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Lemma 1.12. A subset \( S \subset C\left( D\right) \left( {\text{or} \subset \mathcal{O}\left( D\right) }\right) \) is bounded if and only if for every compact \( K \subset D \) one has\n\n\[ \sup \left\{ {{\left| f\right| }_{K}, f \in S}\right\} < \infty . \]\n | The proof is left to the reader. | No |
Theorem 1.13. A subset \( S \subset \mathcal{O}\left( D\right) \) is compact if and only if \( S \) is closed and bounded. | Proof. As the classical proof for \( n = 1 \) generalizes to \( n > 1 \), we only give an outline. Since \( \mathcal{O}\left( D\right) \) is complete metrizable, a closed set \( S \subset \mathcal{O}\left( D\right) \) is compact if and only if every sequence \( \left\{ {f}_{j}\right\} \subset S \) has a convergent subs... | No |
Lemma 1.15. Suppose \( {c}_{v} \in \mathbb{C} \) for \( v \in {\mathbb{N}}^{n} \) and that for some \( w \in {\mathbb{C}}^{n} \)\n\n(1.31)\n\n\[ \mathop{\sup }\limits_{{v \in {\mathbb{N}}^{n}}}\left| {{c}_{v}{w}^{v}}\right| = M < \infty . \]\n\nLet \( r = \tau \left( w\right) = \left( {\left| {w}_{1}\right| ,\ldots ,\l... | Proof. Given \( K \subset \subset P\left( {0, r\right) \), choose \( 0 < \lambda < 1 \), such that \( K \subset P\left( {0,{\lambda r}}\right) \) . For \( z \in P\left( {0,{\lambda r}}\right) \) one obtains from (1.31) that\n\n\[ \left| {{c}_{v}{z}^{v}}\right| \leq \left| {{c}_{v}{w}^{v}}\right| {\lambda }^{\left| v\ri... | Yes |
Theorem 1.17. A power series \( f\left( z\right) = \sum {c}_{v}{z}^{v} \) with nonempty domain of convergence \( \Omega \) defines a holomorphic function \( f \in \mathcal{O}\left( \Omega \right) \) . Moreover, for \( \alpha \in {\mathbb{N}}^{n} \), the series of derivatives \( \sum {c}_{v}\left( {{D}^{\alpha }{z}^{v}}... | Proof. We fix a bijection \( \sigma : \mathbb{N} \rightarrow {\mathbb{N}}^{n} \) . Then\n\n\[ \nf\left( z\right) = \mathop{\lim }\limits_{{k \rightarrow 0}}\mathop{\sum }\limits_{{j = 0}}^{k}{c}_{\sigma \left( j\right) }{z}^{\sigma \left( j\right) }\n\]\n\ncompactly on \( \Omega \) . Since the partial sums are holomorp... | Yes |
Theorem 1.18. Let \( f \in \mathcal{O}\left( {P\left( {a, r}\right) }\right) \) . Then the Taylor series of \( f \) at a converges to \( f \) on \( P\left( {a, r}\right) \), that is,\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{v \in {\mathbb{N}}^{n}}}\frac{{D}^{v}f\left( a\right) }{v!}{\left( z - a\right) }^{v}\;\... | Proof. In the Cauchy integral formula (1.15), applied to \( z \in P\left( {a,\rho }\right) \subset \subset P\left( {a, r}\right) \) , one expands \( {\left( \zeta - z\right) }^{-1} = {\left( {\zeta }_{1} - {z}_{1}\right) }^{-1}\ldots {\left( {\zeta }_{n} - {z}_{n}\right) }^{-1} \) into a multiple geometric series\n\n(1... | Yes |
Theorem 1.19. Let \( D \subset {\mathbb{C}}^{n} \) be connected. If \( f \in \mathcal{O}\left( D\right) \) and there is \( a \in D \), such that \( {D}^{\alpha }f\left( a\right) = 0 \) for all \( \alpha \in {\mathbb{N}}^{n} \), then \( f\left( z\right) = 0 \) for \( z \in D \) . In particular, if there is a nonempty op... | Proof. Theorem 1.18 implies that the set \( \Omega = \left\{ {z \in D : {D}^{\alpha }f\left( z\right) = 0}\right. \) for all \( \left. {\alpha \in {\mathbb{N}}^{n}}\right\} \) is open. By continuity of \( {D}^{\alpha }f,\Omega \) is also closed, and since the hypothesis says that \( \Omega \neq \varnothing \), the conn... | Yes |
Theorem 1.20. Let \( D \) be connected. Then \( \mathcal{O}\left( D\right) \) is an integral domain. | Proof. Suppose \( f, g \in \mathcal{O}\left( D\right) \) and \( f\left( z\right) \cdot g\left( z\right) = 0 \) for \( z \in D \) . If \( f ≢ 0 \), there is \( a \in D \) with \( f\left( a\right) \neq 0 \), and hence \( f\left( z\right) \neq 0 \) in a neighborhood \( U \) of \( a \) . But then \( g\left( z\right) = 0 \)... | Yes |
Theorem 1.21. Let \( D \) be connected and suppose \( f \in \mathcal{O}\left( D\right) \) is not constant. Then \( f\left( \Omega \right) \) is open for any open set \( \Omega \subset D \) . | Proof. It is enough to show that for any ball \( B\left( {a, r}\right) \subset D, f\left( {B\left( {a, r}\right) }\right) \) is a neighborhood of \( f\left( a\right) \) . Theorem 1.19 implies that \( {\left. f\right| }_{B\left( {a, r}\right) } \) is not constant, otherwise \( f \) would have to be constant on \( D \) .... | Yes |
Lemma 2.1. If \( D \subset {\mathbb{C}}^{n} \) and \( F : D \rightarrow {\mathbb{C}}^{n} \) is holomorphic, then\n\n\[ \det {J}_{\mathbb{R}}F\left( z\right) = {\left| \det {F}^{\prime }\left( z\right) \right| }^{2} \geq 0 \]\n\nfor \( z \in D \) . | Proof. After a permutation of the rows and columns one can write\n\n\[ \det {J}_{\mathbb{R}}F = \det \left\lbrack \begin{matrix} \left( \frac{\partial {u}_{k}}{\partial {x}_{j}}\right) & \vdots & \left( \frac{\partial {u}_{k}}{\partial {y}_{j}}\right) \\ \cdots \cdots \cdots & \cdots \cdots \cdots & \\ \left( \frac{\pa... | Yes |
Lemma 2.2. Let \( D \subset {\mathbb{C}}^{n} \) and \( \Omega \subset {\mathbb{C}}^{m} \) be open sets. If \( F = \left( {{f}_{1},\ldots ,{f}_{m}}\right) : D \rightarrow \Omega \) is holomorphic and \( g \in \mathcal{O}\left( \Omega \right) \), then \( g \circ F \in \mathcal{O}\left( D\right) \) ; moreover, for \( a \i... | Proof. We give two proofs of this result. The first one is based on power series, while the second uses a complex version of the real chain rule, which is useful in other contexts as well.\n\nSuppose \( a \in D, F\left( a\right) = b \in \Omega \) . Choose a polydisc \( P\left( {b,\varepsilon }\right) \subset \subset \O... | Yes |
Theorem 2.4. Let \( D \subset {\mathbb{C}}^{n} \) and let \( F = \left( {{f}_{1},\ldots ,{f}_{m}}\right) : D \rightarrow {\mathbb{C}}^{m} \) be holomorphic. Suppose \( m \leq n, F\left( a\right) = 0 \) for some \( a \in D \), and\n\n\[ \det {\left\lbrack \frac{\partial {f}_{k}}{\partial {z}_{j}}\left( a\right) \right\r... | Proof. Lemma 2.1, applied to the map \( \widetilde{F} \), defined by \( \widetilde{F}\left( {z}^{\prime \prime }\right) = F\left( {{a}^{\prime },{z}^{\prime \prime }}\right) \) in a neighborhood of \( {a}^{\prime \prime } \), shows that \( \det {J}_{\mathbb{R}}\widetilde{F}\left( {a}^{\prime \prime }\right) \neq 0 \) .... | Yes |
Theorem 2.5. Suppose \( D \subset {\mathbb{C}}^{n} \) and the holomorphic map \( F : D \rightarrow {\mathbb{C}}^{n} \) is nonsingular at a (i.e., \( \det {F}^{\prime }\left( a\right) \neq 0 \) ). Then there are open neighborhoods \( U \) of a and \( W \) of \( b = F\left( a\right) \), such that \( {\left. F\right| }_{U... | Proof. We introduce the map \( G\left( {w, z}\right) = F\left( z\right) - w \) from \( {\mathbb{C}}^{n} \times D \) into \( {\mathbb{C}}^{n} \) . By hypothesis, \( G\left( {b, a}\right) = 0 \) and\n\n\[ \det {\left\lbrack \frac{\partial {g}_{k}}{\partial {z}_{j}}\right\rbrack }_{\begin{matrix} {k = 1,\ldots, n} \\ {j =... | Yes |
