=z^0,\ v=0\) (\(z^0\) is an arbitrary point of \(D\), the projection of the domain \(T\) in the space of variables \(z\)) intersects it in a connected set.
Let \((z^0,u^0)\) be an arbitrary point of the domain \(B\). We assume everywhere that for the point \((z^0,u^0)\) on the set \(E_0\) of points \((z,u^0)\in B\) there exists a (univalent) ball \(\sigma_{z^0}\) with center at the point \((z^0,u^0)\) such that \(\sigma_{z^0}\subset E_0\), and if \(\sigma'_{z^0}\) is another such ball with center at the point \((z^0,u^0)\) on \(E_0\), then \(\sigma'_{z^0}\subset \sigma_{z^0}\).
Consider the set
\[ G=[(z,u): z\in \sigma_{z^0},\ (z,u)\in B]. \]
This open set may be disconnected. By \(K(B;z^0,u^0)\) we denote that connected component of the set \(G\) which contains the point \((z^0,u^0)\). It is clear that \(K(B;z^0,u^0)\) is a univalent domain (since \(\sigma_{z^0}\) is univalent).
A semitube domain \(T\) is called locally normal if, for any point \((z^0,u^0)\in B\), the domain
\[ T_k=[(z,w):(z,u)\in K(B;z^0,u^0),\ |v|<\infty] \]
is normal. It can be shown (analogously to how this was done in the work \((^1)\)) that for every semitube domain \(T\) there exists a unique locally normal envelope \(T^*\), and that it can be written in the form
\[ T^*=[(z,w): z\in D,\ V_1(z)<u<V_2(z),\ |v|<\infty], \]
where \(D\) is a domain without interior branch points.
2. In the work \((^1)\) the envelope of holomorphy \(H(T)\) of a semitube domain \(T\) was constructed in the case when \(z\) is one variable. The purpose of the present note is to construct the envelope of holomorphy \(H(T)\) of a semitube domain \(T\), when \(z=(z_1,z_2,\ldots,z_n)\).
Namely, the following holds.
Theorem. The envelope of holomorphy \(\Gamma(T)\) of any semitube domain \(T\) can be constructed by the following two steps:
1) by constructing the locally normal envelope \(T^*\) of the domain \(T\);
2) by constructing the envelope of holomorphy \(H(T^*)\) of the domain \(T^*\), which coincides with the domain \(\widetilde{T}\),
\[ \widetilde{T}=[(z,w): z\in H(D),\ \widetilde{V}_1(z)<u<\widetilde{V}_2(z),\ |v|<\infty], \]
where \(H(D)\) is the envelope of holomorphy of \(D\), and \(V_1(z)\) and \(-V_2(z)\) are the greatest pluri-
plurisubharmonic minorants, respectively, of the functions
\[ R_1(z)= \begin{cases} V_1(z), & z\in D,\\ +\infty, & z\in H(D)\setminus D; \end{cases} \]
\[ R_2(z)= \begin{cases} -V_2(z), & z\in D,\\ +\infty, & z\in H(D)\setminus D. \end{cases} \]
3. For the proof of the theorem formulated above, we first give several lemmas.
Lemma 1. The functions \(\widetilde V_1(z)\) and \(-\widetilde V_2(z)\), defined in the theorem, always exist.
This lemma, in the case where \(D\) and \(H(D)\) are univalent, was proved in \((^2)\).
Lemma 2. The semitube domain \(T\)
\[ T=[(z,w): z\in D,\ V_1(z)<u<V_2(z),\ |v|<\infty] \]
is a domain of holomorphy if and only if: 1) \(D\) is a domain of holomorphy, 2) \(V_1(z)\) and \(-V_2(z)\) are plurisubharmonic functions in \(D\).
Lemma 3. The envelope of holomorphy \(H(T)\) of any semitube domain \(T\) is a locally normal semitube domain.
Proof. It is clear that a semitube domain possesses automorphisms of the form \(z=z'\), \(w'=w+it\), where \(t\) is a real parameter. Therefore \(H(T)\) also possesses the indicated automorphisms (cf. \((^3)\), p. 224). Consequently, \(H(T)\) is a semitube domain, i.e.
\[ H(T)=[(z,w): (z,u)\in B^*,\ |v|<\infty]. \]
Now suppose, to the contrary, that \(H(T)\) is not a locally normal semitube domain. This means that there is at least one point \((z^0,u^0)\in B^*\) for which the domain
\[ T_k=[(z,w): (z,u)\in K(B^*;z^0,u^0),\ |v|<\infty] \]
is not normal. Hence there is a point \(z^*\subset \sigma_{z^0}\) such that the line \(z=z^*, v=0\) intersects \(T_k\) (or, equivalently, \(K(B^*;z^0,u^0)\)) not in a connected set. On the line \(z=z^*, v=0\) choose a point \((z^*,u')\in K(B^*;z^0,u^0)\) such that the interval with endpoints at \((z^*,u^0)\) and \((z^*,u')\) contains a point not belonging to \(K(B^*;z^0,u^0)\). Without loss of generality one may assume that \(u^0<u'\). Let \(\gamma\) be a rectifiable curve joining, in the domain \(K(B^*;z^0,u^0)\), the points \((z^*,u^0)\), \((z^*,u')\), and let \(\gamma_z\) be its projection onto the ball \(\sigma_{z^0}\), given by the equations
\[ \gamma_z:\ z=z(t),\ u=u^0, \]
\[ \gamma:\ z=z(t),\ u=u(t),\ 0\leq t\leq 1, \]
where \(z(0)=z^*\), \(z(1)=z^*\), \(u(0)=u^0\), \(u(1)=u'\), \(u(t)>u(0)\). Let \(S(t)\), \(0\leq t\leq 1\), be the set of segments whose endpoints lie respectively on the curves \(\gamma\) and \(\gamma_z\) and are parallel to the \(u\)-axis. Since the point \((z^*,u^0)\in K(B^*;z^0,u^0)\), there exists such a \(t_0\) that for \(0\leq t<t_0\) all the segments \(S(t)\subset K(B^*;z^0,u^0)\). We shall now show that there cannot exist such a first \(t_0\), \(0<t_0\leq 1\), for which, for \(0\leq t<t_0\), all the segments \(S(t)\) are contained in \(K(B^*;z^0,u^0)\), but the segment (interval) \(S(t_0)\) contains a point not belonging to \(K(B^*;z^0,u^0)\).
