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Lemma 1.1.20. We have that\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{\mathrm{e}}^{-\pi {\left| x\right| }^{2}}\mathrm{\;d}x = 1 \n\] | Proof. The case \( N = 1 \) is familiar from calculus (or see [BKR, Sect. 6.6]). For the \( N \) -dimensional case, write\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{\mathrm{e}}^{-\pi {\left| x\right| }^{2}}\mathrm{\;d}x = {\int }_{\mathbb{R}}{\mathrm{e}}^{-\pi {x}_{1}^{2}}\mathrm{\;d}{x}_{1}\cdots {\int }_{\mathbb{R}}{\mathrm{... | Yes |
Lemma 1.1.21. We have\n\n\[ \n{\omega }_{N - 1} = \frac{2{\pi }^{N/2}}{\Gamma \left( {N/2}\right) }\n\]\n\nwhere\n\n\[ \n\Gamma \left( x\right) = {\int }_{0}^{\infty }{t}^{x - 1}{\mathrm{e}}^{-t}\mathrm{\;d}t\n\]\n\nis Euler's gamma function. | Proof. Introducing polar coordinates we have\n\n\[ \n1 = {\int }_{{\mathbb{R}}^{N}}{\mathrm{e}}^{-\pi {\left| x\right| }^{2}}\mathrm{\;d}x = {\int }_{{S}^{N - 1}}\mathrm{\;d}\sigma {\int }_{0}^{\infty }{\mathrm{e}}^{-\pi {r}^{2}}{r}^{N - 1}\mathrm{\;d}r\n\]\n\nor\n\n\[ \n\frac{1}{{\omega }_{N - 1}} = {\int }_{0}^{\inft... | Yes |
Lemma 1.1.22. We have\n\n\\[ \n\\eta \\left( k\\right) = {\\pi }^{n}\\frac{2\\left( {k!}\\right) }{\\left( {k + n - 1}\\right) !},\\;N\\left( k\\right) = {\\pi }^{n}\\frac{k!}{\\left( {k + n}\\right) !}.\n\\] | Proof. Polar coordinates show easily that \\( \\eta \\left( k\\right) = 2\\left( {k + n}\\right) N\\left( k\\right) \\) . So it is enough to calculate \\( N\\left( k\\right) \\) . Let \\( z = \\left( {{z}_{1},{z}_{2},\\ldots ,{z}_{n}}\\right) = \\left( {{z}_{1},{z}^{\\prime }}\\right) \\) . We write\n\n\\[ \nN\\left( k... | Yes |
Lemma 1.1.23. Let \( z \in B \subseteq {\mathbb{C}}^{n} \) and \( 0 < r < 1 \) . The symbol 1 denotes the point \( \left( {1,0,\ldots ,0}\right) \) . Then\n\n\[ \n{K}_{B}\left( {\mathbf{z}, r\mathbf{1}}\right) = \frac{n!}{{\pi }^{n}}\frac{1}{{\left( 1 - r{\mathbf{z}}_{1}\right) }^{n + 1}}.\n\] | Proof. Refer to formula (1.1.1.1) preceding Lemma 1.1.20. Then\n\n\[ \n{K}_{B}\left( {\mathbf{z}, r\mathbf{1}}\right) = \mathop{\sum }\limits_{\alpha }\frac{{\mathbf{z}}^{\alpha }{\left( r\mathbf{1}\right) }^{\alpha }}{{\gamma }_{\alpha }} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\mathbf{z}}_{1}^{k}{r}^{k}}{N\... | Yes |
Theorem 1.1.24. If \( \mathbf{z},\zeta \in B \), then the Bergman kernel for the unit ball in \( {\mathbb{C}}^{n} \) is\n\n\[ \n{K}_{B}\left( {\mathbf{z},\zeta }\right) = \frac{n!}{{\pi }^{n}}\frac{1}{{\left( 1 - \mathbf{z} \cdot \bar{\zeta }\right) }^{n + 1}}, \]\n\nwhere \( \mathbf{z} \cdot \bar{\zeta } = {\mathbf{z}... | Proof. Let \( \mathbf{z} = r\widetilde{\mathbf{z}} \in B \), where \( r = \left| \mathbf{z}\right| \) and \( \left| \widetilde{\mathbf{z}}\right| = 1 \) . Also, fix \( \zeta \in B \) . Choose a unitary rotation \( \rho \) such that \( \rho \widetilde{\mathbf{z}} = \mathbf{1} \) . Then, by Proposition 1.1.14 and Lemma 1... | Yes |
Proposition 1.1.26. The Bergman metric for the ball \( B = B\left( {0,1}\right) \subseteq {\mathbb{C}}^{n} \) is given by\n\n\[ \n{g}_{ij}\left( z\right) = \frac{n + 1}{{\left( 1 - {\left| z\right| }^{2}\right) }^{2}}\left\lbrack {\left( {1 - {\left| z\right| }^{2}}\right) {\delta }_{ij} + {\bar{z}}_{i}{z}_{j}}\right\r... | Proof. Since \(K\left( {z, z}\right) = n!/\left( {{\pi }^{n}{\left( 1 - {\left| z\right| }^{2}\right) }^{n + 1}}\right) \), this is a routine computation that we leave to the reader. | No |
Proposition 1.1.28. The Bergman kernel for the polydisc \( {D}^{n}\left( {0,1}\right) \subseteq {\mathbb{C}}^{n} \) is the product\n\n\[ K\left( {z,\zeta }\right) = \frac{1}{{\pi }^{n}}\mathop{\prod }\limits_{{j = 1}}^{n}\frac{1}{{\left( 1 - {z}_{j}{\bar{\zeta }}_{j}\right) }^{2}}. \] | Proof. Exercise for the reader. Use the uniqueness property of the Bergman kernel. | No |
Lemma 1.2.3. For any \( {z}_{0} \in {\mathbb{C}}^{n} \), any \( \epsilon > 0 \), we have\n\n\[ \n{\int }_{\partial B\left( {{z}^{0},\epsilon }\right) }\eta \left( \bar{z}\right) \land \omega \left( z\right) = n{\int }_{B\left( {{z}^{0},\epsilon }\right) }\omega \left( \bar{z}\right) \land \omega \left( z\right) .\n\] | Proof. Notice that \( \mathrm{d}\eta \left( \bar{z}\right) = \bar{\partial }\eta \left( \bar{z}\right) = {n\omega }\left( \bar{z}\right) \) . Therefore by Stokes’s theorem,\n\n\[ \n{\int }_{\partial B\left( {{z}^{0},\epsilon }\right) }\eta \left( \bar{z}\right) \land \omega \left( z\right) = {\int }_{B\left( {{z}^{0},\... | Yes |