Corollary 2.6. Suppose \( D \subset {\mathbb{C}}^{n} \) and \( F : D \rightarrow {\mathbb{C}}^{m} \) is holomorphic and nonsingular at \( a \in D \) . If \( m \geq n \), then there is a neighborhood \( U \) of a, such that \( {\left. F\right| }_{U} \) is injective. | Proof. Since rank \( {F}^{\prime }\left( a\right) = n \), after renumbering the components of \( F = \) \( \left( {{f}_{1},\ldots ,{f}_{m}}\right) \), one can assume that \( \widetilde{F} = \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) is nonsingular at \( a \) . Theorem 2.5 now implies that \( \widetilde{F} \), and hence... | Yes |
Theorem 2.8. A subset \( M \) of \( {\mathbb{C}}^{n} \) is a complex submanifold if and only if for every \( P \in M \) there are a neighborhood \( U \) of \( P \), an open ball \( {B}^{\left( k\right) }\left( {a,\varepsilon }\right) \subset {\mathbb{C}}^{k} \), and a nonsingular holomorphic map \( H : {B}^{\left( k\ri... | Proof. Suppose first that \( M \) is a complex submanifold, and let \( w = \left( {{w}_{1},\ldots }\right. \) , \( \left. {w}_{n}\right) : U \rightarrow W \) be a coordinate system on the neighborhood \( U \) of \( P \) which satisfies (2.7), with \( k = \dim {M}_{P} \) . By shrinking \( W \) and \( U \) we may assume ... | Yes |
Theorem 2.9. Let \( D \subset {\mathbb{C}}^{n} \) and suppose \( F : D \rightarrow {\mathbb{C}}^{m} \) is nonsingular. Then for every \( a \in D \) the level set\n\n\[ \n{L}_{a}\left( F\right) = \{ z \in D : F\left( z\right) = F\left( a\right) \} \n\]\n\nis a complex submanifold of dimension \( \max \left( {0, n - m}\r... | Proof. Let \( P \in {L}_{a}\left( F\right) \) . If \( m \geq n \), then \( P \) is isolated in \( {L}_{a}\left( F\right) \) by Corollary 2.6, and the theorem is proved. We now assume \( m < n \) . After replacing \( F \) by \( {F}^{\# } = F - F\left( a\right) \) and renumbering the coordinates, the hypotheses of Theore... | Yes |
Theorem 2.12. Let \( M \subset {\mathbb{C}}^{n} \) be a complex submanifold and suppose that \( M \) is compact. Then \( M \) consists of finitely many points. | Proof. It is enough to show: if the given \( M \) is also connected, then \( M \) is a single point. For each \( j = 1,\ldots, n \), the restriction to \( M \) of the coordinate function \( {z}_{j} \) is a holomorphic function on \( M \) . Since \( {z}_{j}\left( M\right) \subset \mathbb{C} \) is compact, the open mappi... | Yes |
Lemma 2.13. Let \( f \) be a holomorphic function on the connected region \( D \) in \( {\mathbb{C}}^{n} \) . Suppose \( Z\left( {f, D}\right) \neq \varnothing \) and \( f ≢ 0 \) . Then there exists an open set \( U \subset D \) such that \( Z\left( {f, U}\right) \) is a nonempty complex submanifold of \( U \) of dimen... | Proof. In case there is a point \( P \in Z\left( {f, D}\right) \) with \( {df}\left( P\right) \neq 0 \), the statement follows immediately from Theorem 2.9. In order to apply this result in the general case, we consider higher order derivatives of \( f \) as follows. Let\n\n\[ \Lambda = \left\{ {\lambda \in \mathbb{N} ... | Yes |
Theorem 2.14. Let \( D \subset {\mathbb{C}}^{n} \) and suppose that the holomorphic map \( F : D \rightarrow {\mathbb{C}}^{n} \) is injective. Then \( \det {F}^{\prime }\left( z\right) \neq 0 \) for all \( z \in D \), and hence \( F \) is biholomorphic from \( D \) onto \( F\left( D\right) \) . | The proof of Theorem 2.14 will involve induction over the number of variables \( n \) . We assume as known the classical case \( n = 1 \) (see [Ahl], Theorem 4.11). Given the induction hypotheses that the theorem has been proved for \( n - 1 > 0 \) variables, we first prove the following technical lemma. | No |
Lemma 2.15. Under the above assumption, if \( F \) is as in Theorem 2.14, then \( {F}^{\prime }\left( a\right) \neq 0 \) at a point \( a \in D \) implies \( \det {F}^{\prime }\left( a\right) \neq 0 \) . | Proof of Lemma 2.15. After renumbering we may assume that \( F = \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) and \( \partial {f}_{n}/\partial {z}_{n}\left( a\right) \neq 0 \) . If \( w\left( z\right) = \left( {{z}_{1},\ldots ,{z}_{n - 1},{f}_{n}\left( z\right) }\right) \), then \( \det \left( {\partial {w}_{k}/\partial ... | Yes |
Lemma 3.1. Suppose \( f \in \mathcal{O}\left( {B\left( {a,\varepsilon }\right) }\right), f\left( a\right) = 0 \), but \( f \) is not identically zero. Then, after a suitable complex linear coordinate change, \( f \) is \( {z}_{n} \) -regular of some order \( k \geq 1 \) at \( a \) . | Proof. By hypothesis there is \( p \in B\left( {a,\varepsilon }\right), p \neq a \), such that \( f\left( p\right) \neq 0 \) . After applying an affine complex linear coordinate change, one may assume that \( p - a \) lies in the \( {z}_{n} \) -axis, i.e., \( p = \left( {{a}^{\prime },{p}_{n}}\right) ,{p}_{n} - {a}_{n}... | Yes |
Lemma 3.2. Suppose \( f \) is holomorphic at \( 0, f\left( 0\right) = 0 \), and \( f \) is \( {z}_{n} \) -regular of order \( k \geq 1 \) at 0 . Then for each sufficiently small \( {\delta }_{n} > 0 \) there is \( {\delta }^{\prime } > 0 \), such that for each fixed \( {z}^{\prime } \in P\left( {{0}^{\prime },{\delta }... | Proof. By hypothesis, for each sufficiently small \( {\delta }_{n} > 0, g\left( {z}_{n}\right) = f\left( {{0}^{\prime },{z}_{n}}\right) \) is holomorphic on \( \left| {z}_{n}\right| \leq {\delta }_{n}, g \) has a zero of order \( k \) at 0, and \( g\left( {z}_{n}\right) \neq 0 \) for \( 0 < \left| {z}_{n}\right| \leq {... | Yes |
Theorem 3.4. Let \( E \) be a thin subset of \( D \subset {\mathbb{C}}^{n} \). Let \( h \in \mathcal{O}\left( {D - E}\right) \) and suppose \( h \) is locally bounded on \( D \) (i.e., for all \( \Omega \subset \subset D, h \) is bounded on \( \Omega - E \) ). Then there is \( H \in \mathcal{O}\left( D\right) \) such t... | Proof. Since \( E \) is nowhere dense, the extension \( H \) -if it exists-is determined uniquely by \( h \). Therefore it is enough to construct a holomorphic extension of \( h \) to a neighborhood of an arbitrary point \( p \in E \). Without loss of generality we may assume \( p = 0 \) and \( E = \{ z : f\left( z\rig... | Yes |
Corollary 3.6. Let \( E \) be a thin subset of \( D \subset {\mathbb{C}}^{n} \). If \( D \) is connected, so is \( D - E \). | Proof. Since \( D - E \) will be connected if \( D - \bar{E} \) is, we may assume that \( E \) is closed. Suppose \( U \neq \varnothing \) is an open and closed subset of \( D - E \). We must show that \( U = D - E \). Define the function \( h \) by setting \( h\left( z\right) = 0 \) for \( z \in U \) and \( h\left( z\... | Yes |
Theorem 3.7. Let \( E \) be a thin subset of \( D \subset {\mathbb{C}}^{n} \). Then the 2n-dimensional Lebesgue measure of \( E \) is zero. In particular, if \( D \) is connected and \( f \in \mathcal{O}\left( D\right) \) vanishes on a set of positive measure, then \( f \equiv 0 \) on \( D \). | The proof is left to the reader. | No |
