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Theorem 11.3. Let \( \\left( {A, X,\\Omega, p}\\right) \) be a maximum modulus algebra over \( \\Omega \) . Fix \( F \\in A \) . Then \( \\lambda \\mapsto \\log {Z}_{F}\\left( \\lambda \\right) \) is subharmonic on \( \\Omega \) . | Proof. In view of Exercise 11.3, it suffices to show that \( \\log {Z}_{F} \) satisfies the inequality (17).\n\nWe fix a disk \( \\Delta = \\left\\{ {\\left| {\\lambda - {\\lambda }_{0}}\\right| \\leq r}\\right\\} \) contained in \( \\Omega \) and apply Theorem 11.2 to the function \( F \), a point \( {x}^{0} \\in {p}^... | Yes |
Theorem 11.4. Let \( \\left( {A, X,\\Omega, p}\\right) \) be a maximum modulus algebra over \( \\Omega \) . Assume that \( \\Delta = \\{ \\left| z\\right| \\leq 1\\} \\subset \\Omega \) . Fix \( F \\in { \\otimes }^{n}A \) and fix \( {x}^{0} \\in {\\Pi }^{-1}\\left( {0,0,\\cdots ,0}\\right) \) . Then\n\n\[ \n\\left| {F... | To prove Theorem 11.4, we need the following.\n\nLemma 11.5. Under the | No |
Lemma 11.5. Under the hypothesis of Theorem 11.4, there exists a function \( G \in \) \( {H}^{\infty }\left( {T}^{n}\right) \) such that\n\n(18)\n\n\[ G\left( {0,0,\cdots ,0}\right) = F\left( {x}^{0}\right) ,\text{ and,}\]\n\nif \( \mathcal{U} \) is any relatively open subset of \( {T}^{n} \),\n\n(19)\n\n\[ \parallel G... | Proof. \( {x}^{0} = \left( {{x}_{1}^{0},\cdots ,{x}_{n}^{0}}\right) \) and \( p\left( {x}_{j}^{0}\right) = 0 \), for all \( j \) . We may choose representing measures \( {\mu }_{k} \) for \( {x}_{k}^{0} \), supported on \( {p}^{-1}\left( {\partial \Delta }\right) \), such that\n\n\[ g\left( {x}_{k}^{0}\right) = {\int }... | No |
Theorem 11.6. \( \left( {{A}^{\left( n\right) },{X}^{\left( n\right) },\Omega ,\pi }\right) \) is a maximum modulus algebra. | Proof. Clearly, \( {A}^{\left( n\right) } \) is an algebra of continuous functions on \( {X}^{\left( n\right) } \) . Let \( \Delta \) be a closed disk in \( \Omega \) with center \( {\lambda }_{0} \) . Fix \( {x}^{0} \in {\pi }^{-1}\left( {\lambda }_{0}\right) \subseteq {X}^{\left( n\right) } \) . We must show that\n\n... | Yes |
Theorem 11.7. \( \lambda \mapsto \log {d}_{n}\left( {g\left( {{f}^{-1}\left( \lambda \right) }\right) }\right) \) is subharmonic on \( \Omega \) . | Proof. For ease of understanding we take \( n = 3 \) . The proof is the same for each \( n \geq 2 \) . \( \log \left( {{d}_{3}\left( {g\left( {{f}^{-1}\left( \lambda \right) }\right) }\right) }\right) = \log \left\lbrack {\max \left| {{z}_{1} - {z}_{2}}\right| {\left| {z}_{1} - {z}_{3}\left| \right| {z}_{2} - {z}_{3} \... | Yes |
Proposition 11.13. Let \( \left( {A, X,\Omega, p}\right) \) satisfy all of the assumptions for a maximum modulus algebra except for (2). Suppose that (34) holds. Then (2) holds as well, and \( \left( {A, X,\Omega, p}\right) \) is a maximum modulus algebra on \( X \) with projection \( p \) . | Proof. Fix \( {\lambda }_{0} \in \Omega \) . Choose \( \epsilon \left( {\lambda }_{0}\right) \) as in (34). Let \( \Delta = \left\{ {\lambda : \left| {\lambda - {\lambda }_{0}}\right| \leq r}\right\} \), where \( r \leq \epsilon \left( {\lambda }_{0}\right) \) . Fix \( g \in A \) and define \( {Z}_{g} \) as in Definiti... | Yes |
Theorem 12.1. If \( \gamma \) is not polynomially convex, then \( \widehat{\gamma } \smallsetminus \gamma \) is a one-dimensional analytic subvariety of \( {\mathbb{C}}^{N} \smallsetminus \gamma \) . | Proof of THEOREM 12.1. First we get the sequence\n\n\[ {U}_{1},{U}_{2},\ldots ,{U}_{\ell } = {U}^{ * } \]\n\nfrom Step 4. By Step 3, \( {f}^{-1}\left( {U}_{0}\right) \) lies at most 0 -sheeted over \( {U}_{0} \) . Then, using Step 2 at each of the \ | No |
Lemma 12.2. Let \( \left( {A, X,\mathcal{M}}\right) \) be a uniform algebra. Fix \( p \in \mathcal{M} \smallsetminus X, f \in A \) , and put \( {\lambda }_{0} = f\left( p\right) \) . Define \( K \), a subset of \( \mathbb{C} \), by\n\n\[ K = \left\{ {\lambda : \left| {\lambda - {\lambda }_{0}}\right| \leq r,\alpha \leq... | Proof. By the Local Maximum Modulus Principle,\n\n\[ \left| {g\left( p\right) }\right| \leq \mathop{\max }\limits_{{\partial \mathcal{N}}}\left| g\right| ,\;g \in A. \]\n\nHence there exists a representing measure \( \mu \) for \( p \) on \( \partial \mathcal{N} \), with\n\n\[ g\left( p\right) = {\int }_{\partial \math... | Yes |
Theorem 12.4. Let \( \gamma \) be a smooth arc in \( {\mathbb{C}}^{N} \) . Then \( \gamma \) is polynomially convex and \( \mathbf{P}\left( \gamma \right) = C\left( \gamma \right) \) | Proof. First we show that \( \gamma \) is polynomially convex. Arguing by contradiction, we suppose that there exists \( {x}^{0} \in \widehat{\gamma } \smallsetminus \gamma \) . By Step 1 in the proof of Theorem 12.1, there exists a polynomial \( f \) such that \( f\left( {x}^{0}\right) = 0 \) and \( f \neq 0 \) on \( ... | Yes |
Theorem 12.5. Let \( \gamma \) be a finite union of smooth compact curves in \( {\mathbb{C}}^{N} \) (as in Theorem 12.1) and let \( K \) be a compact polynomially convex set in \( {\mathbb{C}}^{N} \) . Then \( \widehat{K \cup \gamma } \smallsetminus \left( {K \cup \gamma }\right) \) is a (possibly empty) one-dimensiona... | Sketch of PROOF. Suppose that there exists \( {x}^{0} \in \widehat{K \cup \gamma } \smallsetminus \left( {K \cup \gamma }\right) \) . We must show that \( \widehat{K \cup \gamma } \) is a one-dimensional analytic set near \( {x}^{0} \) . The first step is to produce a polynomial \( f \) in \( {z}_{1},{z}_{2},\cdots ,{z... | Yes |
Theorem 13.1. Assume that \( K \) is given by (1) and satisfies (4),(5), and (6). Then, for each \( f \in A\left( D\right) \cap {\mathcal{C}}^{1}\left( \bar{D}\right) \), we have\n\n\[ \n{c}_{0}f\left( z\right) = {\int }_{\partial D}f\left( \zeta \right) K\left( {\zeta, z}\right) ,\;z \in D.\n\] | Proof. Fix \( \epsilon > 0 \) and put \( {D}_{\epsilon } = D \smallsetminus \{ \left| {\zeta - z}\right| \leq \epsilon \} \) . On \( \overline{{D}_{\epsilon }}, K \) is a smooth \( \left( {n, n - 1}\right) \) -form in \( \zeta \) .\n\nFix \( f \in A\left( D\right) \cap {\mathcal{C}}^{1}\left( \bar{D}\right) \) . We hav... | Yes |
Theorem 13.4 (Leray’s Formula). With \( D, w \) as above, we have\n\n(15)\n\n\[ f\left( z\right) = {a}_{0}{\int }_{\partial D}f\left( \zeta \right) {K}_{w}\left( {\zeta, z}\right) ,\]\n\nfor every \( f \in A\left( D\right) \cap {\mathcal{C}}^{1}\left( \bar{D}\right) \) and \( z \in D \), where \( {a}_{0} = {\left( -1\r... | We shall deduce formula (15) from the corresponding result (12) for the Bochner-Martinelli kernel. To this end, we now prove a number of lemmas.\n\nFix a point \( \left( {{z}_{1},{z}_{2},\ldots ,{z}_{n}}\right) \) in \( {\mathbb{C}}^{n} \) . We use complex coordinates \( {Z}_{1},\ldots {Z}_{n} \) , \( {W}_{1},\ldots {W... | No |