Corollary 1.2.8. In complex dimension 1, the last corollary says that\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\oint }_{\partial \Omega }\frac{f\left( \zeta \right) }{\zeta - z}\mathrm{\;d}\zeta \] | Thus we see that the canonical Szegő integral formula is in fact nothing other than the constructive Cauchy integral formula. But only on the disc! | No |
Proposition 1.2.9. Define\n\n\[ \mathcal{P}\left( {z,\zeta }\right) = \frac{{\left| S\left( z,\zeta \right) \right| }^{2}}{S\left( {z, z}\right) }, z \in \Omega ,\zeta \in \partial \Omega .\n\]\n\nThen, for any \( f \in A\left( \Omega \right) \) and \( z \in \Omega \), it holds that\n\n\[ f\left( z\right) = {\int }_{\p... | Proof. Fix \( z \in \Omega \) and \( f \in A\left( \Omega \right) \) and define\n\n\[ u\left( \zeta \right) = f\left( \zeta \right) \frac{\overline{S\left( {z,\zeta }\right) }}{S\left( {z, z}\right) },\;\zeta \in \partial \Omega .\n\]\n\nThen \( u \in {H}^{2}\left( {\partial \Omega }\right) \) hence\n\n\[ f\left( z\rig... | Yes |
Lemma 1.2.11. The functions \( \left\{ {z}^{\alpha }\right\} \), where \( \alpha \) ranges over multi-indices, are pairwise orthogonal and span \( {H}^{2}\left( {\partial B}\right) \) . | Proof. The orthogonality follows from symmetry considerations. For the completeness, notice that it suffices to see that the span of \( \left\{ {z}^{\alpha }\right\} \) is dense in \( A\left( B\right) \) in the uniform topology on the boundary. By the Stone-Weierstrass theorem, the closed algebra generated by \( \left\... | Yes |
Lemma 1.2.12. Let \( I = \\left( {1,0,\\ldots ,0}\\right) \) . Then\n\n\[ S\\left( {z,1}\\right) = \\frac{\\left( {n - 1}\\right) !}{2{\\pi }^{n}}\\frac{1}{{\\left( 1 - {z}_{1}\\right) }^{n}}. \] | Proof. We have that\n\n\[ S\\left( {z,1}\\right) = \\mathop{\\sum }\\limits_{\\alpha }\\frac{{z}^{\\alpha } \\cdot {1}^{\\alpha }}{{\\begin{Vmatrix}{z}_{1}^{\\alpha }\\end{Vmatrix}}_{{L}^{2}\\left( {\\partial B}\\right) }^{2}} \]\n\n\[ = \\mathop{\\sum }\\limits_{{k = 0}}^{\\infty }\\frac{{z}_{1}^{k}}{\\eta \\left( k\\... | Yes |
Lemma 1.2.13. Let \( \rho \) be a unitary rotation on \( {\mathbb{C}}^{n} \) . For any \( z \in \bar{B},\zeta \in \partial B \), we have that \( S\left( {z,\zeta }\right) = S\left( {{\rho z},{\rho \zeta }}\right) \) . | Proof. This is a standard change of variables argument and we omit it. | No |
Theorem 1.2.14. The Szegő kernel for the ball is\n\n\[ S\left( {z,\zeta }\right) = \frac{\left( {n - 1}\right) !}{2{\pi }^{n}}\frac{1}{{\left( 1 - z \cdot \bar{\zeta }\right) }^{n}}. \] | Proof. Let \( z \in B \) be arbitrary. Let \( \rho \) be the unique unitary rotation such that \( {\rho z} \) is a multiple of 1 . Then, by 1.2.13,\n\n\[ S\left( {z,\zeta }\right) = S\left( {{\rho }^{-1}\mathbf{1},\zeta }\right) \]\n\n\[ = S\left( {\mathbf{1},{\rho \zeta }}\right) = \overline{S\left( {{\rho \zeta },\ma... | Yes |
Theorem 1.14.2. Let \( \Omega \) be a smoothly bounded domain in \( {\mathbb{C}}^{n} \) with boundary having connected components \( {S}_{1},{S}_{2},\ldots ,{S}_{k} \) . For specificity, say that \( {S}_{1} \) is the boundary component that bounds the unbounded portion of the complement of \( \bar{\Omega } \) . Let \( ... | The reader can see that this new theorem is completely analogous to the results presented earlier in the one-variable setting. But it must be confessed that this theorem is something of a canard. For, when \( j \geq 2 \), any function holomorphic on the unbounded domain with boundary \( {S}_{j} \) will (by the Hartogs ... | Yes |
Theorem 2.1.2 (Burns-Shnider-Wells). Let \( k \in \mathbb{N} \) . Let \( \epsilon > 0 \) be small. Let \( {\mathcal{U}}_{\epsilon }^{k} \equiv {\mathcal{U}}_{\epsilon }^{k}\left( B\right) \) be any neighborhood of the ball \( B \subseteq {\mathbb{C}}^{n} \) in the \( {C}^{k} \) topology as defined above. If \( n \geq 2... | The last statement of the theorem perhaps merits some explanation. If \( n = 1 \) and \( \epsilon < 1/5 \), then perforce any equivalence class in \( {\mathcal{U}}_{\epsilon }^{k}/ \sim \) will contain only bounded domains that are simply connected. Thus any such domain will, by the Riemann mapping theorem, be conforma... | No |