Theorem 3.8. Let \( A \) be an analytic set in the connected region \( D \) in \( {\mathbb{C}}^{n} \) . If \( A \neq D \) , then \( A \) is thin, and hence \( D - A \) is connected. | Proof. Since the second statement in the conclusion follows from the first, by Corollary 3.6, it is enough to show that if \( A \) is not thin, then \( A = D \) . For each \( p \in A \) we choose a connected neighborhood \( {U}_{p} \) and a holomorphic map \( {H}_{p} : {U}_{p} \rightarrow {\mathbb{C}}^{{l}_{p}} \) such... | Yes |
Theorem 3.10. Let \( f \) be holomorphic at \( 0, f\left( 0\right) = 0 \), and suppose \( f \) is \( {z}_{n} \) -regular of order \( k \geq 1 \) . Then there is a unique factorization\n\n(3.5)\n\n\[ f = \omega \cdot u \]\n\non some polydisc \( P\left( {0,\delta }\right) \), where \( \omega \in \mathcal{O}\left( {P\left... | Proof. The uniqueness of the factorization (3.5) is obvious in view of the preceding remarks. In order to prove that the coefficients of \( \omega \) are holomorphic, we choose \( P\left( {0,\delta }\right) \) as in Lemma 3.2. Notice that \( {a}_{0}\left( {z}^{\prime }\right) ,\ldots ,{a}_{k - 1}\left( {z}^{\prime }\ri... | Yes |
Lemma 3.11. Suppose \( \omega \in R\left\lbrack X\right\rbrack \) is irreducible in \( Q\left\lbrack X\right\rbrack \) . Let \( E \subset P\left( {0,\delta }\right) \) be the set of points \( z \), such that \( \omega \left( {z, \cdot }\right) \) has at least one zero of multiplicity greater than one. Then \( E \) is t... | Proof. The Euclidean algorithm being valid in \( Q\left\lbrack X\right\rbrack \), the polynomials \( \omega \) and \( \partial \omega /\partial X \) have a greatest common divisor, which must be 1, since \( \omega \) is irreducible. Hence there are \( \varphi ,\psi \in Q\left\lbrack X\right\rbrack \), such that\n\n(3.8... | Yes |
Theorem 3.12. Let \( \omega \in R\left\lbrack X\right\rbrack \) be a Weierstrass polynomial of degree \( k \) which is irreducible in \( Q\left\lbrack X\right\rbrack \), and let \( \pi : {\mathbb{C}}^{n} \times \mathbb{C} \rightarrow {\mathbb{C}}^{n} \) be the projection. Then there is a thin subset \( E \subset P\left... | Proof. Let \( E \subset P\left( {0,\delta }\right) \) be the thin set given by Lemma 3.11. Part (i) is a direct consequence of Lemma 3.11 and Theorem 2.9. For (ii), let \( p \in Z\left( \omega \right) \) with \( a = \pi \left( p\right) \in E \) . Then \( \omega \) is \( {z}_{n + 1} \) -regular at \( p \) . Let \( U \) ... | No |
Lemma 3.14. Suppose \( {\omega }_{1},{\omega }_{2} \in Q\left\lbrack X\right\rbrack \) are monic polynomials such that \( {\omega }_{1} \cdot {\omega }_{2} \in \) \( R\left\lbrack X\right\rbrack \) . Then \( {\omega }_{1} \) and \( {\omega }_{2} \) are in \( R\left\lbrack X\right\rbrack \) . | Proof. Write \( \omega = {\omega }_{1} \cdot {\omega }_{2} = {X}^{k} + {a}_{k - 1}{X}^{k - 1} + \cdots + {a}_{0} \), where \( {a}_{j} \in R = \) \( \mathcal{O}(P\left( {0,\delta }\right) ,0 \leq j \leq k - 1 \) . Since the coefficients of \( \omega \left( {z, X}\right) \) are locally bounded on \( P\left( {0,\delta }\r... | Yes |
Theorem 3.15. Let \( \omega \in R\left\lbrack X\right\rbrack \) be a pseudopolynomial. Then\n\n\[ \omega = {\omega }_{1} \cdot {\omega }_{2} \cdot \cdots \cdot {\omega }_{r} \]\n\nwhere each \( {\omega }_{i} \in R\left\lbrack X\right\rbrack \) is a pseudopolynomial which is irreducible in \( Q\left\lbrack X\right\rbrac... | Proof. Let \( \omega = {\omega }_{1}\cdots \cdots {\omega }_{r} \) be the factorization of \( \omega \) into irreducible monic polynomials in \( Q\left\lbrack X\right\rbrack \) of degree \( \geq 1 \) . Repeated application of Lemma 3.14 gives \( {\omega }_{i} \in R\left\lbrack X\right\rbrack ,1 \leq i \leq r \) . Final... | Yes |
Theorem 1.1. Let \( n \) be \( \geq 2 \) and suppose that \( 0 < {r}_{j} < 1 \) for \( 1 \leq j \leq n \) . Then every function \( f \) holomorphic on the domain\n\n\[ H\left( r\right) = \left\{ {z \in {\mathbb{C}}^{n} : \left| {z}_{j}\right| < 1\text{ for }j < n,\;{r}_{n} < \left| {z}_{n}\right| < 1}\right\} \]\n\n\[ ... | Proof. The uniqueness of the extension is an immediate consequence of the Identity Theorem. In order to obtain the desired extension \( \widehat{f} \) we will write down an integral formula for \( \widehat{f} \) . Fix \( {r}_{n} < \delta < 1 \) ; then\n\n(1.1)\n\n\[ \widehat{f}\left( {{z}^{\prime },{z}_{n}}\right) = {\... | Yes |
Proposition 1.4. Suppose \( 0 < {R}_{j} < \infty \) and \( 0 \leq {r}_{j} \leq {R}_{j} \) for \( 1 \leq j \leq n \), and let \( K\left( {r, R}\right) = \left\{ {z \in {\mathbb{C}}^{n} : {r}_{j} \leq \left| {z}_{j}\right| \leq {R}_{j},1 \leq j \leq n}\right\} \) . Then every \( f \in \mathcal{O}\left( K\right) \), has a... | Proof. We first prove that uniform convergence in (1.2) implies (1.3) and (1.4); in particular, this implies the uniqueness of the Laurent expansion. Given \( \rho \) as in the Theorem, the uniform convergence of (1.2) implies that\n\n(1.5)\n\[ {\int }_{{b}_{0}P\left( {0,\rho }\right) }f\left( \zeta \right) {\zeta }^{\... | Yes |
Theorem 1.5. Let \( D \) be a connected Reinhardt domain with center 0 . Then every \( f \in \mathcal{O}\left( D\right) \) has a Laurent series representation\n\n(1.13)\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{v \in {\mathbb{Z}}^{n}}}{c}_{v}{z}^{v}\;\text{ for }z \in D, \]\n\nwhich converges normally on D. More... | Proof. Suppose \( f \in \mathcal{O}\left( D\right) \) . For \( a \in D \), we choose \( K\left( {r, R}\right) \subset D \) so that \( a \) lies in the interior of \( K\left( {r, R}\right) \) . By Proposition 1.4, \( f\left( z\right) = \sum {c}_{v}\left( a\right) {z}^{v} \) uniformly and absolutely on \( K\left( {r, R}\... | Yes |
Theorem 1.6. Let \( D \) be a connected Reinhardt domain with center 0, and suppose that \( D \cap \left\{ {z \in {\mathbb{C}}^{n}:{z}_{l} = 0}\right\} \neq \varnothing \) for all \( l = 1,2,\ldots, n \) (this holds, in particular, if \( 0 \in D) \) . Then every \( f \in \mathcal{O}\left( D\right) \) has a convergent p... | Proof. Let \( f \in \mathcal{O}\left( D\right) \) . Theorem 1.5 gives a Laurent expansion \( f\left( z\right) = \mathop{\sum }\limits_{{v \in {\mathbb{Z}}^{n}}}{c}_{v}{z}^{v} \) on \( D \), and the hypotheses on \( D \) imply that \( {c}_{v} = 0 \) unless \( v \geq 0 \) . Hence \( f\left( z\right) = \) \( \mathop{\sum ... | Yes |
Lemma 2.2. Let \( \left( {{\Gamma }^{ * },{\widehat{\Gamma }}^{ * }}\right) \) be a Hartogs figure. Then every \( f \in \mathcal{O}\left( {\Gamma }^{ * }\right) \) has a holomorphic extension \( \widehat{f} \in \mathcal{O}\left( {\widehat{\Gamma }}^{ * }\right) \) . | Proof. Let \( F : \widehat{\Gamma } \rightarrow {\widehat{\Gamma }}^{ * } \) be the biholomorphic map given by the hypothesis. If \( f \in \mathcal{O}\left( {\Gamma }^{ * }\right) \), then \( g = f \circ F \in \mathcal{O}\left( \Gamma \right) \), and just as in the proof of Theorem 1.1, one sees that for \( \varepsilon... | Yes |
