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Theorem 6.23. Let \( D \) be Stein. Then every meromorphic function \( m \in \mathcal{M}\left( D\right) \) is the quotient of two global holomorphic functions on \( D \) . | Proof. Without loss of generality we may assume that \( m \in {\mathcal{M}}^{ * }\left( D\right) \) . Define the sheaf homomorphism \( \mu : {\mathcal{M}}_{D} \rightarrow {\mathcal{M}}_{D} \) by \( \mu \left( {\mathbf{s}}_{z}\right) = {\mathbf{m}}_{z}{\mathbf{s}}_{z} \) for \( {\mathbf{s}}_{z} \in {\mathcal{M}}_{z} \) ... | Yes |
Theorem 6.25. Let \( \mathcal{A} \) be a coherent analytic sheaf over the Stein domain D. Let \( {s}_{1},\ldots ,{s}_{l} \in \Gamma \left( {D,\mathcal{A}}\right) \) be global sections which generate the \( \mathcal{O} \) -submodule sheaf \( \mathcal{S} \subset \mathcal{A} \), i.e., \( {s}_{1}\left( z\right) ,\ldots ,{s... | Proof. Consider the homomorphism \( \varphi : {\mathcal{O}}_{D}^{l} \rightarrow \mathcal{A} \) defined by \( {s}_{1},\ldots ,{s}_{l} \) as in (6.22). Then \( \mathcal{S} = {\mathcal{I}}_{\mathcal{M}}\varphi \) is coherent, and \( {\mathcal{K}}_{\mathcal{C}}\varphi \) is coherent as well, by the definition of coherence ... | Yes |
Corollary 6.26. Let \( D \subset {\mathbb{C}}^{n} \) be Stein and let \( A = \{ \left( {z, w}\right) \in D \times D : z = w\} \) . Then for every \( F \in \mathcal{O}\left( {D \times D}\right) \) which vanishes on \( A \) there are functions \( {G}_{j} \in \mathcal{O}\left( {D \times D}\right) \) , \( j = 1,\ldots, n \... | Proof. Consider the sections \( {s}_{j} \in \Gamma \left( {D \times D,{\mathcal{O}}_{D \times D}}\right) ,1 \leq j \leq n \), defined by \( {s}_{j}\left( {z, w}\right) = {z}_{j} - {w}_{j} \) . It is easily seen that \( {s}_{1},\ldots ,{s}_{n} \) generate the ideal sheaf \( \mathcal{I}\left( A\right) \) of \( A \), and ... | Yes |
Lemma 1.1. If \( f \in {A}^{1}\left( D\right) \), then\n\n\[ f\left( z\right) = {\int }_{bD}f\left( \zeta \right) E\left( {\zeta, z}\right) \;\text{ for }z \in D. \] | Proof. This follows from the case \( q = 0 \) of Theorem IV.3.6, with \( W = {L}_{D} \). Notice that \( {T}_{1}\bar{\partial }f \equiv 0 \) since \( f \) is holomorphic.\n\nThus the kernel \( E \) reproduces holomorphic functions from their boundary values, but \( E \) is holomorphic in \( z \) only for \( z \) close t... | Yes |
Lemma 1.2. There are a neighborhood \( {D}^{\# } \) of \( \bar{D} \) and a double form\n\n\[ A\left( {\zeta, z}\right) \in {C}_{n, n - 1}^{k,\infty }\left( {{bD} \times {D}^{\# }}\right) \;\text{such that} \]\n\n\[ {\bar{\partial }}_{z}A\left( {\zeta, z}\right) = {\bar{\partial }}_{z}E\left( {\zeta, z}\right) \;\text{ ... | Proof. We shall use the integral solution operator for \( \bar{\partial } \) on a neighborhood of \( \bar{D} \) given by Theorem V.2.5. Specifically, we choose open neighborhoods \( {D}^{\# } \) and \( V \) of the Stein compactum \( K = \bar{D} \), with \( \bar{D} \subset \subset {D}^{\# } \subset \subset V \subset \su... | Yes |
Lemma 1.5. Let \( \Phi \) be the denominator of the generating form \( {L}_{D} \), and set\n\n\[ \n{J}_{\alpha }\left( z\right) = {\int }_{bD}\frac{dS}{{\left| \Phi \left( \cdot, z\right) \right| }^{n + \alpha }}\;\text{ for }z \in \bar{D}.\n\]\n\nThen\n\n(1.6)\n\[ \n{J}_{\alpha }\left( z\right) \lesssim \left\{ \begin... | Proof. We shall use the special real coordinate system \( t = \left( {{t}_{1},{t}_{2},{t}^{\prime }}\right) \) on the ball \( B\left( {z,\eta }\right) \) introduced in Lemma V.3.4. With the notation chosen there, it is clearly enough to prove (1.6) for \( {\delta }_{D}\left( z\right) \leq a \), and the region of integr... | Yes |
Example 2.2. Let \( D = \left\{ {\left( {z, w}\right) \in {\mathbb{C}}^{2} : 0 < \left| z\right| < \left| w\right| < 1}\right\} \) . It is easy to see that \( D \) is a Stein domain and that \( f\left( {z, w}\right) = {z}^{2}/w \) defines a function in \( A\left( D\right) \) which is completely singular at 0 . Suppose ... | It follows from Theorem I.1.6 that every \( {g}_{i} \) extends holomorphically to \( \bar{P} = \overline{P\left( {0,1}\right) } \), and since \( {b}_{0}P = \) \( \{ \left| z\right| = \left| w\right| = 1\} \subset \bar{D} \), uniform convergence of \( \left\{ {g}_{j}\right\} \) on \( {b}_{0}P \) implies uniform converge... | Yes |
Lemma 2.3. Let \( \\left\\{ {{U}_{v},1 \\leq v \\leq l}\\right\\} \) be a finite open cover of \( {bD} \) . Then every \( f \\in A\\left( D\\right) \) has a decomposition\n\n(2.1)\n\n\[ f = {f}_{1} + \\cdots + {f}_{l} \]\n\nwhere \( {f}_{v} \\in A\\left( D\\right) \) is holomorphic on \( \\bar{D} - {U}_{v} \) for \( 1 ... | Proof. Choose functions \( {\\chi }_{v} \\in {C}_{0}^{\\infty }\\left( {U}_{v}\\right) ,1 \\leq v \\leq l \\), with \( \\mathop{\\sum }\\limits_{{v = 1}}^{l}{\\chi }_{v} = 1 \) on \( {bD} \) . By Theorem 1.3,\n\n\[ f = \\mathbf{C}\\left( f\\right) = \\mathop{\\sum }\\limits_{{v = 1}}^{l}\\mathbf{C}\\left( {{\\chi }_{v}... | Yes |
Lemma 2.4. Let \( D \subset \subset {\mathbb{R}}^{n} \) have \( {C}^{1} \) boundary at \( P \in {bD} \), and let \( \mathbf{n} \) be the unit inner normal to \( {bD} \) at \( P \). Then there are a neighborhood \( U \) of \( P \) and \( {\tau }_{0} > 0 \), such that\n\n\[ z + \tau \mathbf{n} \in D\;\text{ for all }z \i... | Proof of Lemma 2.4. Without loss of generality we may assume that \( P = 0 \), \( \mathbf{n} = \left( {0,\ldots , - 1}\right) \), and that there is a local defining function \( {r}^{\# } \) for \( D \) of the form \( {r}^{\# } = {x}_{n} - \varphi \left( {{x}_{1},\ldots ,{x}_{n - 1}}\right) \) on a neighborhood \( {U}^{... | Yes |
Corollary 2.5. Under the hypothesis of Theorem 2.1, if \( \delta > 0 \) is sufficiently small, then every \( f \in A\left( D\right) \) can be approximated uniformly on \( \bar{D} \) by functions in \( \mathcal{O}\left( {D}_{\delta }\right) \) . | Proof. By the Theorem, \( f \in A\left( D\right) \) can be approximated uniformly by functions in \( \mathcal{O}\left( \bar{D}\right) \) . Now use the Runge property of \( \left( {D,{D}_{\delta }}\right) \) (see VI.§1.6): \( {\bar{D}}_{\mathcal{O}\left( {D}_{\delta }\right) } = \bar{D} \) for \( \delta \) sufficiently ... | Yes |
Corollary 3.2. Given \( D \) as in Proposition 3.1, there is a function \( H \in \) \( {C}^{k + 1}\left( {{bD} \times {D}_{\delta }}\right) \) such that for each \( \zeta \in {bD} \) one has\n\n(i) \( H\left( {\zeta , \cdot }\right) \in \mathcal{O}\left( {D}_{\delta }\right) \), \n\nand \n\n(ii) \( H\left( {\zeta ,\zet... | Proof. \( H\left( {\zeta, z}\right) = \exp \left( {-g\left( {\zeta, z}\right) }\right) \) will do. ∎ | No |
Proposition 3.3. Let \( K \subset {\mathbb{C}}^{n} \) be a Stein compactum and let \( M \) be a \( {C}^{k} \) manifold, \( k \geq 1 \) . Given \( f \in {C}^{k,\infty }\left( {M \times K}\right) \) such that \( f\left( {x, \cdot }\right) \in \mathcal{O}\left( K\right) \) for \( x \in M \), there are functions \( {Q}_{j}... | \[ f\left( {x,\zeta }\right) - f\left( {x, z}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{Q}_{j}\left( {x,\zeta, z}\right) \left( {{\zeta }_{j} - {z}_{j}}\right) \] for \( x \in M \) and \( \left( {\zeta, z}\right) \) in some neighborhood of \( K \times K \) . | Yes |