Lemma 2.1.6. Let \( \Omega \subset {\mathbb{C}}^{n} \) be smoothly bounded and strictly pseudoconvex. Let \( w \in \Omega \) be fixed. Let \( K \) denote the Bergman kernel. There is a constant \( {C}_{w} > 0 \) such that \[ \parallel K\left( {w, \cdot }\right) {\parallel }_{\text{sup }} \leq {C}_{w} \] | Proof. The function \( K\left( {z, \cdot }\right) \) is harmonic. Let \( \phi : \Omega \rightarrow \mathbb{R} \) be a radial, \( {C}_{c}^{\infty } \) function centered at \( w \) . Assume that \( \phi \geq 0 \) and \( \int \phi \left( \zeta \right) \mathrm{d}V\left( \zeta \right) = 1 \) . Then the mean value property i... | Yes |
Lemma 2.1.9. For each \( s \in \mathbb{N} \) we have \( W{H}^{\infty }\left( \Omega \right) \subseteq P\left( {{W}_{0}^{s}\left( \Omega \right) }\right) \) . | Proof. Let \( u \in {C}^{\infty }\left( \bar{\Omega }\right) \) . Choose \( v \) according to Lemma 2.1.7. Then \( u - v \in {W}_{0}^{s} \) and \( {Pu} = P\left( {u - v}\right) \) . Therefore\n\n\[ P\left( {W}_{0}^{s}\right) \supseteq P\left( {{C}^{\infty }\left( \bar{\Omega }\right) }\right) \supseteq P\left( {W{H}^{\... | Yes |
Lemma 2.1.10. For any \( g \in {L}^{2}\left( {\Omega }_{2}\right) \), we have\n\n\[ \n{P}_{1}\left( {u \cdot \left( {g \circ \phi }\right) }\right) = u \cdot \left( {\left( {{P}_{2}\left( g\right) }\right) \circ \phi }\right) .\n\] | Proof. Notice that \( u \cdot \left( {g \circ \phi }\right) \in {L}^{2}\left( {\Omega }_{1}\right) \) by change of variables. Therefore\n\n\[ \n{P}_{1}\left( {u \cdot \left( {g \circ \phi }\right) }\right) \left( z\right) = {\int }_{{\Omega }_{1}}{K}_{1}\left( {z,\zeta }\right) u\left( \zeta \right) g\left( {\phi \left... | Yes |
Lemma 2.1.11. Let \( \psi : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be a \( {C}^{j} \) diffeomorphism that satisfies\n\n\[ \left| {\frac{{\partial }^{\alpha }\psi }{\partial {z}^{\alpha }}\left( z\right) }\right| \leq C \cdot {\left( {\delta }_{1}\left( z\right) \right) }^{-\left| \alpha \right| } \]\n\n(2.1.11.1)\n... | Proof. The subscript 0 causes no trouble by the definition of \( {W}_{0}^{j} \) . Therefore it suffices to prove an estimate of the form\n\n\[ \parallel g \circ \psi {\parallel }_{{W}_{0}^{j}} \leq C\parallel g{\parallel }_{{W}_{0}^{j + J}},\;\text{ all }g \in {C}_{c}^{\infty }\left( \Omega \right) .\n\nBy the chain ru... | Yes |
Lemma 2.1.12. For each \( j \in \mathbb{N} \), there is an integer \( J \) so large that if \( g \in \) \( {W}_{0}^{j + J}\left( {\Omega }_{2}\right) \), then \( g \circ \phi \in {W}_{0}^{j}\left( {\Omega }_{1}\right) \) . | Proof. The Cauchy estimates give (since \( \phi \) is bounded) that\n\n\[ \left| {\frac{{\partial }^{\alpha }{\phi }_{\ell }}{\partial {z}^{\alpha }}\left( z\right) }\right| \leq C \cdot {\left( {\delta }_{1}\left( z\right) \right) }^{-\left| \alpha \right| }\;,\;\ell = 1,\ldots, n \]\n\n(2.1.12.1)\n\n\n\nand\n\n\[ \le... | Yes |
Lemma 2.1.13. The function \( u \) is in \( {C}^{\infty }\left( {\bar{\Omega }}_{1}\right) \) . | Proof. It suffices to show that \( u \in {W}^{j}\left( {\Omega }_{1}\right) \), every \( j \) . So fix \( j \) . Let \( m = \) \( m\left( j\right) \) as in Condition \( R \) . According to (2.1.12.1), \( \left| {u\left( z\right) }\right| \leq C{\delta }_{1}{\left( z\right) }^{-{2n}} \) . Then, by Lemma 2.1.12 and the E... | No |
Lemma 2.1.14. The function \( u \) is bounded from 0 on \( {\bar{\Omega }}_{1} \) . | Proof. By symmetry, we may apply Lemma 2.1.13 to \( {\phi }^{-1} \) and \( \det {J}_{\mathbb{C}}\left( {\phi }^{-1}\right) = 1/u \) . We conclude that \( 1/u \in {C}^{\infty }\left( \overline{{\Omega }_{2}}\right) \) . Thus \( u \) is nonvanishing on \( \bar{\Omega } \) . | Yes |
Lemma 3.1.1. Let \( {\Omega }_{1} \) and \( {\Omega }_{2} \) be two bounded domains in \( {\mathbb{C}}^{n} \) with \( {q}^{1} \in {\Omega }_{1} \) and \( {q}^{2} \in {\Omega }_{2} \) fixed points. Denote by \( {b}_{1}^{1},\ldots ,{b}_{n}^{1} \) the Bergman coordinates as defined near \( {q}^{1} \) in \( {\Omega }_{1} \... | Proof of the Lemma. To avoid confusion, we write \( \left( {{z}_{1},\ldots ,{z}_{n}}\right) \) and \( \left( {{w}_{1},\ldots ,{w}_{n}}\right) \) for the \( {\mathbb{C}}^{n} \) -coordinates in \( {\Omega }_{1} \) and \( \left( {{Z}_{1},\ldots ,{Z}_{n}}\right) \) and \( \left( {{W}_{1},\ldots ,{W}_{n}}\right) \) for the ... | Yes |