Lemma 13.5. Fix positive integers \( N, k \) with \( k < N \). Let \( \sum \) be a \( k \)-dimensional complex submanifold of \( {\mathbb{C}}^{N} \), and let \( \alpha \) be a holomorphic \( k \)-form on \( {\mathbb{C}}^{N} \). Denoting by \( {\left. \alpha \right| }_{\sum } \) the form on \( \sum \) obtained by restri... | Proof. Let \( j : \sum \rightarrow {\mathbb{C}}^{N} \) be the (holomorphic) inclusion map. Then \( {\left. \alpha \right| }_{\sum } \) is just the \ | No |
Lemma 13.6. Fix \( z \in D \), then \( d{K}_{w}\left( {\zeta, z}\right) = 0 \) for \( \zeta \in D \smallsetminus \{ z\} \) . | Proof. Note that (13) yields \( d\left( {\mathop{\sum }\limits_{{j = 1}}^{n}\left( {{w}_{j}\left( {\zeta, z}\right) \left( {{\zeta }_{j} - {z}_{j}}\right) }\right) = 0}\right. \), or\n\n(19)\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\left\lbrack {\bar{\partial }{w}_{j}\left( {{\zeta }_{j} - {z}_{j}}\right) + \partial {w... | Yes |
Lemma 13.7. Fix \( z \in D \), and choose \( \epsilon > 0 \) such that the closed ball \( \{ \left| {\zeta - z}\right| \leq \epsilon \} \) is contained in D. Then\n\n\[ 1 = {a}_{0}{\int }_{\{ \left| {\zeta - z}\right| = \epsilon \} }{K}_{w}\left( {\zeta, z}\right) ,\] \n\nwhere \( {a}_{0} = {\left( -1\right) }^{n\left(... | Proof. Put \( {D}_{\epsilon } = D \smallsetminus \{ \left| {\zeta - z}\right| \leq \epsilon \} .{\int }_{\partial {D}_{\epsilon }}{K}_{w} = {\int }_{{D}_{\epsilon }}d{K}_{w} = 0 \), by Lemma 13.6. So\n\n\[ {\int }_{\partial D}{K}_{w} = {\int }_{\{ \left| {\zeta - z}\right| = \epsilon \} }{K}_{w} \]\n\nBy Leray's formul... | Yes |
Lemma 14.1. Given an integer \( n \geq 1 \), there exists a polynomial \( {P}_{n} \) so that\n\n\[ \left| {{P}_{n}\left( w\right) - \frac{1}{w + \frac{1}{n}}}\right| \leq \frac{1}{n},\;w \in S \]\n\nwhere \( S \) is the closed semidisk\n\n\[ \left\{ {w \in \mathbb{C} : \operatorname{Re}\left( w\right) \geq 0\text{ and ... | Proof. Exercise 14.1. | No |
Lemma 14.2. Let \( S \) be as above. There exists a sequence of polynomials \( \left\{ {P}_{n}\right\} \) such that\n\n(1)\n\n\[ \n{P}_{n}\left( w\right) \rightarrow \frac{1}{w},\;w \in S \smallsetminus \{ 0\} ,\text{ as }n \rightarrow \infty ,\text{ and }\n\]\n\n(2)\n\n\[ \n\left| {{P}_{n}\left( w\right) }\right| \leq... | Proof. Let \( {P}_{n} \) be as in Lemma 14.1. Then\n\n\( \left| {w{P}_{n}\left( w\right) }\right| \)\n\n\[ \n= \left| w\right| \left| {{P}_{n}\left( w\right) - \frac{1}{w + \frac{1}{n}} + \frac{1}{w + \frac{1}{n}}}\right| \leq \left| w\right| \left| {{P}_{n}\left( w\right) - \frac{1}{w + \frac{1}{n}}}\right| \n\]\n\n\[... | Yes |
Theorem 14.3. Assume that there is a constant \( k < 1 \) such that\n\n\[ \left| {R\left( z\right) - R\left( {z}^{\prime }\right) }\right| \leq k\left| {z - {z}^{\prime }}\right| ,\;z,{z}^{\prime } \in D.\]\n\nThen \( \left\lbrack {z,\bar{z} + R\left( z\right) }\right\rbrack \) is dense in \( C\left( D\right) \) . | Proof. Write \( \mathfrak{A} = \left\lbrack {z,\bar{z} + R\left( z\right) }\right\rbrack \) . Fix a point \( a \in \mathbb{C} \) . Let \( \mu \) be a measure on \( D \) with \( \mu \bot \mathfrak{A} \) . If \( \left| a\right| > 1 \), we have\n\n\[ \frac{1}{z - a} = - \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{z}^{... | Yes |
Theorem 14.4. Assume that there exists \( k,0 \leq k < 1 \), with\n\n\[ \left| {R\left( z\right) - R\left( {z}^{\prime }\right) }\right| \leq k\left| {z - {z}^{\prime }}\right| ,\;z,{z}^{\prime } \in N. \]\n\nThen \( \mathfrak{A} = \left\lbrack {{z}_{1},\cdots ,{z}_{n},{\bar{z}}_{1} + {R}_{1},\cdots ,{\bar{z}}_{n} + {R... | The method of proof consists of replacing the Cauchy kernel \( {dz}/\left( {z - a}\right) \), used in the proof of Theorem 14.3, by a suitably constructed Cauchy-Fantappie kernel \( K\left( {\zeta, z}\right) \) . | No |
Lemma 15.2. \( {K}^{\mathcal{U},\mathcal{V}} \) depends only on \( \mathcal{U} \) and \( \mathcal{V} \), not on the choice of \( \phi \) . | For the proofs see H.-W. | No |
Lemma 15.6. Let \( \mathcal{L} \) be a finitely generated uniform algebra on a space \( X \) with \( X = \mathcal{M}\left( \mathcal{L}\right) \) . Then \( \mathcal{L} \) is full [as subalgebra of \( C\left( X\right) \) .] | Proof. By Exercise 7.3 it suffices to assume that \( \mathcal{L} = P\left( X\right), X \) a compact polynomially convex set in \( {\mathbb{C}}^{n} \) . By the Oka-Weil theorem \( \mathcal{H}\left( X\right) \subset P\left( X\right) \) . Fix \( \gamma \in {H}^{1}\left( {X, Z}\right) \) . By the last lemma, \( \exists f \... | "No" |
Lemma 15.7. Let \( \mathcal{L} \) be an arbitrary uniform algebra on a space \( X \) with \( X = \) \( \mathcal{M}\left( \mathcal{L}\right) \) . Then \( \mathcal{L} \) is full. | Proof of Theorem 15.3. Put \( X = \mathcal{M} \) and let \( \mathcal{L} \) be the uniform closure of \( \mathfrak{A} \) on \( X \) . Then \( X = \mathcal{M}\left( \mathcal{L}\right) \) . By Lemma 15.7 \( \mathcal{L} \) is full [as subalgebra of \( C\left( X\right) \) ]. | No |
Theorem 15.8. Let \( \\mathfrak{A} \) be a Banach algebra with \( n \) generators. Then \( {H}^{p}\\left( {\\mathcal{M},\\mathbb{C}}\\right) = \) \( 0, p \\geq n \) . | This result is due to A. Browder, Cohomology of maximal ideal spaces, Bull. Am. Math. Soc. 67 (1961), 515-516. Observe that if \( \\mathfrak{A} \) has \( n \) generators, then \( \\mathcal{M} \) is homeomorphic to a subset of \( {\\mathbb{C}}^{n} \) and hence that the vanishing of \( {H}^{p}\\left( {\\mathcal{M},\\math... | No |
Theorem 16.3. Fix \( f \) in \( {C}_{0,1}^{1}\left( \bar{\Omega }\right) \) . Let \( f \in {\mathcal{D}}_{{T}_{0}^{ * }} \cap {\mathcal{D}}_{{S}_{0}} \) . Then\n\n\( {\left. \left( 7\right) {T}_{0}^{ * }f\right| }_{1}^{2} + {\begin{Vmatrix}{S}_{0}f\end{Vmatrix}}_{3}^{2} = \mathop{\sum }\limits_{{j, k}}{\int }_{\Omega }... | Suppose for the moment that Theorem 16.3 has been established. Put\n\n\[ \phi \left( z\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {z}_{j}\right| }^{2} = {\left| z\right| }^{2} \]\n\nThen \( {\partial }^{2}\phi /\partial {z}_{j}\partial {\bar{z}}_{k} = 0 \) if \( j \neq k, = 1 \) if \( j = k \) . The first inte... | No |
Lemma 16.6. Fix \( f \in {\mathcal{D}}_{S} \cap {C}_{0,1}^{1}\left( \bar{\Omega }\right) \) . Then\n\n(25)\n\n\[ \parallel {Sf}{\parallel }_{3}^{2} = {\int }_{\Omega }\mathop{\sum }\limits_{{j, k}}{\left| \frac{\partial {f}_{k}}{\partial {\bar{z}}_{j}}\right| }^{2} - {\int }_{\Omega }\mathop{\sum }\limits_{{j, k}}\frac... | Proof. Since \( f \in {C}_{0,1}^{1}\left( \bar{\Omega }\right) \) . Then\n\n\[ {Sf} = \bar{\partial }f = \mathop{\sum }\limits_{\alpha }\left( {\mathop{\sum }\limits_{\beta }\frac{\partial {f}_{\alpha }}{\partial {\bar{z}}_{\beta }}d{\bar{z}}_{\beta }}\right) \land d{\bar{z}}_{\alpha }\n\n\[ = \mathop{\sum }\limits_{{\... | Yes |
Put \( B = \left\{ {z \in {\mathbb{C}}^{n}\left| \right| z \mid < 1}\right\} \) . There exists a constant \( K \) such that for \( w \in {C}^{\infty }\left( {\mathbb{C}}^{n}\right) \) ,\n\n\[ \left| {w\left( 0\right) }\right| \leq K\left\{ {\parallel w{\parallel }_{{L}^{2}\left( B\right) } + \mathop{\sup }\limits_{B}\l... | Proof. It is a fact form classical potential theory that if \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{N}\right) \), then\n\n\[ f\left( y\right) = C{\int }_{{\mathbb{R}}^{N}}{\Delta F}\frac{dx}{{\left| x - y\right| }^{N - 2}}, \]\n\nwhere \( C \) is a constant depending on \( N \) and \( {dx} \) is Lebesgue measure... | Yes |