Let \( g \in {C}^{k + 1,\infty }\left( {U \times {D}_{\delta }}\right) \) be the function given by Proposition 3.1 and let \( 0 < \eta < \delta \) . After shrinking \( U \), there are functions \( {g}_{j} \in {C}^{k + 1,\infty }\left( {U \times {D}_{\eta }}\right) \) , \( 1 \leq j \leq n \), with \( {g}_{j}\left( {\zet... | Proof. Since \( {\bar{D}}_{\eta } \) is a Stein compactum, we can apply Proposition 3.3 to \( g \in {C}^{k + 1,\infty }\left( {U \times {\bar{D}}_{\eta }}\right) \), obtaining \[ g\left( {x,\zeta }\right) - g\left( {x, z}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{Q}_{j}\left( {x,\zeta, z}\right) \left( {{\zeta }_{j}... | Yes |
Lemma 3.5. Given \( M \) and \( K \) as in Proposition 3.3, let \( {K}_{1} = K \cap \left\{ {z = \left( {{z}_{1},{z}^{\prime }}\right) \in {\mathbb{C}}^{n}}\right. \) : \( \left. {{z}_{1} = 0}\right\} \) and suppose \( f \in {C}^{k,\infty }\left( {M \times {K}_{1}}\right) \) satisfies \( f\left( {x, \cdot }\right) \in ... | Proof. We proceed as in the proof of Proposition V.2.1, using the same notation and adding the parameter \( x \in M \), as needed. Thus \( \alpha = {\alpha }_{1} = \) \( {\bar{\partial }}_{z}\left\lbrack {\chi \left( z\right) f\left( {x,\pi \left( z\right) }\right) }\right\rbrack /{z}_{1} \) is \( {\bar{\partial }}_{z}... | Yes |
Theorem 3.6. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) is strictly pseudoconvex with boundary of class \( {C}^{k + 2}, k \geq 0 \) . Then \( {\Omega }_{0}\left( {W}^{\mathrm{{HR}}}\right) \) is of class \( {C}^{k,\infty } \) on \( {bD} \times D \), and, for \( \zeta \in {bD} \) fixed, \( {\Omega }_{0}\left( {W}^... | Proof. Notice that the singularities of \( {\Omega }_{0}\left( {W}^{\mathrm{{HR}}}\right) \) are determined by \( {g}^{-n} \), and that for \( \zeta \) close to \( z,{g}^{-n} \) is a nonzero multiple of \( {\Phi }^{-n} \) . Hence the proof of Theorem 3.6 is analogous to the proof of the corresponding statements for the... | No |
Lemma 3.7. At all points \( \left( {\zeta ,\zeta }\right) \in {bD} \times {bD} \) one has\n\n(3.12)\n\n\[ P \land {\left( {\bar{\partial }}_{\zeta }P\right) }^{n - 1} = Q \land {\left( {\bar{\partial }}_{\zeta }Q\right) }^{n - 1} = \partial r \land {\left( \bar{\partial }\partial r\right) }^{n - 1}. \] | Proof. By applying \( \partial /\partial {\zeta }_{j} \) to the equation\n\n\[ g = \mathop{\sum }\limits_{{v = 1}}^{n}{g}_{v}\left( {{\zeta }_{v} - {z}_{v}}\right) = A \cdot {F}^{\# } = A\mathop{\sum }\limits_{{v = 1}}^{n}{P}_{v}\left( {{\zeta }_{v} - {z}_{v}}\right) ,\]\n\nvalid for \( \left| {\zeta - z}\right| \leq \... | Yes |
Proposition 3.8. Suppose \( D \) is strictly pseudoconvex with boundary of class \( {C}^{k + 2} \) , \( k \geq 0 \) . Then\n\n(i) \( {\Omega }_{0}\left( {W}^{\mathrm{{HR}}}\right) = {\Omega }_{0}\left( {\partial r/\Phi }\right) + \frac{O\left( \left| {\zeta - z}\right| \right) }{{\Phi }^{n}} \) .\n\n(ii) \( E = {\Omega... | Proof. (ii) is an immediate consequence of Lemma 3.7. In fact, since \( {\Omega }_{0}\left( P\right) \) is of class \( {C}^{k,\infty } \) on \( {bD} \times {D}_{\delta } \), all partial derivatives of \( {\Omega }_{0}\left( P\right) \) with respect to \( z \) are continuous on \( {bD} \times {D}_{\delta } \) (even if \... | Yes |
Lemma 3.9. If \( \iota : {bD} \rightarrow {\mathbb{C}}^{n} \) is the inclusion, then\n\n\[ {\iota }^{ * }{\Omega }_{0}\left( {\partial r}\right) = \frac{\left( {n - 1}\right) !}{4{\pi }^{n}}\mathcal{D}\left( r\right) \parallel {dr}\parallel {dS} \]\non \( {bD} \), where \( {dS} \) is the surface element of \( {bD} \) . | Proof. It is enough to verify (3.21) at an arbitrary point \( P \in {bD} \) . Fix \( P \) and choose the coordinates of \( {\mathbb{C}}^{n} \) as above, so that \( {\mathcal{L}}_{P}\left( r\right) \) is diagonal. All forms below are evaluated at \( P \) . Then\n\n\[ \bar{\partial }\partial r = \mathop{\sum }\limits_{{j... | Yes |
Theorem 4.2. Let \( D \subset \subset {\mathbb{C}}^{n} \) be strictly pseudoconvex with boundary of class \( \geq 3 \) . There are linear operators \( {L}_{j} : A\left( D\right) \rightarrow \mathcal{O}\left( {D \times D}\right) ,1 \leq j \leq n \), such that for \( f \in A\left( D\right) \) the following hold:\n\n(4.1)... | For fixed \( a \in D \), the functions \( \left( {{L}_{j}f}\right) \left( {a, \cdot }\right) \) and \( \left( {{L}_{j}f}\right) \left( {\cdot, a}\right) \) are in \( A\left( D\right) \) . | No |
Corollary 4.3. If \( f \in A\left( D\right) \) and \( f\left( a\right) = 0 \) at the point \( a \in D \), then there are functions \( {g}_{1},\ldots ,{g}_{n} \in A\left( D\right) \) such that\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{z}_{j} - {a}_{j}}\right) {g}_{j}\left( z\right) \] | Proof. Take \( {g}_{j} = \left( {{L}_{j}f}\right) \left( {\cdot, a}\right) \). | No |
Lemma 5.1. Suppose \( r \) is of class \( {C}^{k + 2} \) . The neighborhood \( U \) of \( {bD} \), and the positive constants \( c,\varepsilon \), and \( \delta \) can be chosen so that\n\n(5.5)\n\n\[ \n\text{for}\zeta \in U,2\operatorname{Re}\widehat{\Phi }\left( {\zeta, z}\right) \geq \left\{ \begin{array}{l} - r\lef... | Proof. Equation (5.5) is an immediate consequence of the estimates V(1.6) and V(1.7) for \( \Phi \) ; (5.7) and (5.8) then follow from (5.5) and (5.4). The fact that \( {\widehat{E}}_{q} \) is nonsingular and extends to \( {\bar{D}}_{\delta } \) if \( q \geq 1 \) follows as the corresponding fact for \( {\Omega }_{q}\l... | Yes |
Lemma 5.2. (i) For \( f \in {C}_{0, q}^{1}\left( \bar{D}\right) \) with \( \bar{\partial }f = 0 \) one has\n\n\[ \n{}_{\bar{\partial }}{\widehat{\mathbf{E}}}_{q}f = {\mathbf{E}}_{q}f\;\text{ and }{}_{\bar{\partial }}{\widehat{\mathbf{T}}}_{q}f = {\mathbf{T}}_{q}f.\n\]\n\n(ii) If \( q \geq 1 \), then \( {}_{\bar{\partia... | Proof. (i) follows from (5.3) and (5.11), and an application of Stokes’ Theorem-note that \( \bar{\partial }f = 0 \) ! (ii) follows from (5.13) and (5.6) by standard estimations. ∎ | Yes |
Proposition 5.3. (ii) The representation (5.16) holds for all \( f \in {C}_{0, q}^{1}\left( D\right) \cap {L}^{1} \) with \( \bar{\partial }f = 0 \) on \( D \) . | Proof of Proposition 5.3. Let us first consider (ii) and fix \( f \in {C}_{0, q}^{1}\left( D\right), q \geq 1 \) , with \( \bar{\partial }f = 0 \) . For \( \eta < 0 \) sufficiently small, the double forms \( {\Omega }_{q}\left( {L}_{D}\right) \) and \( {A}_{q}\left( {{L}_{D}, B}\right) \) are defined on \( b{D}_{\eta }... | Yes |
Theorem 5.4. Suppose \( D \) is strictly pseudoconvex in \( {\mathbb{C}}^{n} \) with boundary of class at least \( {C}^{3} \). Then there is a constant \( C < \infty \) such that the operators \( {}_{\delta }{\widehat{\mathbf{T}}}_{q} \) defined by (5.12) for \( 1 \leq q \leq n \) satisfy the following:\n\n(i) \( {\beg... | The main new result is the \( {L}^{p} \) estimate (i), since (ii) and (iii) are already known for \( {\mathbf{T}}_{q} \) (see the proof of Theorem V.2.7), although a separate proof will be needed for the operator \( {}_{\bar{\partial }}{\widehat{\mathbf{T}}}_{q} \). The proof of (i) will be a consequence of standard re... | No |