Proposition 3.2.1. Let \( \Omega \) be a bounded domain with \( {C}^{2} \) boundary and \( S \) its Szegő kernel. With \( \mathcal{P}\left( {z,\zeta }\right) \) as defined above, and with \( f \in C\left( \bar{\Omega }\right) \) holomorphic on \( \Omega \) , we have\n\n\[ f\left( z\right) = {\int }_{\partial \Omega }\m... | ## Proof: See Proposition 1.2.9. | No |
Proposition 3.2.2. Let \( \Omega \) be a bounded domain and \( K \) its Bergman kernel. With \( \mathcal{B}\left( {z,\zeta }\right) \) as defined above, and with \( f \in C\left( \bar{\Omega }\right) \) holomorphic on \( \Omega \), we have\n\n\[ f\left( z\right) = {\int }_{\partial \Omega }\mathcal{B}\left( {z,\zeta }\... | The proof is just the same as that for Proposition 1.2.9, and we omit the details. | No |
Proposition 3.2.4. Let \( {\Omega }_{1},{\Omega }_{2} \) be domains in \( {\mathbb{C}}^{n} \) . Let \( f : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be biholomorphic. Then\n\n\[ \n{\mathcal{B}}_{{\Omega }_{2}}\left( {f\left( z\right), f\left( \zeta \right) }\right) {\left| \det {J}_{\mathbb{C}}f\left( \zeta \right) \r... | Proof: Of course we use the result of Proposition 3.2.3. Now\n\n\[ \n{\mathcal{B}}_{{\Omega }_{1}}\left( {z,\zeta }\right) = \frac{{\left| {K}_{{\Omega }_{1}}\left( z,\zeta \right) \right| }^{2}}{{K}_{{\Omega }_{1}}\left( {z, z}\right) }\n\]\n\n\[ \n= \frac{{\left| \det {J}_{\mathbb{C}}f\left( z\right) \cdot {K}_{{\Ome... | Yes |
Proposition 3.2.5. The operator\n\n\[ \n\mathcal{B}f\left( z\right) = {\int }_{B}\mathcal{B}\left( {z,\zeta }\right) f\left( \zeta \right) \mathrm{d}V\left( \zeta \right) ,\n\]\n\nacting on \( {L}^{1}\left( B\right) \), is univalent. | Proof: In fact it is useful to take advantage of the symmetry of the ball. We can rewrite the Poisson-Bergman integral as\n\n\[ \n{\int }_{B}f \circ {\Phi }_{z}\left( \zeta \right) \mathrm{d}V\left( \zeta \right)\n\]\n\nwhere \( {\Phi }_{z} \) is a suitable automorphism of the ball. Then it is clear that this integral ... | Yes |
Lemma 3.2.6. Let \( \Omega \) be a bounded domain and \( \mathcal{B} \) its Poisson-Bergman kernel. If \( z \in \Omega \) is fixed, then\n\n\[ \n{\int }_{\Omega }\mathcal{B}\left( {z,\zeta }\right) \mathrm{d}V\left( \zeta \right) = 1 \n\] | Proof: Certainly the function \( f\left( \zeta \right) \equiv 1 \) is an element of the Bergman space on \( \Omega \) . As a result,\n\n\[ \n1 = f\left( z\right) = {\int }_{\Omega }\mathcal{B}\left( {z,\zeta }\right) f\left( \zeta \right) \mathrm{d}V\left( \zeta \right) = {\int }_{\Omega }\mathcal{B}\left( {z,\zeta }\r... | Yes |
Proposition 3.2.7. Let \( \Omega \) be the ball \( B \) in \( {\mathbb{C}}^{n} \) . Then the mapping\n\n\[ f \mapsto {\int }_{\Omega }\mathcal{B}\left( {z,\zeta }\right) f\left( \zeta \right) \mathrm{d}V\left( \zeta \right) \]\n\nsends \( {L}^{p}\left( \Omega \right) \) to \( {L}^{p}\left( \Omega \right) ,1 \leq p \leq... | Proof: We know from the lemma that\n\n\[ \parallel \mathcal{B}\left( {z, \cdot }\right) {\parallel }_{{L}^{1}\left( \Omega \right) } = 1 \]\n\nfor each fixed \( z \) . An even easier estimate shows that\n\n\[ \parallel \mathcal{B}\left( {\cdot ,\zeta }\right) {\parallel }_{{L}^{1}\left( \Omega \right) } \leq 1 \]\n\nfo... | Yes |
Proposition 3.2.8. Let \( \Omega \subseteq {\mathbb{C}}^{n} \) be the unit ball \( B \) . Let \( f \in C\left( \bar{\Omega }\right) \) . Let \( F = \mathcal{B}f \) . Then \( F \) extends to a function that is continuous on \( \bar{\Omega } \) . Moreover, if \( P \in \partial \Omega \), then\n\n\[ \mathop{\lim }\limits_... | Proof: Let \( \epsilon > 0 \) . Choose \( \delta > 0 \) such that if \( z, w \in \bar{\Omega } \) and \( \left| {z - w}\right| < \delta \), then \( \left| {f\left( z\right) - f\left( w\right) }\right| < \epsilon \) . Let \( M = \mathop{\sup }\limits_{{\zeta \in \bar{\Omega }}}\left| {f\left( \zeta \right) }\right| \) .... | Yes |
Proposition 3.2.12. When \( \Omega = B \), the unit ball, then the function \( \rho \) is a metric on \( \partial B \) . For a more general smoothly bounded, strictly pseudoconvex domain, the function \( \rho \) is a pseudometric. That is to say, there is constant \( C \geq 1 \) such that\n\n\[ \rho \left( {z,\zeta }\r... | Proof: The first assertion is Proposition 6.5.1 in [KRA4]. The second assertion is proved in [KRA1, pp. 357-358]. We shall provide the details of this argument in Proposition 3.5.2. | No |