Theorem 2.3. A weak domain of holomorphy is Hartogs pseudoconvex. | Proof. Suppose that \( D \subset {\mathbb{C}}^{n} \) is not \( H \) -pseudoconvex. We will show that there is \( p \in {bD} \) such that no \( f \in \mathcal{O}\left( D\right) \) is completely singular at \( p \), and therefore \( D \) cannot be a weak domain of holomorphy. Let \( \left( {{\Gamma }^{ * },{\widehat{\Gam... | Yes |
Lemma 2.5. Suppose \( {r}_{1} \) and \( {r}_{2} \) are two local defining functions for \( D \) of class \( {C}^{k} \) in a neighborhood \( U \) of \( p \in {bD} \) . Then there exists a positive function \( h \in \) \( {C}^{k - 1}\left( U\right) \) such that\n\n(2.6)\n\n\[ \n{r}_{1} = h \cdot {r}_{2}\text{ on }U \n\]\... | Proof. Clearly the conditions imposed on \( h \) determine \( h \) uniquely, and \( h = \) \( {r}_{1}/{r}_{2} \) is \( {C}^{k} \) and positive on \( U - {bD} \) . Now fix \( q \in U \cap {bD} \) . After a local change of coordinates of class \( {C}^{k} \) near \( q \), one may assume that \( q = 0, U \cap {bD} = \{ x \... | Yes |
Lemma 2.6. If \( D \subset \subset {\mathbb{R}}^{n} \) has \( {C}^{k} \) boundary, then there is a global \( {C}^{k} \) defining function \( r \) for \( D \) . | Proof. By compactness of \( {bD} \) there are finitely many open sets \( {U}_{1},\ldots ,{U}_{l} \) and local defining functions \( {r}_{v} \in {C}^{k}\left( {U}_{v}\right), v = 1,\ldots, l \), so that \( {bD} \subset \mathop{\bigcup }\limits_{{v = 1}}^{l}{U}_{v} \) and (2.3) and (2.4) hold for each \( {r}_{v} \) . Cho... | Yes |
Lemma 2.8. If \( r \) is a local defining function for \( D \subset {\mathbb{C}}^{n} \) at \( p \in {bD} \), then\n\n\[ \n{T}_{p}^{\mathbb{C}}\left( {bD}\right) = \left\{ {t \in {\mathbb{C}}^{n} : \partial {r}_{p}\left( t\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial r}{\partial {z}_{j}}\left( p\right) {t}... | Proof. Since \( r \) is real valued, we have \( d{r}_{p} = \partial {r}_{p} + {\bar{\partial }}_{p}r = 2\operatorname{Re}\partial {r}_{p} \) . According to \( \left( {2.10}\right) \) ,\n\n\[ \n{T}_{p}^{\mathbb{C}}\left( {bD}\right) = \left\{ {t \in {\mathbb{C}}^{n} : d{r}_{p}\left( t\right) = d{r}_{p}\left( {it}\right)... | Yes |
Theorem 2.9. Suppose \( D \subset {\mathbb{C}}^{n} \) is Hartogs pseudoconvex and has \( {C}^{1} \) boundary near \( p \in {bD} \) . Then there is no complex one-dimensional submanifold \( M \) with \( p \) in its interior, and with \( M - \{ p\} \subset D \) . | Proof. The result is clearly true if \( n = 1 \), so we will assume \( n \geq 2 \) . We will show that the existence of \( M \) with the properties stated in the theorem implies that there is a Hartogs figure \( \left( {{\Gamma }^{ * },{\widehat{\Gamma }}^{ * }}\right) \) with \( {\Gamma }^{ * } \subset D \) but \( {\w... | Yes |
Lemma 2.12. Suppose \( D \subset {\mathbb{C}}^{n} \) has \( {C}^{2} \) boundary near \( p \in {bD} \) and \( w = F\left( z\right) \) is a biholomorphic map on a neighborhood \( U \) of \( p \) . Let \( \Omega = F\left( {U \cap D}\right) \) . Then \( \Omega \) is Levi pseudoconvex at \( q = F\left( p\right) \) if and on... | Proof. Let \( r \) be a \( {C}^{2} \) defining function for \( D \) near \( p \) ; then \( \rho = r \circ {F}^{-1} \) is a \( {C}^{2} \) defining function for \( \Omega \) near \( q \) . From \( r\left( z\right) = \rho \circ F\left( z\right) \) one obtains by a straightforward calculation that for all \( t \in {\mathbb... | Yes |
Lemma 2.13. Suppose \( D \) has \( {C}^{2} \) boundary and is Levi pseudoconvex near \( p \in {bD} \) . Then there is a \( {C}^{2} \) defining function \( r \) for \( D \) on a neighborhood \( U \) of \( p \) such that at all points \( z \in U \) one has\n\n(2.25)\n\n\[ \n{L}_{z}\left( {r;t}\right) \geq 0\;\text{ for a... | Proof. Without loss of generality we may assume that \( p = 0 \) and that \( \operatorname{Re}{z}_{1} \) , \( \operatorname{Im}{z}_{1},\ldots ,\operatorname{Re}{z}_{n - 1},\operatorname{Im}{z}_{n - 1} \), and \( {y}_{n} = \operatorname{Im}{z}_{n} \) are tangential coordinates for \( {bD} \) at 0 . Thus there is a local... | Yes |
Proposition 2.14. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) has \( {C}^{2} \) boundary and is strictly Levi pseudo-convex. Then one can find a defining function \( r \) for \( D \) on a neighborhood \( U \) of \( {bD} \) which is strictly plurisubharmonic on \( U \) . | Proof. By Lemma 2.6 there is a global \( {C}^{2} \) defining function \( \varphi \) for \( D \) . We will show that for \( A > 0 \) sufficiently large, the function \( {r}_{A} = \exp \left( {A\varphi }\right) - 1 \) will do. In fact, it is obvious that \( {r}_{A} \) is a defining function for \( D \) . A straightforwar... | Yes |
Proposition 2.16. Let \( U \) be open in \( {\mathbb{C}}^{n} \) and suppose \( r \in {C}^{2}\left( U\right) \) is strictly pluri-subharmonic on \( U \) . Given \( W \subset \subset U \), there are positive constants \( c > 0 \) and \( \varepsilon > 0 \) , such that the function \( {F}^{\left( r\right) }\left( {\zeta, z... | Proof. From formula (2.14), with \( p = \zeta \in U \) and \( t = z - \zeta \), we see that the Taylor expansion of \( r\left( z\right) \) at \( \zeta \) is given by\n\n(2.31)\n\n\[ \nr\left( z\right) = r\left( \zeta \right) - 2\operatorname{Re}{F}^{\left( r\right) }\left( {\zeta, z}\right) + {L}_{\zeta }\left( {r;z - ... | Yes |
Theorem 2.17. Let \( D \subset {\mathbb{C}}^{n} \) have \( {C}^{2} \) boundary near \( p \in {bD} \) . Then \( D \) is strictly Levi pseudoconvex at \( p \) if and only if there is a holomorphic coordinate system \( w = w\left( z\right) \) in a neighborhood of \( p \), so that \( D \) is strictly convex with respect to... | Proof. If there are holomorphic coordinates \( w = w\left( z\right) \) near \( p \in {bD} \), so that \( D \) is strictly convex with respect to \( w \) at \( w\left( p\right) \), then the same computations which proved Lemma 2.10 imply that \( D \) is strictly Levi pseudoconvex in the \( w \) -coordinates, and hence a... | Yes |
Lemma 2.18. A bounded strictly pseudoconvex domain \( D \) in \( {\mathbb{C}}^{n} \) is pseudoconvex. | Proof. Let \( r \) be strictly plurisubharmonic in a neighborhood \( U \) of \( {bD} \), so that \( D \cap U = \{ z \in U : r\left( z\right) < 0\} \) . Without loss of generality we may assume that \( r \in {C}^{2}\left( \bar{D}\right) \) and \( r < 0 \) on \( D \) . A straightforward computation shows that \( \rho = -... | Yes |
Proposition 2.21. If \( D \subset {\mathbb{C}}^{n} \) is pseudoconvex, then there is a strictly pluri-subharmonic exhaustion function \( r \) for \( D \) such that the set of critical points \( \left\{ {z \in D : d{r}_{z} = 0}\right\} \) of \( r \) is discrete in \( D \) . | The proof is based on the following real variable lemma which is a special case of general results in Morse theory. For the reader's convenience we have included a self-contained proof in Appendix A. | No |
Corollary 2.23. Let \( K \) be a compact pseudoconvex set in \( {\mathbb{C}}^{n} \). Then \( K \) has a neighborhood basis of strictly pseudoconvex domains with \( {C}^{2} \) boundary. | Proof. Let \( W \) be a neighborhood of \( K \). By hypothesis, there is an open pseudoconvex neighborhood \( U \) of \( K \) with \( U \subset W \). Let \( r \) be a \( {C}^{2} \) strictly plurisubharmonic exhaustion function for \( U \) whose set \( S \) of critical points is discrete in \( U \), and hence countable.... | Yes |