Suppose, to the contrary, that such a \(t_0\) exists. Since \(H(T)\) and \(\sigma_{z^0}\) are domains of holomorphy, it is easy to show that \(T_k\) is also a domain of holomorphy. Then \(-\ln d_{T_K}(z,w)\) is a plurisubharmonic function in \(T_K\) (cf. \((^4)\)), where \(d_{T_K}(z,w)\) is the distance function in the domain \(T_K\). It is clear that \(d_{T_K}(z,w)=d_{T_K}(z,u)\). Consequently, \(-\ln d_{T_K}(z,u)\) for
for each fixed \(z\in \sigma_{z^0}\) is convex in the intersection of the lines \(z=z'\), \(v=0\), \(z'\in\sigma_{z^0}\), with the domain \(T_K\). Hence it follows that for the segments \(S(t)\subset K(B^*; z^0,u^0)\) the following maximum principle is valid:
\[ -\ln d_{T_K}(z,u) = -\ln d_{T_K}(z,u) \]
\[ S(t)\cup \partial S(t) \qquad\qquad \partial S(t) \]
or
\[ \min_{S(t)\cup \partial S(t)} d_{T_K}(z,u) = \min_{\partial S(t)} d_{T_K}(z,u), \tag{1} \]
where \(\partial S(t)\) is the boundary (the endpoints) of the segment \(S(t)\). Since
\[ S(t)\to S(t_0),\qquad \partial S(t)\to \partial S(t_0)\quad \text{as } t\to t_0 \]
and \(d_{T_K}(z,u)\) is a continuous function (cf. \((4)\)), it follows from (1) that
\[ \min_{S(t_0)\cup \partial S(t_0)} d_{T_K}(z,u) = \min_{\partial S(t_0)} d_{T_K}(z,u). \]
Since \(\partial S(t_0)\subset K(B^*; z^0,u^0)\), we have
\[ \min_{\partial S(t_0)} d_{T_K}(z,u)=m>0. \]
Then
\[ \min_{S(t_0)\cup \partial S(t_0)} d_{T_K}(z,u)=m>0 \]
and, consequently,
\[ S(t_0)\cup \partial S(t_0)\subset K(B^*; z^0,u^0). \]
Thus we have obtained a contradiction to the fact that in the interval \(S(t_0)\) there is a point not belonging to \(K(B; z^0,u^0)\). At the same time Lemma 3 is proved.
- Proof of the theorem. From Lemma 3 it follows that the locally normal envelope \(T^*\) of the domain \(T\) is always contained in \(H(T)\), and
\[ H(T^*)=\bigl[(z,w): z\in D^*,\ \varphi_1(z)<u<\varphi_2(z),\ |v|<\infty\bigr], \]
where \(D^*\subset H(D)\), \(\varphi_1(z)\le V_1(z)\), \(\varphi_2(z)\ge V_2(z)\), \(z\in D\).
Since \(H(T^*)\) is a domain of holomorphy, by Lemma 2, \(D^*\) is a domain of holomorphy and \(\varphi_1(z)\), \(-\varphi_2(z)\) are plurisubharmonic functions in \(D^*\). Consequently, \(D^*=H(D)\). We shall show that
\[ \varphi_1(z)=\widetilde V_1(z),\qquad \varphi_2(z)=\widetilde V_2(z),\qquad z\in H(D). \]
It is clear that \(\varphi_1(z)\ge \widetilde V_1(z)\), \(\varphi_2(z)\le \widetilde V_2(z)\), for \(H(T^*)\subset \widetilde T\). On the other hand, since \(\varphi_1(z)\le R_1(z)\), \(\varphi_2(z)\ge -R_2(z)\), we have \(\varphi_1(z)\le \widetilde V_1(z)\), \(\varphi_2(z)\ge \widetilde V_2(z)\), \(z\in H(D)\).
Consequently, \(\varphi_1(z)=\widetilde V_1(z)\), \(\varphi_2(z)=\widetilde V_2(z)\), and therefore \(H(T^*)=\widetilde T\). The theorem is proved.
Taking this opportunity, I express my sincere gratitude to V. S. Vladimirov for numerous discussions of this work.
Received
23 VI 1964
CITED LITERATURE
- H. J. Bremermann, Math. Ann., 127, 5, 406 (1954).
- V. S. Vladimirov, M. Shirinbekov, Ukr. Mat. Zhurn., No. 2 (1963).
- B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, 1962.
- H. J. Bremermann, Trans. Am. Math. Soc., 82, 17 (1956).