Proposition 3.2.13. The balls\n\n\[ \n{\beta }_{2}\left( {z, r}\right) = \{ \zeta \in \Omega : \rho \left( {z,\zeta }\right) < r\} \n\]\n\ntogether with ordinary Euclidean volume measure \( \mathrm{d}V \), form a space of homogeneous type in the sense of Coifman and Weiss [COW]. | Proof: This is almost immediate from the preceding proposition, but details may be found in [KRA1, Sect. 8.6]. | No |
Theorem 3.2.15. The operator \( \mathcal{M} \) is of weak type \( \left( {1,1}\right) \) and of strong type \( \left( {p, p}\right) \) , \( 1 < p \leq \infty \) . | Proof: Again this is a standard consequence of the previous proposition in the context of spaces of homogeneous type. See [COW]. | No |
Theorem 3.2.16. Let \( \Omega \) be the unit ball \( B \) in \( {\mathbb{C}}^{n} \) . Let \( f \) be a locally integrable function on \( \Omega \) . Then there is a constant \( C > 0 \) such that, for \( z \in \Omega \) ,\n\n\[ \left| {\mathcal{B}f\left( z\right) }\right| \leq C \cdot \mathcal{M}f\left( z\right) \] | Proof: It is easy to see that \( \left| {1 - z \cdot \bar{\zeta }}\right| \geq \left( {1/2}\right) \left( {1 - {\left| z\right| }^{2}}\right) \) . Therefore we may perform these standard estimates:\n\n\[ \left| {\mathcal{B}f\left( z\right) }\right| = \left| {{\int }_{\Omega }\mathcal{B}\left( {z,\zeta }\right) f\left( ... | Yes |
Theorem 3.2.17. Let \( \Omega \) be the unit ball \( B \) in \( {\mathbb{C}}^{n} \) . Let \( f \) be an \( {L}^{p}\left( {\Omega ,\mathrm{d}V}\right) \) function, \( 1 \leq p \leq \infty \) . Then \( \mathcal{B}f \) has radial boundary limits almost everywhere on \( \partial \Omega \) . | Proof: The proof follows standard lines, using Theorems 3.2.15 and 3.2.16. See the detailed argument in [KRA1, Theorem 8.6.11]. | No |
Theorem 3.2.19. Let \( f \) be an \( {L}^{p}\left( B\right) \) function, \( 1 \leq p \leq \infty \) . Then, for almost every \( P \in \partial B \) , | In fact, using the Fefferman asymptotic expansion (as discussed in detail in the next section), we may imitate the development of Theorems 3.2.15 and 3.2.16 and prove a result analogous to Theorem 3.2.17 on any smoothly bounded, strictly pseudoconvex domain. We omit the details, as they would repeat ideas that we prese... | No |
Proposition 3.4.1. The Poisson-Szegő kernel on the ball B solves the Dirichlet problem for the invariant Laplacian \( \mathcal{L} \). That is to say, if \( f \) is a continuous function on \( \partial B \), then the function\n\n\[ u\left( z\right) = \left\{ \begin{array}{l} {\int }_{\partial B}\mathcal{P}\left( {z,\zet... | Sketch of the Proof of Proposition 3.4.1 Now\n\n\[ \mathcal{L}u = \mathcal{L}{\int }_{\partial B}\mathcal{P}\left( {z,\zeta }\right) \cdot f\left( \zeta \right) \mathrm{d}\sigma \left( \zeta \right) = {\int }_{\partial B}\left\lbrack {{\mathcal{L}}_{z}\mathcal{P}\left( {z,\zeta }\right) }\right\rbrack \cdot f\left( \ze... | Yes |
Proposition 3.4.3. Let \( f \) be a \( {C}^{2} \) function on the unit ball that is annihilated by the invariant Laplacian \( \mathcal{L} \) . Then, for any \( 0 < r < 1 \) and \( S \) the unit sphere,\n\n\[ \n{\int }_{S}f\left( {r\zeta }\right) \mathrm{d}\sigma \left( \zeta \right) = c\left( r\right) \cdot f\left( 0\r... | Proof: Replacing \( f \) with the average of \( f \) over the orthogonal group, this just becomes a calculation to determine the exact value of the constant \( c\left( r\right) \) -see [RUD2, p. 51]. | No |
Proposition 3.4.4. Suppose that \( f \) is a \( {C}^{2} \) function on the unit ball \( B \) that is annihilated by the invariant Laplacian \( \mathcal{L} \) . Then \( f \) satisfies the identity \( \mathcal{B}f = f \) . In other words, for any \( z \in B \) , \[ f\left( z\right) = {\int }_{B}\mathcal{B}\left( {z,\zeta... | Proof: We have checked the result when \( z = 0 \) in the last proposition. For a general \( z \), compose with a Möbius transformation and use the biholomorphic invariance of the kernel and the differential operator \( \mathcal{L} \) . | No |