Lemma 3.1. For \( K \subset D \) the following hold:\n\n(i) \( K \subset \widehat{K} \) and \( \widehat{\widehat{K}} = \widehat{K} \) ;\n\n(ii) if \( {K}_{1} \subset K \), then \( {\widehat{K}}_{1} \subset \widehat{K} \) ;\n\n(iii) if \( \Omega \) is open and \( D \subset \Omega \), then \( {\widehat{K}}_{\mathcal{O}\l... | Proof. (i),(ii), and (iii) are straightforward. For (iv), notice that if \( f \in \mathcal{O}\left( D\right) \), the set \( {A}_{f} = \left\{ {z \in D : \left| {f\left( z\right) }\right| \leq {\left| f\right| }_{K}}\right\} \) is closed in \( D\left( {{A}_{f} = D}\right. \) if \( {\left. \left| f{\left. \right| }_{K} =... | Yes |
Lemma 3.2. Let \( D \subset {\mathbb{C}}^{n} \) be holomorphically convex. Then there is a normal exhaustion \( \left\{ {K}_{j}\right\} \) of \( D \) by \( \mathcal{O}\left( D\right) \) -convex sets \( {K}_{j} \) . | Proof. The sequence \( \left\{ {K}_{j}\right\} \) will be constructed inductively. Let \( \left\{ {Q}_{v}\right\} \) be some normal exhaustion. The hypothesis on \( D \) implies that \( {\widehat{Q}}_{v} \) is compact for all \( v \) . We set \( {K}_{1} = {\widehat{Q}}_{1} \) ; then \( {K}_{1} \) is compact and \( {\wi... | Yes |
Lemma 3.3. Let \( \left\{ {K}_{j}\right\} \) be a normal exhaustion of \( D \) by \( \mathcal{O}\left( D\right) \)-convex sets. Suppose \( {p}_{j} \in {K}_{j + 1} - {K}_{j} \) for \( j = 1,2,\ldots \) Then there is \( f \in \mathcal{O}\left( D\right) \) such that \( \mathop{\lim }\limits_{{j \rightarrow \infty }}\left|... | Proof. We construct \( f \) as the limit of a series \( \mathop{\sum }\limits_{{v = 1}}^{\infty }{f}_{v} \), where \( \left\{ {f}_{v}\right\} \subset \mathcal{O}\left( D\right) \) is chosen to satisfy\n\n(3.2)\n\n\[ \n{\left| {f}_{v}\right| }_{{K}_{v}} < {2}^{-v},\;v = 1,2,\ldots \n\] \n\nand \n\n(3.3)\n\n\[ \n\left| {... | Yes |
Proposition 3.4. The region \( D \subset {\mathbb{C}}^{n} \) is holomorphically convex if and only if for every sequence \( \left\{ {{p}_{v} : v = 1,2,3,\ldots }\right\} \subset D \) without accumulation point in \( D \) there is \( f \in \mathcal{O}\left( D\right) \) with \( \mathop{\sup }\limits_{v}\left| {f\left( {p... | Proof. We first show that the given condition is sufficient. Let \( K \subset D \) be compact. We prove that \( {\widehat{K}}_{\mathcal{O}\left( D\right) } \) is compact by showing that every sequence \( \left\{ {p}_{v}\right\} \subset \widehat{K} \) has an accumulation point \( p \in \widehat{K} \) . In fact, if \( \l... | Yes |
Lemma 3.5. Every \( D \subset {\mathbb{C}}^{1} \) is holomorphically convex. | Proof. Suppose \( \left\{ {p}_{v}\right\} \subset D \) has no accumulation point in \( D \) . If \( \left\{ {p}_{v}\right\} \) is unbounded, the function \( f\left( z\right) = z \) will be unbounded on \( \left\{ {p}_{v}\right\} \) . Otherwise \( \left\{ {p}_{v}\right\} \) has an accumulation point \( p \in {bD} \), an... | No |
Lemma 3.7. The intersection of finitely many holomorphically convex open sets is holomorphically convex. | The proof is obvious. | No |
Proposition 3.8. The product \( D = {D}_{1} \times {D}_{2} \) of two holomorphically convex regions \( {D}_{i} \subset {\mathbb{C}}^{{n}_{i}}, i = 1,2 \), is holomorphically convex. In particular, every poly-domain \( \Omega = {\Omega }_{1} \times \cdots \times {\Omega }_{n},{\Omega }_{j} \subset \mathbb{C} \), is holo... | Proof. For the first statement it is clearly enough to show that \( Q = \) \( {\left( {\widehat{{K}_{1} \times K}}_{2}\right) }_{\mathcal{O}\left( {{D}_{1} \times {D}_{2}}\right) } \subset \subset {D}_{1} \times {D}_{2} \) for every pair of compact sets \( {K}_{i} \subset {D}_{i}, i = 1 \) , 2. Since every function \( ... | Yes |
Proposition 3.9. Every analytic polyhedron is holomorphically convex. | Proof. Suppose \( \left\{ {{f}_{1},\ldots ,{f}_{l}}\right\} \subset \mathcal{O}\left( U\right) \) is a frame for the analytic polyhedron \( \Omega \) . If \( K \subset \Omega \) is compact, then \( {r}_{j} = {\left| {f}_{j}\right| }_{K} < 1 \) for \( j = 1,\ldots, l \) . Clearly\n\n\[{\widehat{K}}_{\mathcal{O}\left( \O... | Yes |
Proposition 3.10. If \( K \subset D \) is compact and \( \mathcal{O}\left( D\right) \) -convex, then \( K \) has a neighborhood basis consisting of analytic polyhedra defined by frames of functions holomorphic on \( D \) . | Proof. Let \( U \subset \subset D \) be an open neighborhood of \( K = {\widehat{K}}_{\mathcal{O}\left( D\right) } \) . Since \( {\widehat{K}}_{\mathcal{O}\left( D\right) } \) and \( {bU} \) are disjoint, by using (3.1) and a compactness argument, one can find finitely many open sets \( {W}_{1},\ldots ,{W}_{l} \) and f... | Yes |
Corollary 3.11. Let \( D \subset {\mathbb{C}}^{n} \) be holomorphically convex. Then there is a normal exhaustion \( \left\{ {\bar{\Omega }}_{j}\right\} \) of \( D \), where each \( {\Omega }_{j} \) is an analytic polyhedron defined by a frame of functions in \( \mathcal{O}\left( D\right) \) . | Proof. Combine Lemma 3.2 and Proposition 3.10. | No |
Lemma 3.13. Suppose \( U \) is a connected neighborhood of \( p \in {bD} \) and let \( \Omega \subset \) \( U \cap D \) be a nonempty connected component of \( U \cap D \) . Then \( {b\Omega } \cap \left( {U \cap {bD}}\right) \neq \varnothing \) . | Proof of 3.13. Since \( \Omega \) is a component of the open set \( U \cap D,\Omega \) is open (in \( \left. {\mathbb{C}}^{n}\right) \) and closed in \( U \cap D \) . Since \( U \) is connected and clearly \( \Omega \neq U,\Omega \) cannot be closed in \( U \) . Hence there is \( q \in \left( {{b\Omega } \cap U}\right)... | Yes |
Theorem 3.14. A holomorphically convex domain \( D \) in \( {\mathbb{C}}^{n} \) is a domain of holomorphy. | Proof. By Lemma 3.2 we can choose a normal exhaustion \( \left\{ {K}_{v}\right\} \) of \( D \) with \( {\widehat{K}}_{v} = {K}_{v} \) . We then apply Lemma 3.3 to the sequences \( \left\{ {p}_{j}\right\} \) and \( \left\{ {K}_{{v}_{j}}\right\} \) given by Lemma 3.12 to obtain \( f \in \mathcal{O}\left( D\right) \) with... | Yes |
Lemma 3.16. If \( D \neq {\mathbb{C}}^{n} \), then \( {\delta }_{D}^{\left( r\right) } \) is continuous on \( D \) . Moreover, for \( a \in D \), \[ {\delta }_{D}\left( a\right) = \inf \left\{ {{\delta }_{D}^{\left( r\right) }\left( a\right) : r > 0\text{ and }{\left| r\right| }^{2} = \sum {r}_{j}^{2} = 1}\right\} . \] | Proof. Fix \( a \in D \). One easily checks that for \( 0 < \varepsilon < {\delta }_{D}^{\left( r\right) }\left( a\right) \) one has \( {\delta }_{D}^{\left( r\right) }\left( a\right) - \varepsilon \leq {\delta }_{D}^{\left( r\right) }\left( z\right) \leq {\delta }_{D}^{\left( r\right) }\left( a\right) + \varepsilon \)... | Yes |