Lemma 16.9. Let \( \Omega \) be a bounded domain in \( {\mathbb{C}}^{n} \) and \( u \in {L}^{2}\left( \Omega \right) \) . Assume that for all \( j \) ,\n\n(32)\n\n\[ \frac{\partial u}{\partial {\bar{z}}_{j}} = 0\text{as a distribution on}\Omega \text{.} \]\n\nThen \( u \in H\left( \Omega \right) \) . | Proof. Define \( u = 0 \) outside \( \Omega \) . Then \( u \in {L}^{2}\left( {\mathbb{C}}^{n}\right) \) . By a change of variable, we get\n\n\[ {u}_{\varepsilon }\left( z\right) = \inf u\left( \zeta \right) {\chi }_{\varepsilon }\left( {z - \zeta }\right) {d\zeta }. \]\n\nFix \( j \) . Note that \( \left( {\partial \le... | Yes |
Theorem 17.1. Let \( \sum \) be a \( k \) -dimensional sufficiently smooth submanifold of an open set in \( {\mathbb{C}}^{n} \) . Assume that \( \sum \) has no complex tangents. Let \( X \) be a compact polynomially convex subset of \( \sum \) . Then \( P\left( X\right) = C\left( X\right) \) . | Sketch of Proof. To show that \( P\left( X\right) = C\left( X\right) \) we need only show that \( P\left( X\right) \) contains the restriction to every \( X \) of every \( u \in {C}^{\infty }\left( {\mathbb{C}}^{n}\right) \), since such functions are dense in \( C\left( X\right) \) . Fix \( u \in {C}^{\infty }\left( {\... | No |
Lemma 17.2. Let \( \sum \) be a submanifold of an open set in \( {\mathbb{C}}^{n} \) of class 2 such that \( \sum \) has no complex tangents. Let \( d \) be the distance function to \( \sum \) ; i.e., if \( x \in {\mathbb{C}}^{n}, d\left( x\right) \) is the distance from \( x \) to \( \sum \) . Then \( \exists \) a nei... | Proof of Lemma 17.2. Let \( U \) be a neighborhood of \( \sum \) such that \( {d}^{2} \in {C}^{2}\left( U\right) \) .\n\nFix \( {z}_{0} \in \sum \) . We assert that\n\n(2)\n\n\[ \mathop{\sum }\limits_{{j, k = 1}}^{n}\frac{{\partial }^{2}\left( {d}^{2}\right) }{\partial {z}_{j}\partial {\bar{z}}_{k}}\left( {z}_{0}\right... | No |
Lemma 17.6. \( \sum \) has no complex tangents. | Proof. If \( \sum \) has a complex tangent, then \( \exists \) two tangent vectors to \( \sum \) differing only by the factor \( i \). With \( {d\Phi } \) denoting the differential of the map \( \Phi \), we can hence find \( \xi ,\eta \in {\mathbb{C}}^{n} \) different from 0 so that at some point of \( \Omega \), \[ {d... | Yes |
Lemma 17.7. \( \Phi \left( X\right) \) is a polynomially convex compact set in \( {\mathbb{C}}^{2n} \) . | Proof. Put \( \mathfrak{A} = \left\lbrack {{z}_{1},\ldots ,{z}_{n},{\bar{z}}_{1} + {R}_{1},\ldots ,{\bar{z}}_{n} + {R}_{n} \mid X}\right\rbrack \), \[ {\mathfrak{A}}_{1} = \left\lbrack {{z}_{1},\ldots ,{z}_{2n} \mid {X}_{1}}\right\rbrack ,\;\text{ where }{X}_{1} = \Phi \left( X\right) . \] The map \( \Phi \) induces an... | Yes |
Lemma 18.2. Let \( S \) be a set in \( {\mathbb{C}}^{n} \) homeomorphic to the \( n \) -sphere. Then \( h\left( S\right) \neq S \) . | Proof. \( h\left( S\right) = \mathcal{M}\left( {P\left( S\right) }\right) \) . The algebra \( P\left( S\right) \) has \( n \) generators and hence by Theorem 15.8 the \( n \) ’th cohomology group of \( \mathcal{M}\left( {P\left( S\right) }\right) \) with complex coefficients vanishes. But \( {H}^{n}\left( {S,\mathbb{C}... | Yes |
Lemma 18.4. Let \( {w}_{2},\ldots ,{w}_{n} \) be smooth boundary functions with \( \left| {w}_{j}\right| < \delta \) for all \( j \) and such that \( {w}_{2} \) is schlicht, i.e., its analytic extension is one-one in \( \left| \zeta \right| \leq 1 \) . Put \( A = {A}_{w} \) . Suppose \( {x}^{ * } \in {H}_{1},\left| {x}... | Proof. Since \( A{x}^{ * } = {x}^{ * },{x}^{ * } = - T\left\{ {h\left( {{X}^{ * }, w}\right) }\right\} \), and so \( {x}^{ * } + {ih}\left( {{x}^{ * }, w}\right) \) is a boundary function by (2). Let \( \psi \) be the analytic extension of \( {x}^{ * } + {ih}\left( {{x}^{ * }, w}\right) \) to \( \left| \zeta \right| < ... | Yes |
For all sufficiently small \( M > 0 \) the following holds: if \( \parallel w{\parallel }_{1} < \) \( M \) and \( A = {A}_{w} \), then \( A \) maps \( {B}_{M} \) into \( {B}_{M} \) and \( \exists \alpha ,0 < \alpha < 1 \), such that \( \parallel {Ax} - {Ay}{\parallel }_{1} \leq \alpha \parallel x - y{\parallel }_{1} \)... | Fix \( M \) and choose \( w \) with \( \parallel w{\parallel }_{1} < M \) and choose \( x \in {B}_{M} \) . The map \( \left( {x, w}\right) \) takes \( \Gamma \) into \( \mathbb{R} \times {\mathbb{C}}^{n - 1} = {\mathbb{R}}^{{2n} - 1} \) . If \( M \) is small, \( \parallel \left( {x, w}\right) {\parallel }_{\infty } \le... | Yes |
Theorem 19.1. Let \( \gamma \) be an oriented simple closed curve in \( {\mathbb{C}}^{2} \) with a finite number of self-intersections. Then a necessary and sufficient condition that there exists a bounded analytic variety \( \sum \) in \( {\mathbb{C}}^{2} \) with \( {b\sum } = \pm \gamma \) is that \( \gamma \) satisf... | The complete proof of this theorem involves a considerable number of technical details, and we shall refer the reader to the paper of Harvey and Lawson [HarL2] for these. Here we shall present a sketch that we hope conveys the essential aspects of the construction.\n\nThe orientation of an analytic variety in \( {\math... | No |
Lemma 19.2. Let \( i, j \) be indices such that the regions \( {U}_{i},{U}_{j} \) have a common smooth (open) boundary arc \( \alpha \), with \( \alpha \) positively oriented for \( {U}_{j} \). Then:\n\n(i) \( {F}_{i} \) has a continuous extension to \( \left( {{U}_{i} \cup \alpha }\right) \times \{ \left| w\right| > R... | Proof. We note that the hypothesis, that \( \alpha \) is positively oriented for \( {U}_{j} \), is equivalent to the identity \( {n}_{j} = {n}_{i} + 1 \) for the winding numbers. We represent \( \gamma \) by the equation:\n\n\[ \eta = f\left( \zeta \right) ,\;\zeta \in \pi \left( \gamma \right) . \]\n\nFix \( w \) with... | Yes |
Lemma 19.4. \( F\left( {z, w}\right) = 1 \) for \( z \in {U}_{0},\left| w\right| > R \), where \( {U}_{0} \), as earlier, is the unbounded component of \( \mathbb{C} \smallsetminus \pi \left( \gamma \right) \) . | Proof. Recall that \( R \) can be chosen so that \( \gamma \) is contained in the polydisk \( \Omega = \) \( \{ \left( {\zeta ,\eta }\right) : \left| \zeta \right| < R,\left| \eta \right| < R\} \) . Fix \( z, w \) in \( \mathbb{C} \) such that \( \left| z\right| > R \) and \( \left| w\right| > R \) . Then, with an appr... | Yes |
Lemma 19.5. Fix \( i \geq 0 \) . Then \( {F}_{i} \) is the quotient of two polynomials in \( w \) with coefficients analytic for \( z \in {U}_{i},\left| w\right| > R \) . In particular, \( {F}_{i} \) has a meromorphic continuation to \( {U}_{i} \times \mathbb{C} \) . | Proof. The statement for \( i = 0 \) follows from Lemma 19.4.\n\nNow consider the situation when \( {U}_{k} \) and \( {U}_{j} \) are adjacent components of \( \mathbb{C} \smallsetminus \) \( \pi \left( \gamma \right) \) with common boundary arc \( \alpha \) . Suppose, for definiteness, that \( \alpha \) is positively\n... | Yes |