Theorem 5.6. Let \( D \subset \subset {\mathbb{C}}^{n} \) be strictly pseudoconvex with boundary of class \( {C}^{3} \) . For \( 1 \leq q \leq n \) there are linear (integral) operators\n\n\[ \n{\widehat{\mathrm{S}}}_{q} : {L}_{0, q}^{1}\left( D\right) \rightarrow {L}_{0, q - 1}^{1}\left( D\right) \n\]\nand a constant ... | Proof. By Lemma 5.2(ii), \( {}_{\bar{\delta }}{\widehat{\mathbf{E}}}_{q} : {L}_{0, q}^{1}\left( D\right) \rightarrow {C}_{0, q}^{\infty }\left( {\bar{D}}_{\delta }\right) \) is continuous for \( 1 \leq q \leq n \) . As in the proof of Theorem V.2.7, we introduce the operator \( {\mathbf{T}}_{q}^{V,{V}_{0}} \) given by ... | Yes |
Lemma 6.1. Suppose \( 1 \leq p \leq \infty \) and let \( f \in \mathcal{O}{L}^{p}\left( D\right) \) . For \( 0 < \tau < {\tau }_{0} \) define \( {f}_{0}^{\tau } = f \) and\n\n(6.2)\n\n\[ \n{f}_{j}^{\tau }\left( z\right) = f\left( {z + \tau {\mathbf{n}}_{j}}\right) \;\text{ for }j = 1,\ldots, N.\n\]\n\nThen\n\n(6.3)\n\n... | Proof. Equations (6.3) and (6.4) are obvious from the definitions of \( {U}_{j}^{\tau } \) and \( {f}_{j}^{\tau } \) . Equation (6.5) for \( p < \infty \) follows from the fact that translation defines a\n\ncontinuous operator on \( {L}^{p} \) spaces if \( p < \infty \) ; the case \( p = \infty \) follows from the unif... | Yes |
Then one has \[ \mathop{\lim }\limits_{{\tau \rightarrow 0}}{M}_{p}^{\tau }\left( f\right) = 0\;\text{ if }p < \infty \text{ or if }p = \infty \text{ and }f \in A\left( D\right) ,\] | Proof. Equation (6.9) is obvious. For (6.8), fix \( i \) and \( j \) and let \( W = {U}_{i} \cap {U}_{j} \cap D \) . Then \[ {\begin{Vmatrix}{g}_{ij}^{\tau }\end{Vmatrix}}_{{L}^{p}\left( W\right) } \leq {\begin{Vmatrix}{f}_{j}^{\tau } - f\end{Vmatrix}}_{{L}^{P}\left( W\right) } + {\begin{Vmatrix}{f}_{i}^{\tau } - f\end... | Yes |
Theorem 6.3. Let \( D \) be strictly pseudoconvex with \( {C}^{3} \) boundary and let \( \left\{ {{U}_{j} : j = 0}\right. \) , \( \ldots, N\} \) be an open covering of \( \bar{D} \) . Set \( {V}_{j} = {U}_{j} \cap D \) . Then there is a constant \( C < \infty \) with the following property. If the functions \( {g}_{ij}... | Proof. The proof is analogous to the proof of Theorem VI.4.8, so we will be brief. Choose \( {C}^{\infty } \) functions \( {\chi }_{j} \in {C}_{0}^{\infty }\left( {U}_{j}\right), j = 0,\ldots, N \), such that \( \sum {\chi }_{j} \equiv 1 \) on \( \bar{D} \), and set \( {v}_{j} = \mathop{\sum }\limits_{{v = 0}}^{N}{\chi... | Yes |
Theorem 6.4. Let \( D \) be strictly pseudoconvex with \( {C}^{3} \) boundary. For \( 1 \leq p < \infty \) , every function in \( \mathcal{O}{L}^{p}\left( D\right) \) can be approximated in \( {L}^{p}\left( D\right) \) norm by functions in \( \mathcal{O}\left( \bar{D}\right) \) . Moreover, there is a constant \( C < \i... | Remark 6.5. The question of \ | No |
Theorem 7.1. Suppose the defining function \( r \) for \( D \) is of class \( {C}^{k + 3} \). Then\n\n(i) \( \widehat{A} \) and \( {\bar{\partial }}_{\zeta }\widehat{A} \in {C}^{k + 1,\infty }\left( {\bar{D} \times {D}^{\# }}\right) \);\n\n(ii) \( {\bar{\partial }}_{\zeta }\widehat{C} \in {C}^{k + 1,\infty }\left( {\ba... | Proof. Since \( E = {\Omega }_{0}\left( {L}_{D}\right) = {\Omega }_{0}\left( P\right) /{\Phi }^{n} \), it follows that\n\n\[ {\bar{\partial }}_{\zeta }\widehat{E} = \varphi \left\lbrack {\frac{{\bar{\partial }}_{\zeta }{\Omega }_{0}\left( P\right) }{{\left\lbrack \Phi - r\left( \zeta \right) \right\rbrack }^{n}} - n\fr... | No |
Lemma 7.2. Suppose \( {bD} \) is of class \( {C}^{k + 3} \) . There are \( {\varepsilon }_{0} > 0 \) and functions \( {N}_{0} \) , \( {N}_{1},{N}_{2} \in {C}^{k + 1,\infty }\left( {\bar{D} \times \bar{D}}\right) \) so that\n\n(7.4)\n\n\[ \n{G}_{D}\left( {\zeta, z}\right) = \frac{{N}_{0}}{{\left\lbrack \widehat{\Phi }\l... | Proof. We choose \( {\varepsilon }_{0} > 0 \) as in the construction of \( {\widehat{E}}_{q} \) in \( §{5.1} \), so that \( \varphi \left( \zeta \right) = 1 \) for \( - {\varepsilon }_{0} < r\left( \zeta \right) \leq 0 \) . Since \( \widehat{C} = {\widehat{E}}_{0} - \widehat{A} \), it follows from (5.4) that\n\n\[ \n{\... | Yes |
Lemma 7.3. The function \( \widehat{\Phi } \) satisfies\n\n\[{\int }_{\left\{ \zeta \in D : r\left( \zeta \right) \geq - {\varepsilon }_{0}\right\} }{\left| \widehat{\Phi }\left( \zeta, z\right) \right| }^{-\left( {n + 1 + \alpha }\right) }{dV}\left( \zeta \right) \lesssim \left\{ \begin{array}{l} 1\;\text{ if }\alpha ... | Proof. The proof of Lemma 7.3 is very similar to the one of Lemma 1.5, and the details are left to the reader. The modification required to estimate a volume integral involving \( \widehat{\Phi } \), rather than a boundary integral, has already been used in the proof of Lemma 5.5-see the estimate (5.26) for \( \left| \... | No |
Lemma 7.4. If \( r \) is of class \( {C}^{3} \) and \( F\left( {\zeta, z}\right) \) is the Levi polynomial of \( r \), then | \[ \left\lbrack {F\left( {\zeta, z}\right) - r\left( \zeta \right) }\right\rbrack - \left\lbrack {\overline{F\left( {z,\zeta }\right) } - r\left( z\right) }\right\rbrack = O\left( {\left| \zeta - z\right| }^{3}\right) . \] | Yes |
Corollary 7.5. If \( \Phi \left( {\zeta, z}\right) = F \) locally, then\n\n(i) \( \widehat{\Phi } - {\widehat{\Phi }}^{ * } = O\left( {\left| \zeta - z\right| }^{3}\right) \), and\n\n(ii) \( \left| {\widehat{\Phi }}^{ * }\right| \gtrsim \left| \widehat{\Phi }\right| \) for \( \zeta, z \in \bar{D} \) close to \( {bD} \)... | Proof. (i) is obvious; (ii) is then an immediate consequence, since \( \left| {\widehat{\Phi }}^{ * }\right| \geq \) \( \left| \widehat{\Phi }\right| - \left| {{\widehat{\Phi }}^{ * } - \widehat{\Phi }}\right| \), by the triangle inequality, and \( \left| {\widehat{\Phi }\left( {\zeta, z}\right) }\right| \gtrsim {\left... | Yes |
Theorem 7.6. The kernel\n\n(7.12)\n\n\[ B\left( {\zeta, z}\right) = {G}_{D}\left( {\zeta, z}\right) - {G}_{D}^{ * }\left( {\zeta, z}\right) \]\n\nsatisfies\n\n(7.13)\n\n\[ {\int }_{D}{\left| B\left( \zeta, z\right) \right| }^{s}{dV}\left( \zeta \right) \lesssim 1\;\text{ for all }z \in D \]\n\nand\n\n(7.14)\n\n\[ {\int... | Proof. From (7.4),(7.7), and (7.8) with \( t = n + 1 \) and Corollary 7.5 it follows that\n\n\[ \left| {{G}_{D}\left( {\zeta, z}\right) - {G}_{D}^{ * }\left( {\zeta, z}\right) }\right| \lesssim {\left| \widehat{\Phi }\left( \zeta, z\right) \right| }^{-\left( {n + 1}\right) + 1/2}\;\text{ for }\zeta, z \in D\text{ close... | Yes |
Corollary 7.7. Define the operator \( \mathbf{B} : f \mapsto \mathbf{B}f \) by\n\n\[ \mathbf{B}f\left( z\right) = {\int }_{D}f\left( \zeta \right) B\left( {\zeta, z}\right) {dV}\left( \zeta \right) = {\left( f,\overline{B\left( {\cdot, z}\right) }\right) }_{D}. \]\n\nThen\n\n(i) \( \mathbf{B} \) is a bounded operator f... | Proof. (i) follows from the Theorem and Appendix B, and (ii) follows from the Theorem with \( s = 1 \) and Appendix C. For (iii), define \( {\mathbf{B}}^{ * }f\left( z\right) = {\left( f,\overline{{B}^{ * }\left( {\cdot, z}\right) }\right) }_{D} \) ; if \( f, g \in {C}_{0}\left( D\right) \), it readily follows that \( ... | Yes |
Theorem 7.8. Define the operator \( f \mapsto {\mathbf{G}}_{D}^{ * }{fby} \)\n\n\[ \n{\mathbf{G}}_{D}^{ * }f\left( z\right) = {\int }_{D}f\left( \zeta \right) {G}_{D}^{ * }\left( {\zeta, z}\right) {dV}\left( \zeta \right) = \left( {f,\overline{{G}_{D}^{ * }\left( {\cdot, z}\right) }}\right) .\n\]\n\nThen\n\n(7.18)\n\n\... | Proof. Equation (7.18) is just a reformulation of (7.17). Solving for \( {\mathbf{G}}_{D}^{ * } \) gives\n\n(7.20)\n\n\[ \n{\mathbf{G}}_{D}^{ * } = {\mathbf{P}}_{D} - \mathbf{B} \circ {\mathbf{P}}_{D} = \left( {\mathbf{I} - \mathbf{B}}\right) \circ {\mathbf{P}}_{D}.\n\]\nSince \( \mathbf{B} \) and \( {\mathbf{P}}_{D} \... | Yes |