Proposition 3.4.6. Let \( \Omega = B \), the unit ball in \( {\mathbb{C}}^{n} \), and \( \mathcal{B} = {\mathcal{B}}_{B}\left( {z,\zeta }\right) \) its Poisson-Bergman kernel. Then \( \mathcal{B} \) is plurisubharmonic in the \( \zeta \) variable. | Proof: Fix a point \( \zeta \in B \) and let \( \Phi \) be an automorphism of \( B \) such that \( \Phi \left( \zeta \right) = 0 \) . From Proposition 3.2.4, we then have\n\n\[ \n{\mathcal{B}}_{B}\left( {z,\zeta }\right) = {\mathcal{B}}_{B}\left( {\Phi \left( z\right) ,\Phi \left( \zeta \right) }\right) \cdot {\left| \... | Yes |
Proposition 3.5.2. The binary operator \( \rho \) is a metric on \( \partial B \) . | Proof: Let \( z, w,\zeta \in \partial B \) . We shall show that\n\n\[ \rho \left( {z,\zeta }\right) \leq \rho \left( {z, w}\right) + \rho \left( {w,\zeta }\right) \]\n\nAssume for simplicity that the dimension \( n = 2 \) . After applying a unitary rotation, we may suppose that \( w = \mathbf{1} = \left( {1,0}\right) \... | Yes |
Proposition 3.5.2. Let \( B \subseteq {\mathbb{C}}^{n} \) be the unit ball and \( g \in C\left( {\partial B}\right) \) . Then the function \[ G\left( z\right) = \left\{ \begin{array}{ll} {\int }_{\partial B}\mathcal{P}\left( {z,\zeta }\right) g\left( \zeta \right) \mathrm{d}\sigma \left( \zeta \right) & \text{ if }z \i... | Proof: It is straightforward to calculate that \[ {\bigtriangleup }_{B}G\left( z\right) = {\int }_{\partial B}\left\lbrack {{\bigtriangleup }_{B}\mathcal{P}\left( {z,\zeta }\right) }\right\rbrack g\left( \zeta \right) \mathrm{d}\sigma \left( \zeta \right) \] \[ = 0 \] because \( {\bigtriangleup }_{B}\mathcal{P}\left( {... | Yes |
Lemma 4.2.3. Let \( \\left\\{ {{Y}_{1},\\ldots ,{Y}_{{a}_{k}}}\\right\\} \) be any orthonormal basis for \( {\\mathcal{H}}^{k} \) . The following properties hold for the zonal harmonics:\n\n(a) \( {Z}_{{x}^{\\prime }}^{\\left( k\\right) }\\left( {x}^{\\prime }\\right) = \\frac{{a}_{k}}{\\sigma \\left( {\\sum }_{N - 1}\... | Proof. Let \( {x}_{1}^{\\prime },{x}_{2}^{\\prime } \\in {\\sum }_{N - 1} \) and let \( \\rho \) be a rotation such that \( \\rho {x}_{1}^{\\prime } = {x}_{2}^{\\prime } \). Then, by parts (a) and (c) of 4.2.2, we know that\n\n\\[ \n\\mathop{\\sum }\\limits_{{m = 1}}^{{a}_{k}}{\\left| {Y}_{m}\\left( {x}_{1}^{\\prime }\... | Yes |
Lemma 4.2.5. Let \( P \) be a polynomial in \( {\mathbb{R}}^{N} \) such that\n\n\[ P\left( {\rho x}\right) = P\left( x\right) \]\n\nfor all \( \rho \in O\left( N\right) \) and \( x \in {\mathbb{R}}^{N} \) . Then there exist constants \( {c}_{0},\ldots ,{c}_{p} \) such that\n\n\[ P\left( x\right) = \mathop{\sum }\limits... | Proof. We write \( P \) as a sum of homogeneous terms:\n\n\[ P\left( x\right) = \mathop{\sum }\limits_{{\ell = 0}}^{q}{P}_{\ell }\left( x\right) \]\n\nwhere \( {P}_{\ell } \) is hologeneous of degree \( \ell \) . Now for any \( \epsilon > 0 \) and \( \rho \in O\left( N\right) \), we have\n\n\[ \mathop{\sum }\limits_{{\... | Yes |
Lemma 4.2.8. Fix \( k \) . Let \( {F}_{{y}^{\prime }}\left( {x}^{\prime }\right) \) be defined for all \( {x}^{\prime },{y}^{\prime } \in \sum \) . Assume that\n\n(i) \( {F}_{{y}^{\prime }}\left( \cdot \right) \) is a spherical harmonic of degree \( k \) for every \( {y}^{\prime } \in \sum \) .\n\n(ii) For every rotati... | Proof of the Lemma: Fix \( {y}^{\prime } \in \sum \) and let \( \rho \in O\left( N\right) \) be such that \( \rho \left( {y}^{\prime }\right) = {y}^{\prime } \) . Then\n\n\[ \n{F}_{{y}^{\prime }}\left( {x}^{\prime }\right) = {F}_{\rho {y}^{\prime }}\left( {\rho {x}^{\prime }}\right) = {F}_{{y}^{\prime }}\left( {\rho {x... | Yes |
Proposition 4.2.10. The Gegenbauer polynomials satisfy the following properties:\n\n(1) \( {P}_{0}^{\lambda }\left( t\right) \equiv 1 \) .\n\n(2) \( \frac{\mathrm{d}}{\mathrm{d}t}{P}_{k}^{\lambda }\left( t\right) = {2\lambda }{P}_{k - 1}^{\lambda + 1}\left( t\right) \) for \( k \geq 1 \) .\n\n(3) \( \frac{\mathrm{d}}{\... | Proof. We obtain (1) by simply setting \( r = 0 \) in the defining equation for the Gegenbauer polynomials.\n\nFor (2), note that\n\n\[ \n{2r\lambda }\mathop{\sum }\limits_{{k = 0}}^{\infty }{P}_{k}^{\lambda + 1}\left( t\right) {r}^{k} \equiv {2r\lambda }{\left( 1 - 2rt + {r}^{2}\right) }^{-\left( {\lambda + 1}\right) ... | Yes |