Proposition 19.7. Let \( \psi \left( {\zeta ,\eta }\right) \) be a function analytic on a neighborhood of \( M \) in \( {\mathbb{C}}^{3} \), where \( \zeta = \left( {{\zeta }_{1},{\zeta }_{2}}\right) \) . Put \[ F\left( z\right) = {\int }_{M}\psi \left( {\zeta ,\eta }\right) {\pi }^{ * }\left( {K\left( {\zeta, z}\right... | Proof. We regard \( K \) as defined on \( {\mathbb{C}}^{2} \times {\mathbb{C}}^{2} \smallsetminus \{ z = \zeta \} \) . Here \[ K\left( {\zeta, z}\right) = \left( {\frac{{\bar{\zeta }}_{1} - {\bar{z}}_{1}}{{\left| \zeta - z\right| }^{4}}d{\bar{\zeta }}_{2} - \frac{{\bar{\zeta }}_{2} - {\bar{z}}_{2}}{{\left| \zeta - z\ri... | Yes |
Lemma 19.8. \( - {\bar{\partial }}_{\zeta }{K}_{1}\left( {\zeta, z}\right) = {\bar{\partial }}_{z}K\left( {\zeta, z}\right) \) . | Proof. See Appendix A13. | No |
Lemma 19.9. For \( \left( {\zeta ,\eta }\right) \in M \) (and \( z \in U \) fixed)\n\n\[ \psi \left( {\zeta ,\eta }\right) {\pi }^{ * }\left( {{\bar{\partial }}_{\zeta }{K}_{1}\left( {\zeta, z}\right) }\right) = d\left( {\psi \left( {\zeta ,\eta }\right) {\pi }^{ * }\left( {{K}_{1}\left( {\zeta, z}\right) }\right) }\ri... | Proof. Since the map \( \pi \) and the function \( \psi \) are holomorphic, we have\n\n\[ \psi \left( {\zeta ,\eta }\right) {\pi }^{ * }\left( {{\bar{\partial }}_{\zeta }{K}_{1}\left( {\zeta, z}\right) }\right) = {\bar{\partial }}_{\zeta }\left( {\psi \left( {\zeta ,\eta }\right) {\pi }^{ * }\left( {{K}_{1}\left( {\zet... | Yes |
Lemma 19.10. Let \( \alpha \) be a form of type \( \left( {3,0}\right) \) defined on a neighborhood of a maximally complex real 3-manifold \( M \) in \( {\mathbb{C}}^{3} \). Then the restriction of \( \alpha \) to \( M \) is identically zero. | Proof. We need only verify this at the tangent space \( {T}_{x}\left( M\right) \) at a single point \( x \in M \). By a linear change of variable we may assume, by the maximal complexity of \( M \), that \( {T}_{x}\left( M\right) = \left\{ {\left( {{z}_{1},{z}_{2},{z}_{3}}\right) \in {\mathbb{C}}^{3} : \operatorname{Im... | Yes |
Theorem 20.2. Fix a compact set \( Y \) in \( {\mathbb{C}}^{2} \) lying over \( \Gamma \) . Assume\n\n\[ \n{Y}_{\lambda }\text{is convex for every}\lambda \in \Gamma \text{.\n\]\n\nThen \( \widehat{Y} \smallsetminus Y \) equals the union of all graphs \( \{ \left( {\lambda, f\left( \lambda \right) }\right) : \left| \la... | Proof. Because of the claim just proved, it suffices to show that if \( \left( {{\lambda }_{0},{w}_{0}}\right) \in \) \( \widehat{Y} \smallsetminus Y \), then there exists \( f \in \mathcal{F} \) with \( f\left( {\lambda }_{0}\right) = {w}_{0} \).\n\nWe first take \( {\lambda }_{0} = 0 \) . Since \( \left( {0,{w}_{0}}\... | Yes |
Theorem 20.3. Let \( Y \) be a compact set in \( {\mathbb{C}}^{2} \) lying over \( \Gamma \) . Assume again that each fiber \( {Y}_{\lambda },\lambda \in \Gamma \), is convex. Then each fiber \( {\widehat{Y}}_{\lambda },\left| \lambda \right| < 1 \), is convex. | Proof. Fix \( {\lambda }_{0},\left| {\lambda }_{0}\right| < 1 \), and choose points \( {w}_{1},{w}_{2} \in {\widehat{Y}}_{{\lambda }_{0}} \) . By Theorem 20.2, there exist \( {f}_{1},{f}_{2} \in \mathcal{F} \) with \( {f}_{j}\left( {\lambda }_{0}\right) = {w}_{j}, j = 1,2 \) . Put\n\n\[ f = \frac{1}{2}\left( {{f}_{1} +... | Yes |
Lemma 20.4. \( \mathcal{F} \) consists of all functions \( f \) on \( \{ \left| \lambda \right| < 1\} \) that can be written in the form\n\n(12)\n\n\[ f = \frac{P}{Q} + \frac{B}{{Q}_{0}} \]\n\nwith \( B \in {H}^{\infty },\parallel B{\parallel }_{\infty } \leq 1 \), and\n\n(13)\n\n\[ B\left( {\lambda }_{j}\right) = {w}_... | Proof. Suppose that \( f \in \mathcal{F} \) . Define the function \( \zeta \) by\n\n\[ f = \frac{P}{Q} + \zeta \]\n\nThen \( \zeta \) is meromorphic on \( \{ \left| \lambda \right| < 1\} \) with a simple pole at each \( {\lambda }_{j} \) . Also,\n\n\[ \zeta = \frac{{fQ} - P}{Q} = \left( \frac{{fQ} - P}{R}\right) \frac{... | Yes |
Corollary 20.6. The fibers \( \left\{ {{\widehat{Y}}_{\lambda } : \left| \lambda \right| < 1}\right\} \) form a continuously varying family of nondegenerate closed disks. | Proof. We know that the fibers are nondegenerate since \( {\widehat{Y}}_{\lambda } \supseteq \{ \left| w\right| \leq 1 - k\} \) for all \( \lambda \) with \( \left| \lambda \right| < 1 \) . Fix \( b \) in the open disk and suppose that \( \left\{ {z}_{v}\right\} \) is a sequence\nin\n\n\nin \( \Lambda \) approaching \(... | Yes |
Lemma 21.2. Let \( K \) be a compact subset of the plane, and set\n\n\[ F\left( z\right) = \iint {\int }_{K}\frac{1}{\zeta - z}{dudv},\;\text{ where }\zeta = u + {iv}. \]\n\nThen \( F \) is continuous on the plane and \( \parallel F{\parallel }_{K} \leq {\left( \pi \operatorname{area}\left( K\right) \right) }^{\frac{1}... | Proof. The continuity of \( F \) is a consequence of the fact that the convolution of a local \( {L}^{1} \) function with a bounded function of compact support is continuous. To estimate \( F\left( z\right) \), we may assume, by a translation, that \( z = 0 \) . Then, by a rotation we may assume that \( F\left( 0\right... | Yes |
Lemma 21.3. \( \operatorname{dist}\left( {\bar{z}, R\left( K\right) }\right) \leq {\left( \frac{1}{\pi }\operatorname{area}\left( K\right) \right) }^{\frac{1}{2}} \) | Proof of Lemma 21.3. Let \( \psi \) be a \( {\mathcal{C}}^{\infty } \) function with compact support in the plane such that \( \psi \left( z\right) \equiv \bar{z} \) on a neighborhood of \( K \) . By the generalized Cauchy integral formula,\n\n\[ \psi \left( z\right) = - \frac{1}{\pi }\iint \frac{\partial \psi }{\parti... | Yes |
Proposition 21.4. Let \( \Omega \) be a closed Jordan domain in the plane with a smooth boundary. Let \( A = \operatorname{area}\left( \Omega \right) \) and let \( L = \operatorname{length}\left( {b\Omega }\right) \) . Then \( {2A}/L \leq \) \( \operatorname{dist}\left( {\bar{z}, R\left( \Omega \right) }\right) \) . | Proof. Let \( \epsilon > 0 \) and choose \( g \) a rational function with no poles on \( \Omega \) such that \( \parallel \bar{z} - g{\parallel }_{\Omega } < \operatorname{dist}\left( {\bar{z}, R\left( \Omega \right) }\right) + \epsilon \) . We have, since \( {\int }_{b\Omega }{gdz} = 0 \) by Cauchy,\n\n\[ \left| {{\in... | Yes |
Theorem 21.5. Let \( {f\varepsilon }\mathcal{A} \) and suppose that \( \phi \left( f\right) = 0 \) . Then for all \( t \geq 0 \) , \[ {2\pi t\mu }\{ x \in X : \left| {f\left( x\right) }\right| \geq t\} \leq \ell \left( {{\Gamma }_{t} \cap f\left( M\right) }\right) . \] | Proof of Theorem 21.5. Fix \( t > 0 \) . Let \( \epsilon > 0 \) and put \( \gamma = {\Gamma }_{t} \cap f\left( M\right) \) . Choose a continuous real-valued function \( h \) on \( {\Gamma }_{t} \) such that \( 0 \leq h \leq 1, h \) is identically 1 on a neighborhood of \( \gamma \), and \( {\int }_{0}^{2\pi }h\left( {t... | Yes |