Theorem 7.10. Suppose \( D \) is strictly pseudoconvex with boundary of class \( {C}^{{2k} + 4} \) for some integer \( k \geq 0 \) . Then the Bergman projection \( {\mathbf{P}}_{D} \) maps \( {C}^{k + \alpha }\left( \bar{D}\right) \) boundedly into \( {C}^{k + \alpha /2}\left( \bar{D}\right) \) for any \( 0 < \alpha < ... | Proof of Theorem 7.10. (Assuming the Main Lemma). Suppose \( 0 < \alpha < \frac{1}{2} \) . Then (7.23) implies that \( \mathbf{I} - \mathbf{B} : {C}^{k + \alpha }\left( \bar{D}\right) \rightarrow {C}^{k + \alpha }\left( \bar{D}\right) \) is bounded. Moreover, it is a consequence of the Ascoli-Arzela Theorem that for \(... | No |
Lemma 7.13. Suppose \( {\mathcal{E}}_{j}, j \geq 0 \), is of class \( {C}^{l} \) on \( \bar{D} \times \bar{D}, l \leq k + 2 \), and \( \left| {\mathcal{E}}_{j}\right| \lesssim \) \( {\left| \zeta - z\right| }^{j} \) . Suppose \( {t}_{1},{t}_{2} \in \mathbb{N} \) . Then\n\n(7.30)\n\n\[ \mathcal{A} = \frac{{\mathcal{E}}_... | Proof. By (7.8), for any integer \( t \geq 0 \) we have\n\n(7.31)\n\n\[ \frac{1}{{\widehat{\Phi }}^{*t}} = \frac{1}{{\widehat{\Phi }}^{t}} + \left( {\widehat{\Phi } - {\widehat{\Phi }}^{ * }}\right) \mathop{\sum }\limits_{{v = 0}}^{{t - 1}}{\widehat{\Phi }}^{-\left( {t - v}\right) }{\widehat{\Phi }}^{* - \left( {v + 1}... | Yes |
Corollary 7.15. The kernel \( B = {G}_{D} - {G}_{D}^{ * } \) is admissible of class \( {C}^{k + 1} \) and of order \( \geq 1 \) . | Proof. It is clear from the above that \( B \) is admissible of class \( {C}^{k + 1} \) . From (7.4) and (7.7) it follows that\n\n(7.34)\n\n\[ \nB = {N}_{0}\left( {\zeta ,\zeta }\right) \left\lbrack {\frac{1}{{\widehat{\Phi }}^{n + 1}} - \frac{1}{{\widehat{\Phi }}^{*n + 1}}}\right\rbrack + {\mathcal{A}}_{1}.\n\]\n\nNow... | Yes |
Lemma 7.16. Suppose \( {\mathcal{A}}_{\lambda, l} \) is an admissible kernel on \( D \times D \) of class \( {C}^{l} \) with \( 1 \leq l \leq k + 2 \), and of order \( \geq \lambda \) . Let \( {V}^{\left( z\right) } \) be a vector field of class \( {C}^{l} \) on \( \bar{D} \) acting in the \( z \) -variable. Then | \[ {V}^{\left( z\right) }{\mathcal{A}}_{\lambda, l} = {\mathcal{A}}_{\lambda - 1, l - 1} + {\mathcal{A}}_{\lambda - 2, l} \] | Yes |
Lemma 7.17. Suppose \( D \subset {\mathbb{R}}^{n} \) with \( {bD} \) of class \( {C}^{l + 1}, l \geq 1 \), and let \( L \) be a tangential vector field on \( \bar{D} \) of class \( {C}^{l} \) . Then there is a first order (tangential) partial differential operator \( {L}^{ * } \) on \( \bar{D} \) of class \( {C}^{l - 1... | Proof. We first consider the case that \( L \) is supported in a suitable open set \( U \) . If \( U \subset \subset D \), then the existence of \( {L}^{ * } \) satisfying (7.37) follows by standard integration by parts. No boundary integral appears since \( L \), and hence also \( {L}^{ * } \) are 0 on \( {bD} \) . Ne... | Yes |
Lemma 7.18. The vector field\n\n\[ \nY = {Y}^{\left( r\right) } = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial r}{\partial {\bar{\zeta }}_{j}}\frac{\partial }{\partial {\zeta }_{j}} - \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial r}{\partial {\zeta }_{j}}\frac{\partial }{\partial {\bar{\zeta }}_{j}} \]\n\nof c... | Proof. It is obvious that \( {Yr} = 0 \), so \( Y \) is tangential. Hence \( Y\widehat{\Phi } = {Y\Phi } - {Yr} = \) \( {YF} \) at points \( \left( {\zeta ,\zeta }\right) \), where \( F \) is the Levi polynomial of \( r \) . It is now easy to check that\n\n\[ \n\left( {YF}\right) \left( {\zeta ,\zeta }\right) = \mathop... | Yes |
Proposition 7.20. Suppose \( {bD} \) is of class \( {C}^{k + 4} \), with \( k \geq 1 \), and let \( {\mathcal{A}}_{\lambda, l} \) be admissible of class \( {C}^{l},2 \leq l \leq k + 1 \), and of order \( \geq \lambda \) with \( \lambda = 0 \) or 1 . If \( {V}^{\left( z\right) } \) is a vector field in \( z \) of class ... | Proof. By differentiating under the integral sign and applying Lemma 7.16 to \( {\overline{\mathcal{A}}}_{\lambda, l} \) one obtains\n\n(7.45)\n\n\[ \n{V}^{\left( z\right) }{\int }_{D}f{\mathcal{A}}_{\lambda, l}{dV} = {\left( f,{\mathcal{A}}_{\lambda - 2, l}\right) }_{D} + {\left( f,{\mathcal{A}}_{\lambda - 1, l - 1}\r... | Yes |
Theorem 8.2. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) satisfies condition \( \left( {R}_{k}\right) \) . Then \( {K}_{D}\left( {\cdot, a}\right) \in {C}^{k}\left( \bar{D}\right) \) for every \( a \in D \) . | The proof is an immediate consequence of the following useful representation for the Bergman kernel. | No |
Lemma 8.3. Let \( a \in D \) and suppose \( {\varphi }_{a} \in {C}_{0}^{\infty }\left( D\right) \) is radially symmetric about a (i.e., \( {\varphi }_{a} \) depends only on \( \left| {z - a}\right| \) ), and \( \int {\varphi }_{a}{dV} = 1 \) . Then\n\n\[ \n{K}_{D}\left( {\cdot, a}\right) = {\mathbf{P}}_{D}\overline{{\v... | Proof of Lemma 8.3. We may assume that \( {\varphi }_{a} \) is supported in a ball \( B\left( {a,\varepsilon }\right) \subset \subset \) \( D \) . By the mean value property (see Exercise E.8.2) \( f \in \mathcal{O}\left( D\right) \) satisfies\n\n(8.1)\n\n\[ \nf\left( a\right) {\int }_{{bB}\left( {a,\rho }\right) }{dS}... | Yes |
Theorem 8.4. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) has \( {C}^{\infty } \) boundary and satisfies condition \( \left( {R}_{1}\right) \) . Then \( {A}^{\infty }\left( D\right) \) is contained in the closure of the linear span of \( \left\{ {{K}_{D}\left( {\cdot, a}\right) : a \in D}\right\} \) in the \( {C}^{... | Proof. We shall discuss the proof in case \( D \) is strictly pseudoconvex with boundary of class \( {C}^{6} \), so that \( D \) satisfies condition \( \left( {R}_{1,2}\right) \) by Theorem 8.1. The modifications necessary for the other case will be left to the reader (see also Lemma 8.13). The proof will involve two t... | No |
Theorem 8.7. A strictly pseudoconvex domain \( D \) in \( {\mathbb{C}}^{n} \) with boundary of class \( {C}^{{2k} + 4} \) with \( k \geq 1 \) satisfies condition \( \left( {B}_{k}\right) \) . Moreover, every smoothly bounded domain for which condition \( \left( {R}_{k}\right) \) holds satisfies \( \left( {B}_{k}\right)... | Proof. Part (i) of condition \( \left( {B}_{k}\right) \) is obvious by Theorem 8.1 and Theorem 8.2. To prove (ii), fix \( P \in \bar{D} \) and suppose the determinant (8.17) were 0 for all \( \left( {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right) \in {D}^{n + 1} \) . By using Theorem 8.4 and the multilinearity and continuity ... | Yes |
Lemma 8.8. If the regions \( {D}_{i} \subset \subset {\mathbb{C}}^{{n}_{i}}, i = 1,2 \), satisfy condition \( \left( {B}_{k}\right) \), so does \( D = {D}_{1} \times {D}_{2} \) | The proof is left to the reader (use Theorem IV.4.7!). | No |
Corollary 8.9. Suppose \( D \subset {\mathbb{C}}^{n} \) has \( {C}^{k} \) boundary and satisfies condition \( \left( {B}_{k}\right) \) . Then for each \( P \in {bD} \) there are a neighborhood \( \Omega \) of \( P \) and points \( {a}_{0},{a}_{1},\ldots \) , \( {a}_{n} \in D \), such that the map \( u = \left( {{u}_{1}... | Proof. Given \( P \in {bD} \), choose \( {a}_{0},\ldots ,{a}_{n} \in D \) according to condition \( \left( {B}_{k}\right) \), so that (8.16) and (8.17) hold. By continuity, there is \( \Omega = \Omega \left( P\right) \), so that \( {K}_{D}\left( {z,{a}_{0}}\right) \neq 0 \) for \( z \in \bar{\Omega } \cap \bar{D} \), a... | Yes |