Proposition 3.17. Let \( K \subset D \) be a compact subset of the open set \( D \subset {\mathbb{C}}^{n} \), and fix a positive multiradius \( r > 0 \) . Suppose \( \eta > 0 \) satisfies \( {\delta }_{D}^{\left( r\right) }\left( z\right) \geq \eta \) for all \( z \in K \) . Then for every \( a \in {\widehat{K}}_{\math... | Proof. We fix a function \( f \in \mathcal{O}\left( D\right) \) . We shall estimate the coefficients of the Taylor series of \( f \) by means of Cauchy estimates for derivatives of \( f \) . For this one needs a uniform bound for \( \left| f\right| \) . We therefore fix \( 0 < {\eta }^{\prime } < \eta \) ; then \( Q = ... | Yes |
Theorem 3.18. The following are equivalent for an open set \( D \) in \( {\mathbb{C}}^{n} \) .\n\n(i) \( D \) is a weak domain of holomorphy.\n\n(ii) \( \operatorname{dis}\left( {{\widehat{K}}_{\mathcal{O}\left( D\right) },{bD}}\right) = \operatorname{dist}\left( {K,{bD}}\right) \) for every compact set \( K \subset D ... | Proof. Theorem 3.15 proves (i) \( \Rightarrow \) (ii); (ii) \( \Rightarrow \) (iii) is obvious; (iii) \( \Rightarrow \) (iv) is proved in Theorem 3.14, and (iv) \( \Rightarrow \) (i) is trivial. | No |
Corollary 3.19. The interior \( \Omega \) of the intersection of an arbitrary collection \( \left\{ {{D}_{\alpha } : \alpha \in I}\right\} \) of holomorphically convex domains is holomorphically convex. | Proof. We assume that \( \Omega \neq \varnothing \) . If \( K \subset \Omega \) is compact, \( 0 < d = \operatorname{dist}\left( {K,{b\Omega }}\right) \leq \) \( \operatorname{dist}\left( {K, b{D}_{\alpha }}\right) \) for each \( \alpha \), and by Theorem 3.18, \( d \leq \operatorname{dist}\left( {{\widehat{K}}_{C\left... | Yes |
Proposition 3.21. A Stein compactum \( K \) has a neighborhood basis of analytic polyhedra. | Proof. If \( U \) is an open neighborhood of \( K \), there is a Stein domain \( D \) with \( K \subset D \subset U \) . Then \( {\widehat{K}}_{C\left( D\right) } \) is compact, and \( \mathcal{O}\left( D\right) \) -convex, so, by Proposition 3.10, there is an analytic polyhedron \( \Omega \) with \( {\widehat{K}}_{\ma... | Yes |
Lemma 3.22. If \( K \) is a Stein compactum and \( {h}_{1},\ldots ,{h}_{l} \in \mathcal{O}\left( K\right) \), then\n\n\[ \n{K}_{l} = \left\{ {z \in K : \left| {{h}_{j}\left( z\right) }\right| \leq 1\;\text{ for }1 \leq j \leq l}\right\} \n\]\n\nis Stein. | The proof involves a straightforward application of Proposition 3.9 and Exercise E.3.3, and is left to the reader. | No |
Proposition 3.23. Suppose \( L \subset K \) and \( {\widehat{L}}_{\mathcal{O}\left( K\right) } = L \) . If \( U \) is a neighborhood of \( L \) , then there are finitely many functions \( {h}_{1},\ldots ,{h}_{l} \in \mathcal{O}\left( K\right) \) such that\n\n(3.11)\n\n\[ L \subset \left\{ {z \in K : \left| {{h}_{j}\lef... | Proof. Since \( K - U \) is compact and disjoint from \( {\widehat{L}}_{\mathcal{O}\left( K\right) } \), a simple compactness argument, as in the proof of Proposition 3.10, gives functions \( {h}_{1},\ldots ,{h}_{l} \in \) \( \mathcal{O}\left( K\right) \) which satisfy (3.11). The other statement then follows easily fr... | Yes |
Theorem 3.24. Let \( D \) be a Stein domain. Then there is a real analytic strictly plurisubharmonic exhaustion function for D. In particular, D is pseudoconvex. | Proof. Let \( \left\{ {K}_{j}\right\} \) be a normal exhaustion of \( D \) by \( \mathcal{O}\left( D\right) \) -convex sets. By arguments as those used in the proof of Proposition 3.10, for each \( j = 1,2,\ldots \) we can find an open set \( {\Omega }_{j} \) with \( {K}_{j} \subset {\Omega }_{j} \subset {K}_{j + 1} \)... | Yes |
Lemma 3.26. If \( D \subset {\mathbb{C}}^{n} \) is the region of convergence of a power series \( \mathop{\sum }\limits_{{v \in {\mathbb{N}}^{n}}}{c}_{v}{z}^{v} \) , then \( \log \tau \left( D\right) \) is a convex subset of \( {\mathbb{R}}^{n} \) . | Proof. Suppose \( \xi \) and \( \eta \) are points in \( \log \tau \left( D\right) \) ; we must show that \( {t\xi } + \) \( \left( {1 - t}\right) \eta \in \log \tau \left( D\right) \) for \( 0 \leq t \leq 1 \) . Choose \( p, q \in D \), and \( \lambda > 1 \), so that \( {\xi }_{j} = \log \left| {p}_{j}\right| \) and \... | Yes |
Theorem 3.28. The following are equivalent for a complete Reinhardt domain \( D \) in \( {\mathbb{C}}^{n} \) with center 0 .\n\n(i) \( D \) is the region of convergence of a power series.\n\n(ii) \( \log \tau \left( D\right) \subset {\mathbb{R}}^{n} \) is convex.\n\n(iii) \( D \) is \( \mathcal{M} \) -convex.\n\n(iv) \... | Proof. The implications (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) are proved in Lemma 3.26 and Lemma 3.27. (iii) \( \Rightarrow \) (iv) is trivial, since \( \mathcal{M} \subset \mathcal{O}\left( D\right) \), and (iv) \( \Rightarrow \) (v) was proved in \( §{3.4} \n\n![5e807d86-5f6d-4104-b03f-a2cbf7f2570a_97_0.... | Yes |
Lemma 4.1. Let \( D \subset \mathbb{C} \) be open.\n\n(i) If \( u \) is subharmonic on \( D \), so is cu for \( c > 0 \) .\n\n(ii) If \( \left\{ {{u}_{\alpha } : \alpha \in A}\right\} \) is a family of subharmonic functions on \( D \) such that \( u = \sup {u}_{\alpha } \) is finite and upper semicontinuous, then \( u ... | Proof. (i) and (ii) are obvious consequences of the definitions. In order to prove (iii), suppose \( K \subset D \) is compact and \( h \in C\left( K\right) \) is harmonic on int \( \left( K\right) \) with \( h \geq u = \lim {u}_{j} \) on \( {bK} \) . Given \( \varepsilon > 0,{E}_{j} = \left\{ {z \in {bK} : {u}_{j}\lef... | Yes |
For every open set \( D \) in \( \mathbb{C} \) the function \( u\left( z\right) = - \log {\delta }_{D}\left( z\right) \) is subharmonic on \( D \) . | Proof. If \( D = \mathbb{C} \), then \( u \equiv - \infty \), and there is nothing to prove. If \( D \neq \mathbb{C} \), then \( u\left( z\right) \) is continuous, and for \( z \in D \) one has \( u\left( z\right) = \sup \{ - \log \left| {z - \zeta }\right| : \zeta \in {bD}\} \) ; since \( - \log \left| {z - \zeta }\ri... | Yes |
Lemma 4.5. An upper semicontinuous function \( u \) which satisfies the submean value property satisfies the strong maximum principle (4.3.) | Proof. The argument is identical to the one which is often used to prove the maximum principle for harmonic functions. Suppose \( u \) satisfies the submean value property and \( u \) has a local maximum at \( a \in D \), i.e., there is \( \rho > 0 \) such that \( u\left( z\right) \leq u\left( a\right) \) for all \( z ... | Yes |
Proposition 4.6. If \( f \) is holomorphic on \( D \), then \( {\left| f\right| }^{\alpha } \) for \( \alpha > 0 \) and \( \log \left| f\right| \) are subharmonic on \( D \) . | The proof is left to the reader. | No |
Lemma 4.7. If \( u \) is subharmonic on the disc \( \{ \left| {z - a}\right| < \rho \} \), then\n\n\[ A\left( {u;r}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\left( {a + r{e}^{i\theta }}\right) {d\theta } \]\n\nis a nondecreasing function for \( 0 < r < \rho \) . | Proof. Let \( \Delta \left( r\right) = \{ \left| {z - a}\right| < r\} \) and suppose \( 0 < {r}_{1} < {r}_{2} < \rho \) . Let \( \varphi \in \) \( C\left( {{b\Delta }\left( {r}_{2}\right) }\right) \) satisfy \( \varphi \geq u \) on \( {b\Delta }\left( {r}_{2}\right) \) . By taking the Poisson integral of \( \varphi \),... | Yes |