Lemma 21.6. Let \( V \) be a one-dimensional analytic subvariety of an open subset of \( {\mathbb{C}}^{n} \) . Then\n\n\[ \operatorname{area}\left( V\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\text{ area-with-multiplicity }\left( {{z}_{j}\left( V\right) }\right) . \]\n\nHere, by the area of \( V \) we understand the ... | Proof. This is a local result and so we can assume that \( V \) can be parameterized by a one-one analytic map \( f : W \rightarrow V \subseteq {\mathbb{C}}^{n} \), where \( W \) is a domain in the complex plane. Let \( \zeta \in W \) be \( \zeta = s + {it} \) and let \( f = \left( {{f}_{1},{f}_{2},\ldots ,{f}_{n}}\rig... | Yes |
Corollary 21.10. Let \( X \) be a compact subset of \( {\mathbb{C}}^{n} \). Suppose that \( p \in \widehat{X} \) and that \( B\left( {p, r}\right) \subseteq {\mathbb{C}}^{n} \smallsetminus X \). Then \( {\mathcal{H}}^{2}\left( {\widehat{X} \cap B\left( {p, r}\right) }\right) \geq \pi {r}^{2} \). | Proof of the Corollary. If \( {\mathcal{H}}^{2}\left( {\widehat{X} \cap B\left( {p, r}\right) }\right) < \infty \), then the theorem implies that \( \widehat{X} \cap B\left( {p, r}\right) \) is a one-dimensional analytic set and so Rutishauser’s theorem applies. If \( {\mathcal{H}}^{2}\left( {\widehat{X} \cap B\left( {... | No |
Lemma 22.1. Let \( p \) be an isolated nondegenerate critical point of \( \psi \) . Then the index of \( \psi \) at \( p \) is \( \leq n \) . | Proof. We can assume that \( p = 0 \) and that \( \psi \left( z\right) = \psi \left( 0\right) + \sum {c}_{i\bar{j}}{z}_{i}{\bar{z}}_{j} + \) \( \operatorname{Re}\sum {a}_{ij}{z}_{i}{z}_{j} + O\left( {\left| z\right| }^{3}\right) \), where the first sum is positive definite, and so we can further assume that \( {c}_{i\b... | Yes |
Lemma 22.4. Let \( K \) be polynomially convex in \( {\mathbb{C}}^{n} \) with \( K \subseteq U \), with \( U \) open and bounded. Then there exists a smooth strictly plurisubharmonic function \( \rho \) : \( {\mathbb{C}}^{n} \rightarrow \mathbb{R} \) and \( R > 0 \) such that:\n\n(i) \( \rho < 0 \) on \( K \) and \( \r... | Proof of THE LEMMA. We first construct a smooth strictly plurisubharmonic function \( v \) on \( {\mathbb{C}}^{n} \) satisfying (i). Choose \( C > \) the maximum of \( {\left| z\right| }^{2} \) on \( K \) . Let \( L = \left\{ {z \in {\mathbb{C}}^{n} : {\left| z\right| }^{2} - C \leq 0}\right\} \smallsetminus U \) . The... | Yes |
If \( K \) in \( {\mathbb{C}}^{n} \) is polynomially convex, then \( K \) is a decreasing limit of Runge domains. More precisely, if \( U \) is an open set containing \( K \), then there exists a bounded Runge domain \( \Omega \) such that | Proof. Choose a constant \( C > 0 \) such that \( \left| {z}_{j}\right| \leq C \) on \( K \) for \( 1 \leq j \leq n \) and set \( L = \left\{ {z \in {\mathbb{C}}^{n} : \left| {z}_{j}\right| \leq C\text{for}1 \leq j \leq n}\right\} \smallsetminus U \) . Then \( L \) is compact and disjoint from \( K \) . For all \( q \i... | Yes |
Theorem 22.6. Let \( f \) be a continuous complex-valued function defined on \( b{\mathbb{B}}_{n} \) , \( n \geq 2 \) . Let \( G\left( f\right) \) be the graph of \( f \) in \( {\mathbb{C}}^{n + 1} \) . Then \( \widehat{G\left( f\right) } \) covers \( {\mathbb{B}}_{n} \) ; i.e., the projection of the set \( \widehat{G\... | Proof. Let \( \pi : {\mathbb{C}}^{n + 1} \rightarrow {\mathbb{C}}^{n} \) be the projection \( \pi \left( {z,{z}_{n + 1}}\right) = z \) for \( z \in {\mathbb{C}}^{n} \) . We argue by contradiction and suppose that \( \pi \left( \widehat{G\left( f\right) }\right) \) does not contain \( {\mathbb{B}}_{n} \) . By applying a... | Yes |
Theorem 23.1. Fix a compact set \( Y \) in \( {\mathbb{C}}^{2} \) and fix a point \( {p}^{0} \in \widehat{Y} \smallsetminus Y \) . Let \( B \) be an open ball in \( {\mathbb{C}}^{2} \), centered at \( {p}^{0} \), such that \( \bar{B} \) does not meet \( Y \) . Then each connected component of \( B \smallsetminus \wideh... | Why should we expect this theorem to be true? Let \( Y \) be a smooth closed curve in \( {\mathbb{C}}^{2} \) with \( \widehat{Y} \smallsetminus Y \) nonempty. Then \( \widehat{Y} \smallsetminus Y \) is a one-dimensional analytic variety by Chapter 12. Fix \( {p}^{0} \in \widehat{Y} \smallsetminus Y \) . In a small ball... | Yes |
Lemma 23.3. Suppose that \( \partial \Omega \) is strictly pseudoconvex at \( {z}^{0} \), where \( \Omega \subseteq {\mathbb{C}}^{n} \) . Then there exists an open neighborhood \( V \) of \( {z}^{0} \) in \( {\mathbb{C}}^{n} \) and a biholomorphism \( \phi \) of \( V \) onto an open set in \( {\mathbb{C}}^{n} \) such t... | Proof of Lemma 23.3. Without loss of generality we may assume that \( {z}^{0} = 0 \) and that \( {T}_{{z}^{0}}\left( {\partial \Omega }\right) = \left\{ {{x}_{n} = 0}\right\} \) . We write\n\n\[ \rho \left( z\right) = {x}_{n} + \operatorname{Re}\left( {\mathop{\sum }\limits_{{1 \leq j, k \leq n}}{\alpha }_{jk}{z}_{j}{z... | Yes |
Theorem 23.5. Let \( \Omega \) be a region in \( {\mathbb{C}}^{2} \) and fix \( {z}^{0} \in \partial \Omega \) with \( \partial \Omega \) smooth near \( {z}^{0} \) . Assume that there exists an analytic disk, \( z = f\left( \lambda \right) \), with \( f\left( 0\right) = {z}^{0} \), which is contained in \( \partial \Om... | Proof. Let \( \Omega \) be given by \( \{ \rho \left( z\right) < 0\} \) with \( \rho \) smooth in a neighborhood of \( {z}^{0} \) . Then\n\n\[ \rho \left( {f\left( \lambda \right) }\right) = 0,\left| \lambda \right| < 1. \]\n\nDifferentiating, we get, writing \( {f}_{j}^{\prime } = d{f}_{j}/{d\lambda },\partial /\parti... | Yes |
Theorem 23.6. Let \( Y \) be a compact set in \( {\mathbb{C}}^{2} \) lying over the circle \( \left\{ {\left| {z}_{1}\right| = 1}\right\} \) . Let \( \Omega \) be the interior of \( \widehat{Y} \) ( \( \Omega \) is contained in \( \left\{ {\left| {z}_{1}\right| < 1}\right\} \) ). Then \( \partial \Omega \) is Levi-flat... | Proof. Fix \( {z}^{0} \in \partial \Omega \) with \( \left| {z}_{1}^{0}\right| < 1 \) . Choose a defining function \( \rho \) for \( \Omega \) such that\n\n\[ \Omega = \{ \rho \left( z\right) < 0\} \]\n\n\( \rho \) is smooth in a neighborhood of \( {z}^{0} \) in \( {\mathbb{C}}^{2} \), and \( {\left( \nabla \rho \right... | Yes |
Claim 1. Assume that \( S \) has no interior. Then \( {h}_{r}\left( X\right) \) contains no analytic disk. | Proof. Suppose that \( E \) is an analytic disk contained in \( {h}_{r}\left( {X}_{S}\right) \) . Either \( z \) or \( w \) is not a constant on \( E \) . Suppose that \( z \) is not constant. Then \( z\left( E\right) \) contains interior in \( \mathbb{C} \) and so the \( z \) -projection of \( {h}_{r}\left( {X}_{S}\ri... | Yes |