Theorem 8.10. Let \( {D}_{1} \) and \( {D}_{2} \) be bounded domains in \( {\mathbb{C}}^{n} \) with boundary of class \( {C}^{k} \) which satisfy condition \( \left( {B}_{k}\right) \) . Then every biholomorphic map \( F : {D}_{1} \rightarrow {D}_{2} \) is in \( {C}^{k}\left( {\bar{D}}_{1}\right) \) . | The statement \( F \in {C}^{k}\left( {\bar{D}}_{1}\right) \) means, of course, that every component of \( F = \) \( \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) is in \( {C}^{k}\left( {\bar{D}}_{1}\right) \) . It follows that \( F \) has a \( {C}^{k} \) extension to a neighborhood of \( {\bar{D}}_{1} \) (see Exercise E.I... | No |
Lemma 8.13. For each positive integer \( s \) there are an integer \( {N}_{s} \) and a linear operator\n\n\[ \n{T}^{\left( s\right) } : {C}^{{N}_{s}}\left( \bar{D}\right) \rightarrow {C}^{s}\left( \bar{D}\right) \]\n\nwith the following properties:\n\n(i) \( {T}^{\left( s\right) } \) is bounded, i.e., \( {\left| {T}^{\... | The proof of Lemma 8.5 gives Lemma 8.13 in case \( s = 2 \), with \( {N}_{s} = 6 \), by setting \( {T}^{\left( 2\right) }f = f - q \) . The general case is proved by the same method. The details are left to the reader. | No |
Lemma 8.14. Let \( D \) be a complete Reinhardt domain with center 0 . Then\n\n(8.30)\n\n\[ {K}_{D}\left( {{\lambda \zeta }, z}\right) = {K}_{D}\left( {\zeta ,\bar{\lambda }z}\right) \]\n\nfor all \( \lambda ,\zeta, z \in {\mathbb{C}}^{n} \) with \( \zeta, z,{\lambda \zeta },\bar{\lambda }z \in D \) . | Proof. The formula (8.30) is an obvious consequence of the representation\n\n(8.31)\n\n\[ {K}_{D}\left( {\zeta, z}\right) = \mathop{\sum }\limits_{{v \in {\mathbb{N}}^{n}}}{c}_{v}{\zeta }^{v}{\bar{z}}^{v}\;\text{ on }D \times D, \]\n\nwith suitable constants \( {c}_{v} \), valid for the Bergman kernel of a complete Rei... | No |
Theorem 9.1. If \( \gamma \subset \mathbb{C} \) is a connected real analytic arc, then there are a connected neighborhood \( \Omega \) of \( \gamma \) and a diffeomorphism \( {\sigma }_{\gamma } : \Omega \rightarrow \Omega \) with the following properties:\n\n(i) \( \overline{{\sigma }_{\gamma }} \) is holomorphic on \... | The properties (i) and (ii) determine \( {\sigma }_{\gamma } \) uniquely in the following sense: if \( {\sigma }^{\left( j\right) } : {\Omega }_{j} \rightarrow \) \( {\Omega }_{j}, j = 1,2 \), are two maps which satisfy (i) and (ii), then \( {\sigma }^{\left( 1\right) } \equiv {\sigma }^{\left( 2\right) } \) on the con... | Yes |
Lemma 9.2. If \( \gamma \) is defined near \( P \in \gamma \) by \( \{ z : r\left( {z,\bar{z}}\right) = 0\} \) as above, then the reflection \( {\sigma }_{\gamma } \) is the (locally) unique solution of \( r\left( {z,{\bar{\sigma }}_{\gamma }\left( z\right) }\right) = 0 \) in a neighborhood of \( \left( {P,\bar{P}}\rig... | Example. Suppose \( \gamma \) is a circle with center \( a \) and radius \( R \), i.e., \( \gamma = \{ z \in \mathbb{C} \) : \( r\left( {z,\bar{z}}\right) = 0\} \), where \( r\left( {z,\bar{z}}\right) = \left( {z - a}\right) \left( {\bar{z} - \bar{a}}\right) - {R}^{2} \) . Then \( t\left( z\right) = \bar{a} + {R}^{2}/\... | No |
Theorem 9.3. Let \( D \) be open in \( \mathbb{C} \) and \( P \in {bD} \) . Suppose there is an open disc centered at \( P \) such that \( \gamma = \Delta \cap {bD} \) is a real analytic arc which bounds \( D \) from one side. If \( f \in \mathcal{O}\left( D\right) \) extends continuously to \( D \cup \gamma \) and \( ... | Proof. We leave it to the reader to verify that \( \widehat{f} \) in (9.2) is defined for \( z \in \Delta \) provided \( \Delta \) is sufficiently small, and that \( \widehat{f} \) is holomorphic on \( \Delta - \gamma \) . The hypothesis on \( f \) implies that \( \widehat{f} \) extends continuously from \( \Delta - \b... | No |
For \( \varepsilon \) sufficiently small the reflection \( {\sigma }_{S} \) is defined and real analytic on the polydisc \( P\left( {0,\varepsilon }\right) \) . Moreover, \( {\sigma }_{S}\left( z\right) = z \) for \( z \in S \), and \( \overline{{\sigma }_{S}}\left( {{z}^{\prime },{z}_{n}}\right) \) is holomorphic in \... | Proof. By Lemma 9.2, \( {\bar{\sigma }}_{{z}^{\prime }}\left( {z}_{n}\right) = {w}_{n}\left( {{z}^{\prime },{z}_{n},{\bar{z}}^{\prime }}\right) \), where \( {w}_{n}\left( {{z}^{\prime },{z}_{n},{w}^{\prime }}\right) \) is the unique holomorphic solution of the equation \( r\left( {{z}^{\prime },{z}_{n},{w}^{\prime },{w... | No |
Lemma 9.7. Suppose \( {\sigma }_{S} \) is defined on \( P\left( {0,\varepsilon }\right) \) and let \( \varphi \) be real analytic in a neighborhood of \( S \) . Then, after perhaps shrinking \( \varepsilon \), there is a real analytic function \( \widehat{\varphi } \) on \( P\left( {0,\varepsilon }\right) \) which is h... | Proof. We may write \( \varphi = \varphi \left( {\left( {{z}^{\prime },{z}_{n}}\right) ,\left( {{\bar{z}}^{\prime },{\bar{z}}_{n}}\right) }\right) \), where \( \varphi \left( {z, w}\right) \) is holomorphic on \( P\left( {0,\varepsilon }\right) \times P\left( {0,\varepsilon }\right) \) (shrink \( \varepsilon \), if nec... | Yes |
Proposition 9.8. Let \( F : D \rightarrow {\mathbb{C}}^{n} \) be a holomorphic map which extends continuously to \( \bar{D} \cap P\left( {0,\varepsilon }\right) = \{ z \in P\left( {0,\varepsilon }\right) : r\left( {z,\bar{z}}\right) \leq 0\} \) . Suppose there is a continuous map \( T : \bar{D} \cap P\left( {0,\varepsi... | Proof. Recalling the discussion of the one variable case it is clear how one should define the extension \( \widehat{F} \) of \( F \) . For \( z \in P\left( {0,\varepsilon }\right) \) -shrink \( \varepsilon \), if necessary-we set\n\n\[ \widehat{F}\left( z\right) = \left\{ \begin{array}{ll} F\left( z\right) & \text{ if... | Yes |
Lemma 9.9. Suppose \( f \) is a continuous function on the polydisc \( P\left( {0,\left( {{\varepsilon }^{\prime },{\varepsilon }_{n}}\right) }\right) \) in \( {\mathbb{C}}^{n} \) which satisfies\n\n9.8) for each fixed \( {z}^{\prime } \in {P}^{\prime }\left( {0,{\varepsilon }^{\prime }}\right), f\left( {{z}^{\prime },... | Proof. By Corollary I.1.5 and by (9.8) it is enough to show that \( f \) is separately holomorphic on \( P\left( {0,\left( {{\varepsilon }^{\prime },{\varepsilon }_{n}}\right) }\right) \) in each of the other variables \( {z}_{1},\ldots ,{z}_{n - 1} \) . This will follow from Morera’s Theorem if we show for \( 1 \leq j... | Yes |
Lemma 9.10. Suppose\n\n(9.18)\n\n\[ \det \left( {\frac{\partial {H}_{j}}{\partial {v}_{k}}\left( {0,0,{w}^{\left( 0\right) }}\right) }\right) \neq 0 \]\n\nThen there are \( \varepsilon > 0 \) and a continuous map \( T \) on \( \bar{D} \cap P\left( {0,\varepsilon }\right) \) which satisfies the hypotheses (9.6) and (9.7... | Proof. Application of the Implicit Function Theorem I.2.4 to the holomorphic system of equations \( {H}_{j}\left( {u, v, w}\right) = 0,1 \leq j \leq n \) gives \( {\delta }^{\prime },\delta > 0 \) such that if \( \left( {u, w}\right) \in P\left( {0,{\delta }^{\prime }}\right) \times P\left( {{w}^{\left( 0\right) },{\de... | Yes |