Proposition 4.8. A real valued function \( u \in {C}^{2}\left( D\right) \) is subharmonic on \( D \) if and only if \( {\Delta u} \geq 0 \) on \( D \) . | Proof. We first show that \( {\Delta u} > 0 \) implies that \( u \) is subharmonic. Let \( K \subset D \) be compact, \( h \in C\left( K\right) \) harmonic on int \( K \), and suppose \( v = u - h \leq 0 \) on \( {bK} \) . If \( v\left( z\right) > 0 \) for some \( z \in K \), then \( v \) would take on its maximum at a... | Yes |
Proposition 4.9. Let \( D \subset {\mathbb{C}}^{n} \) and suppose \( u \in {C}^{2}\left( D\right) \) is real valued. Then \( u \in {PS}\left( D\right) \) if and only if the complex Hessian of \( u \) ,\n\n\[ \n{L}_{z}\left( {u;w}\right) = \mathop{\sum }\limits_{{j, k = 1}}^{n}\frac{{\partial }^{2}u}{\partial {z}_{j}\pa... | Proof. A straightforward computation gives\n\n(4.8)\n\n\[ \n\frac{{\partial }^{2}}{\partial \lambda \partial \bar{\lambda }}u\left( {a + {\lambda w}}\right) = {L}_{a + {\lambda w}}\left( {u;w}\right) \n\]\n\nfor \( w \in {\mathbb{C}}^{n} \) and \( a + {\lambda w} \in D \) . By Proposition 4.8, \( u\left( {a + {\lambda ... | No |
Corollary 4.10. Suppose \( \Omega \subset {\mathbb{C}}^{n} \) and \( D \subset {\mathbb{C}}^{m} \) are open, and \( F : D \rightarrow \Omega \) is holomorphic. Then \( u \circ F \in {PS}\left( D\right) \) if \( u \in {PS}\left( \Omega \right) \cap {C}^{2}\left( \Omega \right) \) . | Proof. A computation gives \( {L}_{a}\left( {u \circ F;w}\right) = {L}_{F\left( a\right) }\left( {u;{F}^{\prime }\left( a\right) w}\right) \) . Now use the Proposition. | No |
Lemma 4.11. Let \( D \subset {\mathbb{C}}^{n} \) be connected. If \( u \in {PS}\left( D\right) \) and \( u ≢ - \infty \) on \( D \), then \( u \in {L}_{\text{loc }}^{1}\left( D\right) \) . In particular, \( \{ z \in D : u\left( z\right) = - \infty \} \) has Lebesgue measure 0 . | Proof. We first show that if \( u\left( a\right) > - \infty \) at some point \( a \in D \), then \( u \in \) \( {L}^{1}\left( {P\left( {a, r}\right) }\right) \) for every polydisc \( P\left( {a, r}\right) \subset \subset D \) . Since \( u \) is bounded from above on such a polydisc, it is enough to show \( {\int }_{P\l... | Yes |
Theorem 4.13. If \( \Omega \subset {\mathbb{C}}^{n}, D \subset {\mathbb{C}}^{m} \) and \( F : D \rightarrow \Omega \) is holomorphic, then \( u \circ F \in \) \( {PS}\left( D\right) \) for every \( u \in {PS}\left( \Omega \right) \) . | Proof. Without loss of generality we may assume that \( \Omega \) is connected and that \( u \in {PS}\left( \Omega \right) \) is \( ≢ - \infty \) . Choose a decreasing sequence \( \left\{ {u}_{j}\right\} \) with \( \lim {u}_{j} = u \) as in Theorem 4.12. If \( {D}^{\prime } \subset \subset D \), then \( {u}_{j} \circ F... | Yes |
Proposition 4.14. Suppose \( u \) is a continuous plurisubharmonic exhaustion function for \( D \subset {\mathbb{C}}^{n} \). Given a compact set \( K \) in \( D \) and \( \varepsilon > 0 \), there is a \( {C}^{\infty } \) strictly plurisubharmonic exhaustion function \( \varphi \) for \( D \) such that\n\n\[ u \leq \va... | Proof. For \( j = 0,1,2,\ldots \) we set \( {\Omega }_{j} = \{ z \in D : u\left( z\right) < j\} \). Then \( {\Omega }_{j} \subset \subset D \), and by adding a suitable constant to \( u \) we may assume that \( K \subset {\Omega }_{0} \). By Theorem 4.12 there are functions \( {u}_{j} \in {C}^{\infty }\left( D\right), ... | Yes |
Lemma 5.1. Suppose there is a plurisubharmonic exhaustion function for the region \( D \) in \( {\mathbb{C}}^{n} \) . Then \( D \) is PS-convex. | Proof. Let \( u \in {PS}\left( D\right) \) be the given exhaustion function. If \( K \subset D \) is compact, let \( c = \mathop{\max }\limits_{K}u < \infty \) . Then \( u \leq c \) on \( {\widehat{K}}_{{PS}\left( D\right) } \), so \( {\widehat{K}}_{{PS}\left( D\right) } \subset \{ z \in D : u\left( z\right) \leq c\} \... | Yes |
Lemma 5.2. Every PS-convex domain satisfies the continuity principle. | Proof. If \( \varphi : \bar{\Delta } \rightarrow D \) defines an analytic disc \( S \) in \( D \) and \( u \in {PS}\left( D\right) \), then \( u \circ \varphi \in \) \( {PS}\left( \Delta \right) \), by Theorem 4.13. By the maximum principle, \( u\left( z\right) \leq \mathop{\max }\limits_{{\partial S}}u \) for \( z \in... | Yes |
Proposition 5.3. If D satisfies the continuity principle, then D is Hartogs pseudo-convex. | Proof. Suppose \( \left( {{\Gamma }^{ * },{\widehat{\Gamma }}^{ * }}\right) \) is a Hartogs figure defined by the biholomorphic map \( F : \widehat{\Gamma } \rightarrow {\widehat{\Gamma }}^{ * } \), such that \( {\Gamma }^{ * } \subset D \) . Let \( \Delta \) be the open unit disc in \( \mathbb{C} \) ; for \( \tau \in ... | Yes |
Lemma 5.4. \( {\delta }_{D, u} : D \rightarrow \mathbb{R} \cup \{ \infty \} \) is lower semicontinuous on \( D \) (i.e., \( - {\delta }_{D, u} \) is upper semicontinuous), and\n\n(5.4)\n\n\[{\delta }_{D}\left( z\right) = \inf \left\{ {{\delta }_{D, u}\left( z\right) : u \in {\mathbb{C}}^{n}\text{ with }\left| u\right| ... | The proof is left to the reader. | No |
Theorem 5.6. If \( D \subset {\mathbb{C}}^{n} \) is Hartogs pseudoconvex, then \( - \log {\delta }_{D} \) is pluri-subharmonic on \( D \) . | Proof. From Lemma 5.4 it follows that\n\n\[ \n- \log {\delta }_{D} = \sup \left\{ {-\log {\delta }_{D, u} : \left| u\right| = 1}\right\} .\n\]\n\nThe conclusion then follows from Proposition 5.5 and Lemma 4.1. | No |
Theorem 5.8. The following properties are equivalent for an open set \( D \) in \( {\mathbb{C}}^{n} \) :\n\n(i) There is a \( {C}^{2} \) strictly plurisubharmonic exhaustion function for \( D \) (i.e., \( D \) is pseudoconvex according to the definition in \( §{2.10} \) ).\n\n(ii) There is a plurisubharmonic exhaustion... | Proof. We have, essentially, proved everything, except the equivalence of (iv) with the other properties. In fact,(i) \( \Rightarrow \) (ii) is trivial, and the sequence of implications (ii) \( \Rightarrow \) (iii) \( \Rightarrow \) (v) \( \Rightarrow \) (vi) \( \Rightarrow \) (vii) was proved in \( §{5.1} \) - \( §{5.... | Yes |
Theorem 5.9. (a) If \( \left\{ {{D}_{\alpha },\alpha \in I}\right\} \) is a collection of pseudoconvex domains, then the interior \( \Omega \) of \( \mathop{\bigcap }\limits_{{\alpha \in I}}{D}_{\alpha } \) is pseudoconvex. | Proof. Since (a) for finite collections and (b) hold for \( H \) -pseudoconvexity, Theorem 5.8 gives the desired result in this case. In order to prove (a) in general we use the characterization (iv) in Theorem 5.8. Let \( S \) be an analytic disc in \( \Omega \) . Then \( \operatorname{dist}\left( {\partial S,{b\Omega... | No |