Theorem 24.3. There exists a compact subset \( Y \) of \( {\mathbb{C}}^{2} \) with \( \pi \left( Y\right) = \{ \left| z\right| = 1\} \) and \( \pi \left( \widehat{Y}\right) = \{ \left| z\right| \leq 1\} \) such that \( \widehat{Y} \smallsetminus Y \) contains no analytic disk. | Proof of Theorem 24.3. We denote by \( {a}_{1},{a}_{2},\ldots \) the points in the disk \( \{ \left| z\right| \leq \) \( 1/2\} \), both of whose coordinates are rational. For \( j = 1,2\ldots \), we denote by \( {B}_{j} \) the algebraic function\n\n\[ \n{B}_{j}\left( z\right) = \left( {z - {a}_{1}}\right) \left( {z - {... | Yes |
Lemma 24.5. A point \( \left( {z, w}\right) \) belongs to \( X \) if and only if there exists a sequence \( \left\{ \left( {z,{w}_{n}}\right) \right\} \) with \( \left( {z,{w}_{n}}\right) \in {\sum }_{n} \) for each \( n \) and \( {w}_{n} \rightarrow w \) as \( n \rightarrow \infty \) . | Proof. Fix \( \left( {z, w}\right) \) and assume that such a sequence \( \left\{ \left( {z,{w}_{n}}\right) \right\} \) exists. Fix \( {n}_{0} \) . Because of (12), \[ \left\{ {\left| {P}_{k}\right| \leq {\epsilon }_{k}}\right\} \subseteq \left\{ {\left| {P}_{{n}_{0}}\right| \leq {\epsilon }_{{n}_{0}}}\right\} \] if \( ... | Yes |
Lemma 24.6. Let \( \Omega \) be a region contained in \( \{ \left| z\right| < 1/2\} \) . There does not exist a continuous function \( f \) on \( \Omega \) whose graph \( \{ \left( {z, f\left( z\right) }\right) : z \in \Omega \} \) is contained in \( X \) . | Proof. Suppose that such a function \( f \) exists. We choose a rectangle: \( {s}_{1} \leq \operatorname{Re}z \leq \) \( {s}_{2},{t}_{1} \leq \operatorname{Im}z \leq {t}_{2} \) contained in \( \Omega \), with \( {s}_{1},{s}_{2},{t}_{1},{t}_{2} \) irrational numbers. Let \( \gamma \) denote the boundary of this rectangl... | No |
Lemma 24.9. Put\n\n\[ Y = X \cap \left\{ {\left| z\right| = \frac{1}{2}}\right\} \]\n\nThen \( X = \widehat{Y} \) . | Proof. Since \( X \) is polynomially convex, \( \widehat{Y} \subseteq X \). \n\nNow fix \( \left( {z, w}\right) \in X \). Choose a sequence \( \left\{ \left( {z,{w}_{k}}\right) \right\} \) converging to \( \left( {z, w}\right) \) such that \( \left( {z,{w}_{k}}\right) \in {\sum }_{k} \) for each \( k \). For each \( k ... | Yes |
Proposition 24.10. \( \sum \) is totally real if and only if \( {L\sigma } \neq 0 \) at every point of \( {S}^{3} \) . | Proof. Fix \( \left( {a, b}\right) \in {S}^{3} \) . Then the complex line tangent \( \ell \) to \( {S}^{3} \) at \( \left( {a, b}\right) \) can be parameterized by\n\n\[ \lambda \rightarrow \left( {a + \bar{b}\lambda, b - \bar{a}\lambda }\right) \]\n\nIt is straightforward to check that\n\n\[ \sigma \left( {a + \bar{b}... | Yes |
Lemma 24.11. Let \( \Omega \) be a bounded domain in \( \mathbb{C} \) such that \( 0 \in {b\Omega } \) and such that \( \Omega \) is disjoint from the negative real axis. Let \( {\zeta }_{0} \in \Omega \) . For \( 0 < r < \left| {\zeta }_{0}\right| \), let \( {\Omega }_{r} = \{ \zeta \in \Omega : \left| \zeta \right| >... | Proof. Denote by \( \sqrt{\zeta } \) the principal value of the square root on the plane cut by the negative real axis. In the right half-plane define a nonnegative harmonic function\n\n\[ H\left( z\right) = \frac{2}{\pi }\arg \left( \frac{z + i\sqrt{r}}{z - i\sqrt{r}}\right) \]\n\nWe have \( H\left( z\right) \equiv 1 ... | Yes |
Lemma 24.13. \( \log {Z}_{i}\left( \zeta \right) \leq {U}_{i}\left( \zeta \right) \) for all \( \zeta \in \Omega, i = 1,2,3 \) . | Proof. Fix \( {\zeta }_{0} \in \Omega \) and let \( 0 < r < \left| {\zeta }_{0}\right| \) . Let \( {\mu }^{r} \) be harmonic measure for \( {\zeta }_{0} \) on \( b{\Omega }_{r} \) . In \( \Omega ,\log {Z}_{1} \) is a subharmonic function bounded above by \( M \) . Since \( \log {Z}_{1} \) is subharmonic and has continu... | Yes |
Lemma 24.14. If \( \gamma \) has positive plane measure, then \( {\mathfrak{A}}_{\gamma } \) contains three functions that separate the points on \( {S}^{2} \) . | Proof. Put\n\n\[ F\left( \zeta \right) = {\int }_{\gamma }\frac{dxdy}{z - \zeta }.\]\n\n\( F\left( \zeta \right) \rightarrow 0 \) as \( \zeta \rightarrow \infty \) and \( \mathop{\lim }\limits_{{\zeta \rightarrow \infty }}\zeta \cdot F\left( \zeta \right) \neq 0 \) . Hence \( F \) is not a constant. Fix \( {\zeta }_{0}... | Yes |
Theorem 24.15. \( {J}_{0} \) is not polynomially convex in \( {\mathbb{C}}^{3} \) . Hence \( P\left( {J}_{0}\right) \neq \mathrm{C}\left( {J}_{0}\right) \) . | Proof. Fix \( {\zeta }_{0} \in {S}^{2} \smallsetminus \gamma \) . Then \( {x}^{0} = \left( {F\left( {\zeta }_{0}\right) ,{F}_{2}\left( {\zeta }_{0}\right) ,{F}_{3}\left( {\zeta }_{0}\right) }\right) \notin {J}_{0} \) . Yet, if \( P \) is any polynomial in \( {\mathbb{C}}^{3} \) ,\n\n\[ \left| {P\left( {x}^{0}\right) }\... | Yes |
Theorem 2.2.2 Let \( f : X \rightarrow Y \) and \( g : Y \rightarrow X \) be two functions. Then there exist sets \( A, B, C, D \), such that\n\n\[ A \cup B = X, C \cup D = Y, A \cap B = \varnothing, C \cap D = \varnothing ,\]\n\n\[ f\left( A\right) = C, g\left( D\right) = B. \] | Proof: Consider the empty set, \( \varnothing \subseteq X \) . If \( y \in Y \smallsetminus f\left( \varnothing \right) \), then \( g\left( y\right) \notin \varnothing \) because \( \varnothing \) has no elements. Also, if \( A, B, C \), and \( D \) are as described above, \( A \) also would have this same property tha... | Yes |
Theorem 2.2.3 (Schroder Bernstein) If \( f : X \rightarrow Y \) and \( g : Y \rightarrow X \) are one to one, then there exists \( h : X \rightarrow Y \) which is one to one and onto. | Proof: Let \( A, B, C, D \) be the sets of Theorem2.2.2 and define\n\n\[ \nh\left( x\right) \equiv \left\{ \begin{matrix} f\left( x\right) \;\text{ if }x \in A \\ {g}^{-1}\left( x\right) \text{ if }x \in B \end{matrix}\right.\n\]\n\nThen \( h \) is the desired one to one and onto mapping. | No |
Corollary 2.2.5 If \( f : X \rightarrow Y \) is onto and \( g : Y \rightarrow X \) is onto, then there exists \( h : X \rightarrow Y \) which is one to one and onto. | Proof: For each \( y \in Y,{f}^{-1}\left( y\right) \equiv \{ x \in X : f\left( x\right) = y\} \neq \varnothing \) . Therefore, by the axiom of choice, there exists \( {f}_{0}^{-1} \in \mathop{\prod }\limits_{{y \in Y}}{f}^{-1}\left( y\right) \) which is the same as saying that for each \( y \in Y,{f}_{0}^{-1}\left( y\r... | Yes |
Theorem 2.2.7 If \( X \) and \( Y \) are both at most countable, then \( X \times Y \) is also at most countable. If either \( X \) or \( Y \) is countable, then \( X \times Y \) is also countable. | Proof: It is given that there exists a mapping \( \eta : \mathbb{N} \rightarrow X \) which is onto. Define \( \eta \left( i\right) \equiv {x}_{i} \) and consider \( X \) as the set \( \left\{ {{x}_{1},{x}_{2},{x}_{3},\cdots }\right\} \) . Similarly, consider \( Y \) as the set \( \left\{ {{y}_{1},{y}_{2},{y}_{3},\cdots... | Yes |