Proposition 4.3.2. The spaces \( {\mathcal{H}}^{p, q} \) enjoy the following properties:\n\n(1) \( D\left( {p, q;n}\right) \equiv {\dim }_{\mathbb{C}}{\mathcal{H}}^{p, q} = \frac{\left( {p + q + n - 1}\right) \left( {p + n - 2}\right) !\left( {q + n - 2}\right) !}{p!q!\left( {n - 1}\right) !\left( {n - 2}\right) !} \) ... | Proof. We leave the proofs of parts (1) and (2) as exercises.\n\nTo prove (3), notice that if \( \phi \in {\mathcal{H}}^{p, q} \), then we may write \( \phi = \mathop{\sum }\limits_{{j = 1}}^{D}{a}_{j}{f}_{j} \) . Then\n\n\[ {\int }_{\sum }{H}_{n}^{p, q}\left( {\zeta ,\eta }\right) \phi \left( \eta \right) \mathrm{d}\s... | No |
Lemma 4.3.5 (Dini-Kummer). For \( j = 1,2,\ldots \) let \( {a}_{j},{b}_{j} > 0 \) and put\n\n\[ \n{D}_{j} = {b}_{j} - {b}_{j + 1}\frac{{a}_{j + 1}}{{a}_{j}}.\n\]\n\nIf \( \lim \mathop{\inf }\limits_{{j \rightarrow \infty }}{D}_{j} > 0 \), then the series \( \mathop{\sum }\limits_{j}{a}_{j} \) converges. | Proof. By hypothesis, we may find a \( \beta > 0 \) and an integer \( {j}_{0} > 0 \) such that, if \( j \geq {j}_{0} \), then \( {D}_{j} > \beta \) . Thus\n\n\[ \n\beta < {b}_{j} - {b}_{j + 1}\frac{{a}_{j + 1}}{{a}_{j}}\n\]\n\nso that\n\n\[ \n0 < {a}_{j} < \frac{{a}_{j}{b}_{j} - {b}_{j + 1}{a}_{j + 1}}{\beta }\n\]\n\n(... | Yes |
Corollary 4.3.7 (Raabe). If \( {a}_{j} > 0 \) for \( j = 1,2,\ldots \), we set \( {Q}_{j} = j(1 - \) \( \left. {{a}_{j + 1}/{a}_{j}}\right) \) . If it holds that\n\n\[ \liminf {Q}_{j} > 1 \]\n\n(4.3.7.1)\n\nthen \( \mathop{\sum }\limits_{j}{a}_{j} \) converges. | Proof. Let \( {b}_{1} = 1 \) and \( {b}_{j} = j - 1 \) for \( j \geq 2 \) . Then\n\n\[ {Q}_{j} - 1 = j\left( {1 - \frac{{a}_{j + 1}}{{a}_{j}}}\right) - 1 \]\n\n\[ = \left( {j - 1}\right) - j\frac{{a}_{j + 1}}{{a}_{j}} \]\n\n\[ = {D}_{j}\text{,} \]\n\nwhere we are using the notation of the lemma. Then \( \lim \mathop{\i... | No |
Proposition 4.3.8. Take\n\n\\[ \nF\\left( {a, b, c;x}\\right) = \\mathop{\\sum }\\limits_{{j = 0}}^{\\infty }\\frac{\\Gamma \\left( {a + j}\\right) \\Gamma \\left( {b + j}\\right) \\Gamma \\left( c\\right) }{\\Gamma \\left( a\\right) \\Gamma \\left( b\\right) \\Gamma \\left( {c + j}\\right) } \\cdot \\frac{{x}^{j}}{j!}... | Proof. We want to apply Raabe’s test. Thus we need to calculate the terms \\( {Q}_{j} \\) . Denote the absolute value of the \\( j \\) th summand by \\( {\\alpha }_{j} \\) . Then, since \\( \\left| x\\right| = 1 \\), we have\n\n\\[ \n\\frac{{\\alpha }_{j + 1}}{{\\alpha }_{j}} = \\frac{\\left( {a + j}\\right) \\left( {b... | Yes |
Theorem 4.3.12. Let \( 0 \leq r < 1 \) and \( \eta ,\zeta \in \partial B \) . Then the Poisson-Szegő kernel for the ball \( B \subseteq {\mathbb{C}}^{n} \) is given by the formula\n\n\[ \mathcal{P}\left( {{r\eta },\zeta }\right) = \mathop{\sum }\limits_{{p, q = 0}}^{\infty }{S}_{n}^{p, q}\left( r\right) {H}_{n}^{p, q}\... | Proof. Recall that, if \( g \in C\left( {\partial B}\right) \), then\n\n\[ G\left( z\right) = \left\{ \begin{array}{ll} {\int }_{\partial B}\mathcal{P}\left( {z,\zeta }\right) g\left( \zeta \right) \mathrm{d}\sigma \left( \zeta \right) & \text{ on }B \\ g\left( z\right) & \text{ on }\partial B \end{array}\right. \]\n\n... | Yes |
Theorem 4.4.5. Let \( \Omega \) be a smoothly bounded, strictly pseudoconvex domain in \( {\mathbb{C}}^{n} \). Then, for each \( s \in \mathbb{R} \), there is a constant \( C = C\left( s\right) \) such that \[ \parallel {Pf}{\parallel }_{{W}^{s}} \leq C \cdot \parallel f{\parallel }_{{W}^{s}}. \] | Proof of the Theorem: We have already observed that the Bergman projection of a strictly pseudoconvex domain maps functions in \( {C}^{\infty }\left( \bar{\Omega }\right) \) to functions in \( {C}^{\infty }\left( \bar{\Omega }\right) \). Thus it suffices to prove our estimate (4.4.5.1) for \( f \in {C}^{\infty }\left( ... | Yes |
Lemma 5.1.3. Let \( \Omega \subset {\mathbb{C}}^{n} \) and \( f : \Omega \rightarrow {\mathbb{C}}^{n} \) be a holomorphic mapping. Suppose that \( \det {\left( \mathrm{d}f\right) }_{P} \neq 0 \) for some \( P \in \Omega \) . Then there exist neighborhoods \( U \) of \( P \) and \( V \) of \( f\left( P\right) \) such th... | Of course this lemma is nothing other than a holomorphic version of the inverse function theorem in several complex variables. We have proved it in Theorem 1.1.12. | No |
Lemma 5.1.4. Let \( \left\{ {g}_{j}\right\} \) be a sequence of continuous open mappings of \( \Omega \subset {\mathbb{C}}^{n} \) into \( {\mathbb{C}}^{n} \) . Suppose that the \( {g}_{j} \) converge, uniformly on compact sets, to a limit mapping \( g : \Omega \rightarrow {\mathbb{C}}^{n} \) . Further suppose that, for... | Proof of Lemma 5.1.4: Seeking a contradiction, we suppose the assertion to be false. Passing to a subsequence, we may suppose that \( U \) is such that \( \bar{U} \) is compact, \( g\left( P\right) \notin {g}_{j}\left( \bar{U}\right) \), for \( j = 1,2,\ldots \), and \( \bar{U} \cap {g}^{-1}\left( {g\left( P\right) }\r... | Yes |