Corollary 9.11. If (9.18) holds, then \( F \) has a holomorphic extension across 0 . | Let us analyze the condition (9.18) in more detail. From (9.16) and (9.17) one obtains\n\n\[ \left( {\partial {H}_{j}/\partial {v}_{k}}\right) \left( {0,0,{w}^{\left( 0\right) }}\right) = \mathop{\sum }\limits_{{v = 1}}^{n}\frac{{\partial }^{2}\rho }{\partial {u}_{v}\partial {\bar{u}}_{k}}\left( 0\right) {L}_{j}{F}_{v}... | Yes |
Lemma 9.12. Suppose the Levi form of the hypersurface \( {S}^{\# } \) is nondegenerate at 0 and that the map \( F \) satisfies \( \det {F}^{\prime }\left( 0\right) \neq 0 \) . Then the hypothesis (9.18) in Lemma 9.10 is satisfied. | Proof. The first hypothesis means that the Levi form\n\n\[ \left( {t,{t}^{\prime }}\right) \mapsto {L}_{0}\left( {\rho ;t,{t}^{\prime }}\right) = \mathop{\sum }\limits_{{v, k = 1}}^{n}\frac{{\partial }^{2}\rho }{\partial {u}_{v}\partial {\bar{u}}_{k}}\left( 0\right) {t}_{v}{\bar{t}}_{k}^{\prime } \]\n\nat 0 of the defi... | Yes |
Theorem 9.15. Let \( {D}_{1} \) and \( {D}_{2} \) be bounded strictly pseudoconvex domains in \( {\mathbb{C}}^{n} \) with real analytic boundary and let \( F : {D}_{1} \rightarrow {D}_{2} \) be biholomorphic. Then \( F \) has a holomorphic extension to a neighborhood of \( \overline{{D}_{1}} \) . | Proof. By Fefferman’s Mapping Theorem, \( F \) extends as a \( {C}^{\infty } \) map to \( {\bar{D}}_{1} \), and we saw in Step 1 of the Proof of Theorem 8.13 that \( \left| {\det {F}^{\prime }\left( z\right) }\right| \geq c > 0 \) on \( {D}_{1} \) , hence det \( {F}^{\prime }\left( z\right) \neq 0 \) for \( z \in b{D}_... | Yes |
Lemma 1. Let \( D \subset {\mathbb{R}}^{n} \) be open and suppose \( F : D \rightarrow {\mathbb{R}}^{n} \) is of class \( {C}^{1} \) . Let \( A \subset D \) be the set of critical points of \( F \), i.e., the set of those points \( a \in D \), for which \( \det {J}_{\mathbb{R}}F\left( a\right) = 0 \) . Then \( F\left( ... | Proof. It is clearly enough to show that \( F\left( {Q \cap A}\right) \) has measure 0 for any compact cube \( Q \subset D \) . For all \( x, y \in Q \) we have\n\n\[ F\left( x\right) - F\left( y\right) = d{F}_{y}\left( {x - y}\right) + r\left( {x, y}\right) ,\]\n\nwhere \( r\left( {x, y}\right) = o\left( \left| {x - y... | Yes |
Lemma 2. Suppose \( D \subset {\mathbb{R}}^{n} \) is open and \( f : D \rightarrow \mathbb{R} \) is of class \( {C}^{2} \) . If \( a \in D \) is a nondegenerate critical point of \( f \) (i.e., \( \det \left\lbrack {{\partial }^{2}f/\partial {x}_{j}\partial {x}_{k}\left( a\right) }\right\rbrack \neq 0 \) ), then there ... | Proof. After an orthogonal change of coordinates we may assume that \( \left( {{\partial }^{2}f/\partial {x}_{j}\partial {x}_{k}\left( a\right) }\right) \) is diagonal with nonzero entries \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) in the diagonal. In a neighborhood \( U \) of \( a \) one then has (notice that \( d{f}... | Yes |
Lemma 3. Let \( D \subset {\mathbb{R}}^{n} \) be open and suppose \( g \in {C}^{2}\left( D\right) \) is real valued. Then there is a set \( E \subset {\mathbb{R}}^{n} \) of measure 0, such that for all \( u \in {\mathbb{R}}^{n} - E \) the set \( {A}_{u} \) of critical points of \( {g}_{u} : D \rightarrow \mathbb{R} \),... | Proof. Define \( F : D \rightarrow {\mathbb{R}}^{n} \) by\n\n\[ F\left( x\right) = \left( {\frac{\partial g}{\partial {x}_{1}}\left( x\right) ,\ldots ,\frac{\partial g}{\partial {x}_{n}}\left( x\right) }\right) .\n\]\nThen \( F \) is of class \( {C}^{1} \) . By Lemma 1, there is \( E \subset {\mathbb{R}}^{n} \) of meas... | Yes |
Theorem 1.1. Let \( f : A \rightarrow {A}^{\prime } \) be a sense-preserving homeomorphism which leaves invariant the modules of the quadrilaterals of the domain \( A \) . Then \( f \) is conformal. | We sketch a proof. Map a quadrilateral of \( A \) and its image in \( {A}^{\prime } \) canonically onto identical rectangles \( R \) and \( {R}^{\prime } \) whose sides are parallel to the coordinate axes. Given a point \( z = x + {iy} \) of \( R \), we consider the two rectangles \( {R}_{1} \) and \( {R}_{2} \) onto w... | Yes |
Lemma 2.1. Let \( F \) be a family of \( K \) -quasiconformal mappings of a domain \( A \) . If every \( f \in F \) omits two values which have a mutual spherical distance \( \geq d > 0 \) , then \( F \) is equicontinuous in \( A \) . | Proof. Let \( s \) denote the spherical distance. Given an \( \varepsilon ,0 < \varepsilon < d \), and a point \( {z}_{0} \in A \), we consider a ring domain \( B = \left\{ {z \mid \delta < s\left( {z,{z}_{0}}\right) < r}\right\} \) with \( \left\{ {z \mid s\left( {z,{z}_{0}}\right) < r}\right\} \subset A \), and choos... | Yes |
Theorem 2.1. A family \( F \) of \( K \) -quasiconformal mappings of a domain \( A \) is equicontinuous and normal, if for three fixed points \( {z}_{1},{z}_{2},{z}_{3} \) of \( A \) and for every \( f \in F \), the distances \( s\left( {f\left( {z}_{i}\right), f\left( {z}_{j}\right) }\right) \) are uniformly bounded a... | Proof. By Lemma 2.1, the family \( F \) is equicontinuous in \( A \smallsetminus \left\{ {{z}_{i},{z}_{j}}\right\}, i, j = \) \( 1,2,3, i \neq j \), and hence throughout \( A \) . | No |
Theorem 2.2. The limit function \( f \) of a sequence \( \left( {f}_{n}\right) \) of K-quasiconformal mappings of a domain \( A \), locally uniformly convergent in \( A \), is either a constant or a \( K \) -quasiconformal mapping. | Proof. If \( f \) is a homeomorphism, then it follows easily from the definition of \( K \) -quasiconformality and from the continuity of the module of quadrilaterals that \( f \) is \( K \) -quasiconformal ([LV], p. 29). A continuous injective map of an open set of the plane into the plane is a homeomorphism (Newman [... | Yes |
Theorem 2.3. Let \( A \) be a domain with at least two boundary points and \( \left( {f}_{n}\right) \) a sequence of \( K \) -quasiconformal mappings of \( A \) onto a fixed domain \( {A}^{\prime } \) . If the sequence \( \left( {f}_{n}\right) \) converges in \( A \), then the limit function is either a \( K \) -quasic... | Here we need not assume that \( \left( {f}_{n}\right) \) is locally uniformly convergent, because we conclude from Lemma 2.1 that \( \left\{ {f}_{n}\right\} \) is a normal family. The theorem follows from equicontinuity and normal family arguments ([LV], p. 78). | No |
Theorem 2.4. Let \( f \) be a \( K \) -quasiconformal mapping of the plane fixing 0 and \( \infty \) . Then for every \( r > 0 \) , \[ \frac{\mathop{\max }\limits_{\varphi }\left| {f\left( {r{e}^{i\varphi }}\right) }\right| }{\mathop{\min }\limits_{\varphi }\left| {f\left( {r{e}^{i\varphi }}\right) }\right| } \leq c\le... | There are many ways to prove this important theorem. A normal family argument shows that a finite bound \( c\left( K\right) \) must exist. A quantitative estimate is obtained as follows. Let \( {z}_{1} \) and \( {z}_{2} \) be points on the circle \( \left| z\right| = \) \( r \) at which the minimum and maximum of \( \l... | Yes |
Theorem 3.1. Let \( f : A \rightarrow {A}^{\prime } \) be a sense-preserving diffeomorphism with the property\n\n\[ \n{D}_{f}\left( z\right) \leq K \n\]\n\nfor every \( z \in A \) . Then \( f \) is a \( K \) -quasiconformal mapping. | Proof. We pick an arbitrary quadrilateral \( Q \) of \( A \) . Let \( w \) be the mapping which is induced from the canonical rectangle \( R\left( {0, M, M + i, i}\right) \) of \( Q \) onto the canonical rectangle \( {R}^{\prime }\left( {0,{M}^{\prime },{M}^{\prime } + i, i}\right) \) of \( f\left( Q\right) \) . Becaus... | Yes |