Theorem 5.10. A region \( D \subset {\mathbb{C}}^{n} \) is pseudoconvex if and only if every point \( \zeta \in \bar{D} \) has a neighborhood \( {U}_{\zeta } \) such that \( {U}_{\zeta } \cap D \) is pseudoconvex. | Proof. One implication is obvious. For the other implication, we assume first that \( D \) is bounded. If \( \zeta \in {bD} \) and \( {U}_{\zeta } \cap D \) is pseudoconvex, then \( - \log {\delta }_{{U}_{\zeta } \cap D} \) is plurisubharmonic. But \( {\delta }_{{U}_{\zeta } \cap D}\left( z\right) = {\delta }_{D}\left(... | Yes |
Corollary 5.12. If \( D \subset {\mathbb{C}}^{n} \) is pseudoconvex and \( K \subset D \) is compact, then\n\n\[ {\widehat{K}}_{{PS}\left( D\right) } = {\widehat{K}}_{{PS}\left( D\right) \cap {C}^{\infty }\left( D\right) }.\]\n\nIn particular, \( {\widehat{K}}_{{PS}\left( D\right) } \) is closed in \( D \) (and hence c... | Proof. In order to prove (5.9) it is enough to prove \( {\widehat{K}}_{{PS}\left( D\right) \cap {C}^{\infty }\left( D\right) } \subset {\widehat{K}}_{{PS}\left( D\right) } \), the opposite inclusion being trivial. For this, note that if \( a \in D - {\widehat{K}}_{{PS}\left( D\right) } \), then by applying the theorem ... | Yes |
Theorem 5.13. A bounded domain in \( {\mathbb{C}}^{n} \) with \( {C}^{2} \) boundary is pseudoconvex if and only if it is Levi pseudoconvex. | Proof. We only need to show that a Levi pseudoconvex domain \( D \) is pseudo-convex. This has already been done in Lemma 2.19 in case \( D \) has boundary of class \( {C}^{3} \) . Even though the extension to \( {C}^{2} \) boundaries looks like just a minor technical improvement, the proof is somewhat deeper, as it ma... | No |
Theorem 1.1. The tangent space \( {T}_{P}M \) carries a unique structure of a real \( n \) -dimensional vector space such that for every coordinate system \( \left( {U,\varphi }\right) \) around \( P \) the differential \( d{\varphi }_{P} : {T}_{P}M \rightarrow {T}_{\varphi \left( P\right) }{\mathbb{R}}^{n} \) is an is... | Proof. If \( \left( {U,\varphi }\right) \) is a fixed coordinate system, then \( d{\varphi }_{P} : {T}_{P}M \rightarrow {T}_{\varphi \left( P\right) }{\mathbb{R}}^{n} \) is clearly one-to-one and onto (with inverse \( d{\left( {\varphi }^{-1}\right) }_{\varphi \left( P\right) } \) ), so there is a unique vector space s... | Yes |
Corollary 1.4. The exterior derivative is a local operator, i.e., if \( {\omega }_{1} = {\omega }_{2} \) on an open set \( U \), then \( d{\omega }_{1} = d{\omega }_{2} \) on \( U \) . | Proof. It is enough to show that if \( \omega \in {C}_{r}^{1}\left( M\right) \) is 0 on \( U \), then \( {d\omega } = 0 \) on \( U \) . If \( P \in U \), choose \( f \in {C}^{1}\left( M\right) \) such that \( f\left( P\right) = 0 \) and \( f = 1 \) in a neighborhood of \( M - U \) . Then \( \omega = {f\omega } \) ; by ... | Yes |
Corollary 1.5. If \( 2 \leq l < k \), then\n\n\[ d\left( {d\omega }\right) = 0\;\text{ for }\omega \in {C}_{r}^{l}\left( M\right) . \] | Proof. In a local coordinate system \( \left( {U,\varphi }\right) \) one represents \( {d\omega } \) as in (1.17). The conclusion then follows by (1.16) and (1.15). | No |
Theorem 1.6. Suppose \( M, N \), and \( F : M \rightarrow N \) are of class \( {C}^{k} \) .\n\n(i) The pull back \( {F}^{ * } \) is an algebra homomorphism\n\n\[ \n{F}^{ * } : {\mathcal{G}}^{0}\left( N\right) \rightarrow {\mathcal{G}}^{0}\left( M\right) \n\]\n\nwhich satisfies \( {F}^{ * }\left( {{C}_{r}^{l}\left( N\ri... | Proof. We first verify (1.20) for \( \omega = f \in {C}^{l}\left( N\right) ,1 \leq l \leq k \) . Since \( {F}^{ * }f = f \circ F \in \) \( {C}^{l}\left( M\right) \), and since \( {d}_{M} \) and \( {d}_{N} \) agree on functions with the usual differentials, one obtains\n\n\[ \n{d}_{M}\left( {{F}^{ * }f}\right) = d\left(... | Yes |
Theorem 1.7. Let \( M \) be an oriented \( {C}^{k} \) manifold, \( k \geq 2 \), of dimension \( n \), and let \( D \subset \subset M \) be an open set with \( {C}^{1} \) boundary \( {bD} \) . If \( \omega \in {C}_{n - 1}^{1}\left( \bar{D}\right) \), then\n\n\[{\int }_{bD}\omega = {\int }_{D}{d\omega }\] | Proof. By compactness of \( \bar{D} \) there are finitely many positively oriented coordinate systems \( \left( {{U}_{i},{\varphi }_{i}}\right) ,1 \leq i \leq l \), such that \( \bar{D} \subset \mathop{\bigcup }\limits_{{i = 1}}^{l}{U}_{i} \), and, if \( {U}_{i} \cap {bD} \neq \) \( \varnothing \) for some \( i \), the... | Yes |
Theorem 2.1. Let \( M \) be a complex manifold. For every \( P \in M \) there is a unique \( \mathbb{R} \) -linear map \( J = {J}_{P} : {T}_{P}M \rightarrow {T}_{P}M \) such that for all functions f holomorphic at \( P \) one has\n\n\[ d{f}_{P}\left( {Jv}\right) = {id}{f}_{P}\left( v\right) \;\text{ for all }v \in {T}_... | Proof. We first prove uniqueness. Let \( \left( {{z}_{1},\ldots ,{z}_{n}}\right) \) be a holomorphic coordinate system near \( P \) with underlying real coordinates \( \left( {{x}_{1},{y}_{1},\ldots ,{x}_{n},{y}_{n}}\right) \) ; then \( d{x}_{j} = \operatorname{Re}d{z}_{j} \) and \( d{y}_{j} = \operatorname{Im}d{z}_{j}... | Yes |
Proposition 2.2. The operators \( \partial \) and \( \bar{\partial } \) on a complex manifold \( M \) have the following properties\n\n(a) \( d = \partial + \bar{\partial } \) on \( {\mathcal{G}}^{1}\left( M\right) \).\n\n(b) \( \partial \circ \partial = 0,\bar{\partial } \circ \bar{\partial } = 0,\partial \circ \bar{\... | Proof. (a) is obvious since \( d = \partial + \bar{\partial } \) on \( {C}_{p, q}^{1} \) for all \( p, q \geq 0 \) . For (b), we recall that \( d \circ d = 0 \) on \( {\mathcal{G}}^{2}\left( M\right) \) ; combined with (a), one obtains\n\n(2.17)\n\n\[ 0 = \left( {\partial + \bar{\partial }}\right) \circ \left( {\partia... | Yes |
Theorem 3.2. Suppose \( \dim M = n \) and let \( d{V}_{P} \) be a volume form at \( P \in M \) . There is a unique \( \mathbb{C} \) -linear map \( * : {\mathcal{G}}_{P}\left( M\right) \rightarrow {\mathcal{G}}_{P}\left( M\right) \) with the following properties.\n\n(3.12)\n\n\[ * \varphi \in {\Lambda }^{n - r}\mathbb{C... | Proof. We first prove uniqueness. Choose an orthonormal basis \( {\omega }_{1},\ldots ,{\omega }_{n} \) of \( \mathbb{C}{T}_{P}^{ * }M \) such that \( {\omega }_{1} \land \ldots \land {\omega }_{n} = d{V}_{P} \) . By linearity of \( * \) it is enough to verify that the properties stated in the theorem determine \( * {\... | Yes |
Lemma 3.3. The \( * \) -operator on \( {\mathbb{C}}^{n} \) has the following additional properties.\n\n(3.18)\n\n\[ * \left( {{\Lambda }_{P}^{p, q}\left( {\mathbb{C}}^{n}\right) }\right) \subset {\Lambda }_{P}^{n - q, n - p}\left( {\mathbb{C}}^{n}\right) \;\text{ for all }0 \leq p, q \leq n\text{ and }P \in {\mathbb{C}... | Proof. If \( \varphi \) is a \( \left( {p, q}\right) \) form, then \( \langle \psi ,\varphi \rangle \neq 0 \) only for \( \psi \) of type \( \left( {p, q}\right) \) . From (3.16) it follows that \( * \bar{\varphi } \) is of type \( \left( {n - p, n - q}\right) \), i.e., \( * \varphi \) is of type \( (n - q \) , \( n - ... | Yes |
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