Theorem 2.2.8 If \( X \) and \( Y \) are at most countable, then \( X \cup Y \) is at most countable. If either \( X \) or \( Y \) are countable, then \( X \cup Y \) is countable. | Proof: As in the preceding theorem, \[ X = \left\{ {{x}_{1},{x}_{2},{x}_{3},\cdots }\right\} \] and \[ Y = \left\{ {{y}_{1},{y}_{2},{y}_{3},\cdots }\right\} . \] Consider the following array consisting of \( X \cup Y \) and path through it.  is sequentially compact. | Proof: Let \( \left\{ {x}_{n}\right\} \subseteq \left\lbrack {a, b}\right\rbrack \equiv {I}_{0} \) . Consider the two intervals \( \left\lbrack {a,\frac{a + b}{2}}\right\rbrack \) and \( \left\lbrack {\frac{a + b}{2}, b}\right\rbrack \) each of which has length \( \left( {b - a}\right) /2 \) . At least one of these int... | Yes |
Theorem 3.0.4 Let \( f : K \rightarrow \mathbb{R} \) be continuous where \( K \) is a sequentially compact set in \( \mathbb{R} \). Then \( f \) is uniformly continuous on \( K \). | Proof: If this is not true, there exists \( \varepsilon > 0 \) such that for every \( \delta > 0 \) there exists a pair of points, \( {x}_{\delta } \) and \( {y}_{\delta } \) such that even though \( \left| {{x}_{\delta } - {y}_{\delta }}\right| < \delta ,\left| {f\left( {x}_{\delta }\right) - f\left( {y}_{\delta }\rig... | No |
The function, \( {af} + {bg} \) is continuous at \( x \) when \( f, g \) are continuous at \( x \) \( \in D\left( f\right) \cap D\left( g\right) \) and \( a, b \in \mathbb{R} \) . | First consider 1.) Let \( \varepsilon > 0 \) be given. By assumption, there exist \( {\delta }_{1} > 0 \) such that whenever \( \left| {x - y}\right| < {\delta }_{1} \), it follows \( \left| {f\left( x\right) - f\left( y\right) }\right| < \frac{\varepsilon }{2\left( {\left| a\right| + \left| b\right| + 1}\right) } \) a... | Yes |
Theorem 3.2.2 Suppose \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is continuous and suppose \( f\left( a\right) < c < \) \( f\left( b\right) \) . Then there exists \( x \in \left( {a, b}\right) \) such that \( f\left( x\right) = c \) . | Proof: Let \( d = \frac{a + b}{2} \) and consider the intervals \( \left\lbrack {a, d}\right\rbrack \) and \( \left\lbrack {d, b}\right\rbrack \) . If \( f\left( d\right) \geq c \) , then on \( \left\lbrack {a, d}\right\rbrack \), the function is \( \leq c \) at one end point and \( \geq c \) at the other. On the other... | Yes |
Lemma 3.2.3 Let \( \phi : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be a continuous function and suppose \( \phi \) is \( 1 - 1 \) on \( \left( {a, b}\right) \) . Then \( \phi \) is either strictly increasing or strictly decreasing on \( \left\lbrack {a, b}\right\rbrack \) . | Proof: First it is shown that \( \phi \) is either strictly increasing or strictly decreasing on \( \left( {a, b}\right) \) .\n\nIf \( \phi \) is not strictly decreasing on \( \left( {a, b}\right) \), then there exists \( {x}_{1} < {y}_{1},{x}_{1},{y}_{1} \in \left( {a, b}\right) \) such that\n\n\[ \left( {\phi \left( ... | Yes |
Corollary 3.2.4 Let \( f : \left( {a, b}\right) \rightarrow \mathbb{R} \) be one to one and continuous. Then \( f\left( {a, b}\right) \) is an open interval, \( \left( {c, d}\right) \) and \( {f}^{-1} : \left( {c, d}\right) \rightarrow \left( {a, b}\right) \) is continuous. | Proof: Since \( f \) is either strictly increasing or strictly decreasing, it follows that \( f\left( {a, b}\right) \) is an open interval, \( \left( {c, d}\right) \) . Assume \( f \) is decreasing. Now let \( x \in \left( {a, b}\right) \) . Why is \( {f}^{-1} \) is continuous at \( f\left( x\right) \) ? Since \( f \) ... | Yes |
Lemma 4.1.2 If \( P \subseteq Q \) then\n\n\[ U\left( {f, Q}\right) \leq U\left( {f, P}\right) ,\text{ and }L\left( {f, P}\right) \leq L\left( {f, Q}\right) . | Proof: This is verified by adding in one point at a time. Thus let\n\n\[ P = \left\{ {{x}_{0},\cdots ,{x}_{n}}\right\} \]\n\nand let\n\n\[ Q = \left\{ {{x}_{0},\cdots ,{x}_{k}, y,{x}_{k + 1},\cdots ,{x}_{n}}\right\} . \]\n\nThus exactly one point, \( y \), is added between \( {x}_{k} \) and \( {x}_{k + 1} \) . Now the ... | No |
Lemma 4.1.3 If \( P \) and \( Q \) are two partitions, then\n\n\[ L\left( {f, P}\right) \leq U\left( {f, Q}\right) \] | Proof: By Lemma 4.1.2,\n\n\[ L\left( {f, P}\right) \leq L\left( {f, P \cup Q}\right) \leq U\left( {f, P \cup Q}\right) \leq U\left( {f, Q}\right) . \] | Yes |
Theorem 4.1.5 \( \underline{I} \leq \bar{I} \) . | Proof: From Lemma 4.1.3,\n\n\[ \underline{I} = \sup \{ L\left( {f, P}\right) \text{ where }P\text{ is a partition }\} \leq U\left( {f, Q}\right) \]\n\nbecause \( U\left( {f, Q}\right) \) is an upper bound to the set of all lower sums and so it is no smaller than the least upper bound. Therefore, since \( Q \) is arbitr... | Yes |
Theorem 4.1.8 A bounded function \( f \) is Riemann integrable if and only if for all \( \varepsilon > 0 \), there exists a partition \( P \) such that\n\n\[ U\left( {f, P}\right) - L\left( {f, P}\right) < \varepsilon . \]\n\n(4.1.3) | Proof: First assume \( f \) is Riemann integrable. Then let \( P \) and \( Q \) be two partitions such that\n\n\[ U\left( {f, Q}\right) < \bar{I} + \varepsilon /2, L\left( {f, P}\right) > \underline{I} - \varepsilon /2. \]\n\nThen since \( \underline{I} = \bar{I} \),\n\n\[ U\left( {f, Q \cup P}\right) - L\left( {f, P \... | Yes |
Theorem 4.3.1 Let \( f, g \) be bounded functions and let\n\n\[ f\left( \left\lbrack {a, b}\right\rbrack \right) \subseteq \left\lbrack {{c}_{1},{d}_{1}}\right\rbrack, g\left( \left\lbrack {a, b}\right\rbrack \right) \subseteq \left\lbrack {{c}_{2},{d}_{2}}\right\rbrack . \]\n\nLet \( H : \left\lbrack {{c}_{1},{d}_{1}}... | Proof: In the following claim, \( {M}_{i}\left( h\right) \) and \( {m}_{i}\left( h\right) \) have the meanings assigned above with respect to some partition of \( \left\lbrack {a, b}\right\rbrack \) for the function, \( h \) .\n\nClaim: The following inequality holds.\n\n\[ \left| {{M}_{i}\left( {H \circ \left( {f, g}\... | Yes |
Theorem 4.3.2 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be either increasing or decreasing on \( \left\lbrack {a, b}\right\rbrack \) and suppose \( F \) is continuous. Then \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) . | Proof: Let \( \varepsilon > 0 \) be given and let\n\n\[ \n{x}_{i} = a + i\left( \frac{b - a}{n}\right), i = 0,\cdots, n.\n\]\n\nSince \( F \) is continuous, it follows from Corollary 3.0.5 on Page 38 that it is uniformly continuous. Therefore, if \( n \) is large enough, then for all \( i \) ,\n\n\[ \nF\left( {x}_{i}\r... | Yes |
Corollary 4.3.3 Let \( \left\lbrack {a, b}\right\rbrack \) be a bounded closed interval and let \( \phi : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be Lipschitz continuous and suppose \( F \) is continuous. Then \( \phi \in R\left( \left\lbrack {a, b}\right\rbrack \right) . | Proof: Let \( f\left( x\right) = x \) . Then by Theorem 4.3.2, \( f \) is Riemann Stieltjes integrable. Let \( H\left( {a, b}\right) \equiv \phi \left( a\right) \) . Then by Theorem 4.3.1 \( H \circ \left( {f, f}\right) = \phi \circ f = \phi \) is also Riemann Stieltjes integrable. This proves the corollary. | Yes |