Theorem 3.2. Let \( f : A \rightarrow {A}^{\prime } \) be a K-quasiconformal mapping. If \( f \) is differentiable at \( {z}_{0} \in A \), then\n\n\[ \mathop{\max }\limits_{\alpha }\left| {{\partial }_{\alpha }f\left( {z}_{0}\right) }\right| \leq K\mathop{\min }\limits_{\alpha }\left| {{\partial }_{\alpha }f\left( {z}_... | The idea of the proof is to consider a small square \( Q \) centered at \( {z}_{0} \) and regard it as a quadrilateral with the vertices at distinguished points. The area and the distance of the sides of \( f\left( Q\right) \) can be approximated by expressions involving the partial derivatives of \( f \) at \( {z}_{0}... | No |
Theorem 3.3. A quasiconformal mapping is absolutely continuous on lines. | This result was first established by Strebel (1955) and Mori. A later proof by Pfluger, which uses Rengel's inequality and a minimum of real analysis, is presented in [LV], p. 162. | No |
Theorem 3.4. A K-quasiconformal mapping \( f \) of a domain \( A \) is differentiable and satisfies the dilatation condition (3.1) almost everywhere in \( A \) . | Differentiability a.e. of quasiconformal mappings was first proved by Mori [1] with the aid of the Rademacher-Stepanoff theorem and Theorem 2.4. | No |
Theorem 3.5. A sense-preserving homeomorphism \( f \) of a domain \( A \) is \( K \) -quasi-conformal if\n\n\( {1}^{ \circ }f \) is ACL in \( A \) ;\n\n\( {2}^{ \circ }\mathop{\max }\limits_{\alpha }\left| {{\partial }_{\alpha }f\left( z\right) }\right| \leq K\mathop{\min }\limits_{\alpha }\left| {{\partial }_{\alpha }... | Proof. We first note that being ACL, the mapping \( f \) has partial derivatives a.e. and, as a homeomorphism, is therefore differentiable a.e. Thus condition \( {2}^{ \circ } \) makes sense. As above, we conclude that \( f \) has \( {L}^{2} \) -derivatives. After this, we can follow the proof of Theorem 3.1, apart fro... | No |
Theorem 4.1. A homeomorphism \( f \) is \( K \) -quasiconformal if and only if \( f \) is an \( {L}^{2} \) -solution of an equation \( \bar{\partial }f = \mu \partial f \), where \( \mu \) satisfies (4.2) for almost all \( z \) . | Proof. The necessity follows from Theorem 3.4 and the sufficiency from Theorem 3.5, when we note that \( \parallel \mu {\parallel }_{\infty } < 1 \) implies that \( f \) is sense-preserving. | Yes |
Theorem 4.2 (Uniqueness Theorem). Let \( f \) and \( g \) be quasiconformal mappings of a domain \( A \) whose complex dilatations agree a.e. in \( A \) . Then \( f \circ {g}^{-1} \) is a conformal mapping. | Proof. By (4.4), the complex dilatation of \( f \circ {g}^{-1} \) vanishes a.e. From Theorem 3.5 we deduce that \( f \circ {g}^{-1} \) is 1-quasiconformal. Hence, by Theorem 1.1, it is conformal.\n\nConversely, if \( f \circ {g}^{-1} \) is conformal, we conclude from (4.4) that \( f \) and \( g \) have the same complex... | Yes |
Theorem 4.3. Let \( f \) be a quasiconformal mapping of the plane whose complex dilatation \( \mu \) has a bounded support and which satisfies the condition \( \mathop{\lim }\limits_{{z \rightarrow \infty }}\left( {f\left( z\right) - z}\right) = 0 \) . Then\n\n\[ f\left( z\right) = z + \mathop{\sum }\limits_{{i = 1}}^{... | Proof. By (4.7) we have \( f\left( z\right) = z + T\bar{\partial }f\left( z\right) \) . By (4.14), \( T\bar{\partial }f\left( z\right) = \left( {T\sum {\varphi }_{i}}\right) \left( z\right) \) . For \( p > 2 \), it follows from Hölder’s inequality that \( \left| {T{\varphi }_{i}\left( z\right) }\right| \leq {c}_{p}{\be... | Yes |
Theorem 4.4 (Existence Theorem). Let \( \mu \) be a measurable function in a domain A with \( \parallel \mu {\parallel }_{\infty } < 1 \) . Then there is a quasiconformal mapping of \( A \) whose complex dilatation agrees with \( \mu \) a.e. | The proof can be divided into three parts. One first shows that if \( \mu \in {C}_{0}^{\infty } \) , the Beltrami equation \( \bar{\partial }w = \mu \partial w \) has a locally injective solution. This can be so constructed that a topological argument shows it to be in fact globally injective. Another way to obtain fro... | No |
Theorem 4.6. Let \( \mu \) and \( {\mu }_{n}, n = 1,2,\ldots \), be measurable functions in the plane such that \( {\begin{Vmatrix}{\mu }_{n}\end{Vmatrix}}_{\infty } \leq k < 1 \) and \( \lim {\mu }_{n}\left( z\right) = \mu \left( z\right) \) a.e. If \( f \) and \( {f}_{n} \) are the quasi-conformal mappings of the pla... | Proof. By Theorems 4.4 and 4.2, the mappings \( f \) and \( {f}_{n} \) exist and are uniquely determined. By Theorem 2.1, \( \left\{ {f}_{n}\right\} \) is a normal family. By Theorems 4.5 and 4.2, every convergent subsequence \( \left( {f}_{{n}_{i}}\right) \) tends to \( f \) . Then the sequence \( \left( {f}_{n}\right... | Yes |
Theorem 4.7. Let \( f \) be a quasiconformal mapping with maximal dilatation \( K \), and \( 0 < t < 1 \) . Then \( f = {f}_{2} \circ {f}_{1} \), where \( {f}_{1} \) is \( {K}^{t} \) -quasiconformal and \( {f}_{2} \) is \( {K}^{1 - t} \) -quasiconformal. | Proof. Let \( \mu \) denote the complex dilatation of \( f \) . We choose the complex dilatation \( {\mu }_{1} \) of \( {f}_{1} \) as follows: \( {\mu }_{1}\left( z\right) \) is the point on the line segment from 0 to\n\n\( \mu \left( z\right) \) for which \( h\left( {0,{\mu }_{1}\left( z\right) }\right) = \operatornam... | Yes |
Theorem 5.1. The boundary values \( h \) of a K-quasiconformal self-mapping \( f \) of the upper half-plane, \( f\left( \infty \right) = \infty \), satisfy the double inequality\n\n\[ 1/\lambda \left( K\right) \leq \frac{h\left( {x + t}\right) - h\left( x\right) }{h\left( x\right) - h\left( {x - t}\right) } \leq \lambd... | Proof. Choose \( {x}_{1} = x - t,{x}_{2} = x,{x}_{3} = x + t,{x}_{4} = \infty \), and denote the middle term in (5.1) by \( \alpha \) . By the considerations in 2.4, we then have \( M\left( {H\left( {{x}_{1},{x}_{2},{x}_{3},{x}_{4}}\right) }\right) = 1 \) and\n\n\[ M\left( {H\left( {h\left( {x}_{1}\right), h\left( {x}_... | Yes |
Lemma 5.1. The family of k-quasisymmetric functions \( h \) which keep 0 and 1 fixed is equicontinuous at every point of the real axis. | Proof. We first conclude from \( h\left( {2}^{-n + 1}\right) - h\left( {2}^{-n}\right) \geq h\left( {2}^{-n}\right) /k \) that\n\n\[ h\left( {2}^{-n}\right) \leq {\left( \frac{k}{k + 1}\right) }^{n} \]\n\n(5.2)\n\nfor every non-negative integer \( n \) . In looking for a bound for \( h\left( {a + x}\right) - h\left( a\... | Yes |
Lemma 5.2. Let \( \left\lbrack {a, b}\right\rbrack \) be a closed interval on the real axis, and \( \varepsilon > 0 \) . Then there is a \( \delta > 0 \) such that for a normalized quasisymmetric function \( h \) ,\n\n\[ \left| {h\left( x\right) - x}\right| < \varepsilon ,\;x \in \left\lbrack {a, b}\right\rbrack ,\]\n\... | Proof. If the lemma is not true, there is an \( \varepsilon > 0 \) and a sequence of normalized \( \left( {1 + 1/n}\right) \) -quasisymmetric functions \( {h}_{n}, n = 1,2,\ldots \), such that\n\n\[ \mathop{\sup }\limits_{{a \leq x \leq b}}\left| {{h}_{n}\left( x\right) - x}\right| \geq \varepsilon \]\n\nfor every \( n... | Yes |
Lemma 5.3. Let \( {h}_{1} \) and \( {h}_{2} \) be \( k \) -quasisymmetric functions and \( {f}_{1} \) and \( {f}_{2} \) their Beurling-Ahlfors extensions. If the quasisymmetry constant of \( {h}_{2} \circ {h}_{1}^{-1} \) tends to 1, then the maximal dilatation of \( {f}_{2} \circ {f}_{1}^{-1} \) converges to 1 . | Proof. Suppose the lemma is not true. Making use again of the linear transformations \( {A}_{j} \), appealing to Lemmas 5.1 and 5.2, and reasoning as in the proof of Theorem 5.2, we would arrive at the following situation. There exist normalized \( k \) -quasisymmetric mappings \( {h}_{n1} \) and \( {h}_{n2}, n = 1,2,\... | Yes |
Lemma 5.4. The Beurling-Ahlfors extension of a quasisymmetric function is a quasi-isometry in the hyperbolic metric of the upper half-plane. | Proof. Let \( h \) be a \( k \) -quasisymmetric function and \( f \) its Beurling-Ahlfors extension (5.4). We have to prove the existence of a constant \( c \) depending only on \( k \), such that\n\n\[ \frac{1}{c}\frac{\left| dz\right| }{\operatorname{Im}z} \leq \frac{\left| df\left( z\right) \right| }{\operatorname{I... | Yes |
Theorem 5.3. For every quasisymmetric function, the boundary value problem has a real-analytic solution. | Proof. The result can be proved with the aid of the decomposition formula (4.17) in 4.7 which makes it possible to express a \( k \) -quasisymmetric function as a composition of \( \left( {1 + \varepsilon }\right) \) -quasisymmetric functions. The proof based on this method will be given in II.5.2. A more direct proof ... | No |
Lemma 5.5. Let \( {K}^{ * } \) be the maximal dilatation of the quasisymmetric function \( h \) and \( K \) the minimal maximal dilatation of the quasiconformal self-mappings of \( H \) with boundary values \( h \) . Then \( K \rightarrow 1 \) as \( {K}^{ * } \rightarrow 1 \) . | Proof. The function \( h \) is \( \lambda \left( {K}^{ * }\right) \) -quasisymmetric by the remark in 5.2 preceding Lemma 5.1. Here \( \lambda \left( {K}^{ * }\right) \rightarrow 1 \) as \( {K}^{ * } \rightarrow 1 \), as we noted at the end of 2.4. By Theorem 5.2, the maximal dilatation of the Beurling-Ahlfors extensio... | Yes |
Lemma 6.1. Let \( C \) be a quasicircle and \( f \) a homeomorphism of the plane which is \( K \) -quasiconformal in the complement of \( C \) . Then \( f \) is a \( K \) -quasiconformal mapping of the plane. | Proof. It follows from the definition of a quasicircle that there is a quasi-conformal mapping \( w \) of the plane which maps the real axis onto \( C \) . Then \( f \circ w \) is quasiconformal in the plane, as is also \( f = \left( {f \circ w}\right) \circ {w}^{-1} \) . Since the area of \( C \) is zero, we conclude ... | Yes |
Theorem 6.1. A Jordan curve admits a quasiconformal reflection if and only if it is a quasicircle. | Proof. Suppose first that \( C \) is a quasicircle. Let \( f \) be a quasiconformal mapping of the plane which maps \( {A}_{1} \) onto the upper half-plane \( H \) . Then the mapping \( \varphi = {f}^{-1} \circ j \circ f \), where \( j\left( z\right) = \bar{z} \), is a quasiconformal reflection in \( C \) .\n\nConverse... | Yes |
Lemma 6.2. A K-quasidisc A has the following properties:\n\n\( {1}^{ \circ } \) Every quasiconformal reflection in \( \partial A \) is of the form \( f \circ j \circ {f}^{-1} \), where \( f \) is a quasiconformal mapping of the plane which maps the upper half-plane \( H \) conformally onto \( A \), and \( j \) denotes ... | Proof. Let \( \varphi \) be an arbitrary quasiconformal reflection in \( \partial A \) . If \( f \) is constructed as in the second part of the proof of Theorem 6.1, with \( A \) replacing \( {A}_{1} \), we obtain \( {1}^{ \circ } \) . Since \( \partial A \) is a \( K \) -quasicircle, there exists a \( {K}^{2} \) -quas... | Yes |
Lemma 6.3. Let \( \varphi \) be a K-quasiconformal reflection in \( C \) which passes through \( \infty \) . Then\n\n\[ \left| {\varphi \left( z\right) - z}\right| \leq \frac{{c}_{1}\left( K\right) }{{\eta }_{2}\left( {\varphi \left( z\right) }\right) },\;z \in {A}_{1}, \]\n\nwhere \( {\eta }_{2} \) is the Poincaré den... | Proof. Let \( h : H \rightarrow {A}_{1} \) be a conformal mapping, \( h\left( \infty \right) = \infty \), and set again \( f = h \) in the closure of \( H \) and \( f = \varphi \circ h \circ j \) in the lower half-plane. Then \( f \) is a \( K \) -quasiconformal mapping of the plane.\n\nFix \( z \in {A}_{1} \) and \( {... | Yes |
Lemma 6.4. Let \( C \) be a \( K \) -quasicircle bounding the domains \( {A}_{1} \) and \( {A}_{2} \) and passing through \( \infty \) . Then there exists a \( {c}_{2}\left( K\right) \) -quasiconformal reflection \( \varphi \) in \( C \), continuously differentiable in \( {A}_{1} \) and \( {A}_{2} \), such that \[ \lef... | Proof. Let \( {h}_{1} \) be a conformal mapping of \( {A}_{1} \) onto the upper half-plane \( H \) and \( {h}_{2} \) a conformal mapping of \( {A}_{2} \) onto the lower half-plane, both fixing \( \infty \) . Since \( {h}_{1} \) and \( {h}_{2} \) have homeomorphic extensions to the boundary, we can form the function \( ... | Yes |
Lemma 6.5. Let \( A \) be a \( K \) -quasidisc with \( \infty \in \partial A \), and \( f : H \rightarrow A \) a conformal mapping satisfying \( f\left( \infty \right) = \infty \) . Then\n\n\[ \n{\int }_{0}^{y}\left| {{f}^{\prime }\left( {it}\right) }\right| {dt} \leq {c}_{1}\left( K\right) d\left( {f\left( {iy}\right)... | Proof. By Lemma 6.2, \( f \) has a \( {K}^{2} \) -quasiconformal extension to the plane. We may assume without loss of generality that \( f\left( 0\right) = 0 \) . For \( y > 0 \), the mapping \( f \) satisfies the inequality\n\n\[ \n\left| {{f}^{\prime }\left( {iy}\right) }\right| \leq \frac{{4d}\left( {f\left( {iy}\r... | Yes |
Theorem 6.4. A uniform domain \( A \) is linearly locally connected. | Proof. Fix a finite \( {z}_{0} \) and \( r > 0 \), and suppose that \( {z}_{1},{z}_{2} \in A \cap \overline{D\left( {{z}_{0}, r}\right) } \) . Since \( A \) is uniform, there exists an arc \( \alpha \) joining \( {z}_{1} \) and \( {z}_{2} \) in \( A \) such that \( l\left( \alpha \right) \leq a\left| {{z}_{1} - {z}_{2}... | Yes |
Theorem 6.5. Let \( A \) be a simply connected domain whose boundary contains more than one point. If \( A \) is linearly locally connected, then \( \partial A \) is a Jordan curve which satisfies the arc condition. | Proof. In 6.4, after defining the notion of linear local connectivity, we proved that \( \partial A \) is a Jordan curve.\n\nChoose two finite points \( {z}_{1},{z}_{2} \in \partial A \) and set \( {z}_{0} = \left( {{z}_{1} + {z}_{2}}\right) /2, r = \) \( \left| {{z}_{1} - {z}_{2}}\right| /2 \) . The theorem follows if... | Yes |
Lemma 6.6. Let \( C \) be a Jordan curve such that for all conjugate quadrilaterals \( {A}_{1},{A}_{2} \) with \( M\left( {A}_{1}\right) = 1 \) we have \( M\left( {A}_{2}\right) \leq K \) . Then \( C \) is a \( c\left( K\right) \) -quasicircle, where \( c\left( K\right) \) depends only on \( K \) . | Proof. Let \( {g}_{1} : {A}_{1} \rightarrow H \) and \( {g}_{2} : {A}_{2} \rightarrow {H}^{\prime } \) be conformal mappings, where \( {H}^{\prime } \) is the lower half-plane. Consider the increasing homeomorphism \( x \rightarrow h\left( x\right) = \) \( {g}_{2}\left( {{g}_{1}^{-1}\left( x\right) }\right) \) of the r... | Yes |
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