Theorem 4.3.4 Suppose \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is continuous and \( F \) is just an increasing function defined on \( \left\lbrack {a, b}\right\rbrack \) . Then \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) . | Proof: By Corollary 3.0.5 on Page 38, \( f \) is uniformly continuous on \( \left\lbrack {a, b}\right\rbrack \) . Therefore, if \( \varepsilon > 0 \) is given, there exists a \( \delta > 0 \) such that if \( \left| {{x}_{i} - {x}_{i - 1}}\right| < \delta \), then \( {M}_{i} - {m}_{i} < \frac{\varepsilon }{F\left( b\rig... | Yes |
Lemma 4.4.1 Let \( S \) be a nonempty set which is bounded above and below. Then if \( - S \equiv \{ - x : x \in S\} \), \[ \sup \left( {-S}\right) = - \inf \left( S\right) \] \[ \left( {4.4.5}\right) \] and \[ \inf \left( {-S}\right) = - \sup \left( S\right) \] \[ \left( {4.4.6}\right) \] | Proof: Consider 4.4.5. Let \( x \in S \) . Then \( - x \leq \sup \left( {-S}\right) \) and so \( x \geq - \sup \left( {-S}\right) \) . It follows that \( - \sup \left( {-S}\right) \) is a lower bound for \( S \) and therefore, \( - \sup \left( {-S}\right) \leq \) \( \inf \left( S\right) \) . This implies \( \sup \left(... | No |
Lemma 4.4.2 If \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) then \( - f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) and\n\n\[ - {\int }_{a}^{b}f\left( x\right) {dF} = {\int }_{a}^{b} - f\left( x\right) {dF}. \] | Proof: The first part of the conclusion of this lemma follows from Theorem 4.3.2 since the function \( \phi \left( y\right) \equiv - y \) is Lipschitz continuous. Now choose \( P \) such\n\nthat\n\[ {\int }_{a}^{b} - f\left( x\right) {dF} - L\left( {-f, P}\right) < \varepsilon . \]\n\nThen since \( {m}_{i}\left( {-f}\r... | Yes |
Theorem 4.4.3 The integral is linear,\n\n\[ \n{\int }_{a}^{b}\left( {{\alpha f} + {\beta g}}\right) \left( x\right) {dF} = \alpha {\int }_{a}^{b}f\left( x\right) {dF} + \beta {\int }_{a}^{b}g\left( x\right) {dF}. \]\n\nwhenever \( f, g \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) and \( \alpha ,\beta \in \ma... | Proof: First note that by Theorem 4.3.1, \( {\alpha f} + {\beta g} \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) . To begin with, consider the claim that if \( f, g \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) then\n\n\[ \n{\int }_{a}^{b}\left( {f + g}\right) \left( x\right) {dF} = {\int }_{a}^{b}f... | Yes |
Theorem 4.4.4 If \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) and \( f \in R\left( \left\lbrack {b, c}\right\rbrack \right) \), then \( f \in R\left( \left\lbrack {a, c}\right\rbrack \right) \) and\n\n\[{\int }_{a}^{c}f\left( x\right) {dF} = {\int }_{a}^{b}f\left( x\right) {dF} + {\int }_{b}^{c}f\left( ... | Proof: Let \( {P}_{1} \) be a partition of \( \left\lbrack {a, b}\right\rbrack \) and \( {P}_{2} \) be a partition of \( \left\lbrack {b, c}\right\rbrack \) such that\n\n\[U\left( {f,{P}_{i}}\right) - L\left( {f,{P}_{i}}\right) < \varepsilon /2, i = 1,2.\]\n\nLet \( P \equiv {P}_{1} \cup {P}_{2} \) . Then \( P \) is a ... | Yes |
Corollary 4.4.5 Let \( F \) be continuous and let \( \left\lbrack {a, b}\right\rbrack \) be a closed and bounded interval and suppose that\n\n\[ a = {y}_{1} < {y}_{2}\cdots < {y}_{l} = b \]\n\nand that \( f \) is a bounded function defined on \( \left\lbrack {a, b}\right\rbrack \) which has the property that \( f \) is... | Proof: This follows from Theorem 4.4.4 and Theorem 4.3.2. | No |
Theorem 4.4.7 Assuming all the integrals make sense,\n\n\\[ \n{\\int }_{a}^{b}f\\left( x\\right) {dF} + {\\int }_{b}^{c}f\\left( x\\right) {dF} = {\\int }_{a}^{c}f\\left( x\\right) {dF}.\n\\] | Proof: This follows from Theorem 4.4.4 and Definition 4.4.6. For example,\n\nassume\n\n\\[ \nc \\in \\left( {a, b}\\right) \\text{.}\n\\]\n\nThen from Theorem 4.4.4,\n\n\\[ \n{\\int }_{a}^{c}f\\left( x\\right) {dF} + {\\int }_{c}^{b}f\\left( x\\right) {dF} = {\\int }_{a}^{b}f\\left( x\\right) {dF}\n\\]\n\nand so by Def... | Yes |
Theorem 4.5.1 Let \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) and let\n\n\[ F\left( x\right) \equiv {\int }_{a}^{x}f\left( t\right) {dt} \]\n\nThen if \( f \) is continuous at \( x \in \left( {a, b}\right) \) ,\n\n\[ {F}^{\prime }\left( x\right) = f\left( x\right) . \] | Proof: Let \( x \in \left( {a, b}\right) \) be a point of continuity of \( f \) and let \( h \) be small enough that \( x + h \in \left\lbrack {a, b}\right\rbrack \) . Then by using 4.4.13,\n\n\[ {h}^{-1}\left( {F\left( {x + h}\right) - F\left( x\right) }\right) = {h}^{-1}{\int }_{x}^{x + h}f\left( t\right) {dt}. \]\n\... | Yes |
Theorem 4.5.2 Let \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) and suppose there exists an antiderivative for \( f, G \), such that\n\n\[ \n{G}^{\prime }\left( x\right) = f\left( x\right) \n\]\n\nfor every point of \( \left( {a, b}\right) \) and \( G \) is continuous on \( \left\lbrack {a, b}\right\rbra... | Proof: Let \( P = \left\{ {{x}_{0},\cdots ,{x}_{n}}\right\} \) be a partition satisfying\n\n\[ \nU\left( {f, P}\right) - L\left( {f, P}\right) < \varepsilon .\n\]\n\nThen\n\n\[ \nG\left( b\right) - G\left( a\right) = G\left( {x}_{n}\right) - G\left( {x}_{0}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1}}^{n}G\lef... | Yes |
Proposition 4.5.4 Suppose \( f \in R\left( \left\lbrack {a, b}\right\rbrack \right) \) . Then there exists a partition, \( P \equiv \) \( \left\{ {{x}_{0},\cdots ,{x}_{n}}\right\} \) with the property that for any choice of \( {z}_{k} \in \left\lbrack {{x}_{k - 1},{x}_{k}}\right\rbrack \) , | Proof: Choose \( P \) such that \( U\left( {f, P}\right) - L\left( {f, P}\right) < \varepsilon \) and then both \( {\int }_{a}^{b}f\left( x\right) {dx} \) and \( \mathop{\sum }\limits_{{k = 1}}^{n}f\left( {z}_{k}\right) \left( {{x}_{k} - {x}_{k - 1}}\right) \) are contained in \( \left\lbrack {L\left( {f, P}\right), U\... | Yes |
Lemma 5.1.2 A set of vectors \( \left\{ {{\mathbf{x}}_{1},\cdots ,{\mathbf{x}}_{p}}\right\} \) is linearly independent if and only if none of the vectors can be obtained as a linear combination of the others. | Proof: Suppose first that \( \left\{ {{\mathbf{x}}_{1},\cdots ,{\mathbf{x}}_{p}}\right\} \) is linearly independent. If\n\n\[ \n{\mathbf{x}}_{k} = \mathop{\sum }\limits_{{j \neq k}}{c}_{j}{\mathbf{x}}_{j} \n\]\n\nthen\n\n\[ \n\mathbf{0} = 1{\mathbf{x}}_{k} + \mathop{\sum }\limits_{{j \neq k}}\left( {-{c}_{j}}\right) {\... | Yes |
Theorem 5.1.3 (Exchange Theorem) Let \( \left\{ {{\mathbf{x}}_{1},\cdots ,{\mathbf{x}}_{r}}\right\} \) be a linearly independent set of vectors such that each \( {\mathbf{x}}_{i} \) is in \( \operatorname{span}\left( {{\mathbf{y}}_{1},\cdots ,{\mathbf{y}}_{s}}\right) \) . Then \( r \leq s \) . | Proof: Define \( \operatorname{span}\left\{ {{\mathbf{y}}_{1},\cdots ,{\mathbf{y}}_{s}}\right\} \equiv V \), it follows there exist scalars, \( {c}_{1},\cdots ,{c}_{s} \) such that \[ {\mathbf{x}}_{1} = \mathop{\sum }\limits_{{i = 1}}^{s}{c}_{i}{\mathbf{y}}_{i} \] (5.1.1) Not all of these scalars can equal zero because... | Yes |
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