Lemma 5.1.5 (Hurwitz’s theorem). Let \( \Omega \) be an open, connected set in \( {\mathbb{C}}^{n} \) and let \( \left\{ {f}_{j}\right\} \) be a sequence of holomorphic functions on \( \Omega \) . We assume that the \( {f}_{j} \) converge uniformly on compact sets to a holomorphic function \( f \) . Then, if \( {f}_{j}... | Proof. Seeking a contradiction, we suppose that \( f\left( P\right) = 0 \) for some \( P \in \Omega \) . Let \( \mathcal{Q} \) be a small open polydisc about \( P \) . Then \( f ≢ 0 \) on \( \mathcal{Q} \) (since, if it were identically 0, then \( f \) would be identically 0 on \( \Omega \) since \( \Omega \) is connec... | Yes |
Theorem 5.2.1 (Ebin). If \( \left( {M,{g}_{0}}\right) \) is a \( {C}^{\infty } \) compact Riemannian manifold, then there is a neighborhood \( \mathcal{G} \) of \( {g}_{0} \) in the \( {C}^{\infty } \) topology on the \( {C}^{\infty } \) Riemannian metrics such that: If \( g \in \mathcal{G} \) then there is a diffeomor... | Ebin's original proof of the theorem just stated involved infinite-dimensional manifolds and the construction of \ | No |
Corollary 5.2.2. If \( {G}_{j} \times M \rightarrow M \) is a sequence of actions on a compact manifold \( M \) by compact Lie groups \( {G}_{j} \) and if the \( {G}_{j} \) -actions sub-converge in the \( {C}^{\infty } \) topology to a compact Lie group action \( {G}_{0} \times M \rightarrow M \), then for all \( j \) ... | This corollary follows from the proof of Ebin's theorem (Theorem 5.2.1) by averaging a fixed Riemannian metric over the group actions to produce \( {G}_{j} \) -invariant metrics \( {g}_{j} \) converging in the \( {C}^{\infty } \) topology to a \( {G}_{0} \) -invariant metric \( {g}_{0} \ | Yes |
Theorem 5.2.3 ([GRK2]). If \( {\Omega }_{0} \) is a bounded, \( {C}^{\infty } \), strictly pseudoconvex domain in \( {\mathbb{C}}^{n} \) that is not biholomorphic to the ball, then there is a neighborhood \( \mathcal{U} \) of \( {\Omega }_{0} \) in the \( {C}^{\infty } \) topology (on bounded domains with the \( {C}^{\... | The essential idea of the proof of this theorem is to note, from the Lu Qi-Keng theorem (Theorem 5.8.1), that the Bergman metric of \( {\Omega }_{0} \) does not have constant holomorphic sectional curvature, while at the same time the holomorphic sectional curvature is asymptotically constant at the boundary. So far, t... | No |
Lemma 5.2.6. If \( {\Omega }_{j}, j = 1,2,\ldots \), converge in the \( {C}^{\infty } \) topology to \( {\Omega }_{0} \) (with \( {\Omega }_{0} \) being \( {C}^{\infty } \), strictly pseudoconvex, and not biholomorphic to the ball), and if \( {g}_{j} \in \operatorname{Aut}\left( {\Omega }_{j}\right) \), then there are ... | Proof of the Lemma: Fix a point \( p \) and a compact set \( {K}_{0} \) as in Proposition 5.2.4. Then, for \( j \) large, \( {g}_{j}\left( p\right) \in {K}_{0} \subset {\Omega }_{j} \) . By normal families, there is a subsequence \( {g}_{{j}_{k}} \) which converges uniformly on each compact subset of \( {\Omega }_{0} \... | Yes |
Lemma 5.2.7. There is a neighborhood \( \mathcal{V} \) of \( {\Omega }_{0} \) in the \( {C}^{\infty } \) topology on domains and a family \( {g}_{\Omega },\Omega \in \mathcal{V} \), with \( {g}_{\Omega } \) a \( {C}^{\infty } \) Riemannian metric on \( \operatorname{cl}\left( \Omega \right) \) such that (1) if \( \oper... | Proof. Set \( {g}_{{\Omega }_{0}} \) equal to the average with respect to \( \operatorname{Aut}\left( {\Omega }_{0}\right) \) of the Euclidean metric on \( \operatorname{cl}\left( {\Omega }_{0}\right) \) . For each \( \Omega \neq {\Omega }_{0} \), choose diffeomorphisms \( {F}_{\Omega } : \operatorname{cl}\left( \Omega... | Yes |
Lemma 5.2.8. The metrics \( {g}_{\Omega } \) in Lemma 5.2.7 can be chosen to be product metrics near the boundary. | Proof of Lemma 5.2.8: An \( \operatorname{Aut}\left( \Omega \right) \) product metric of this sort at and near the boundary is easily obtained using the map\n\n\[ \partial \Omega \times \lbrack 0,\delta ) \rightarrow \Omega \]\n\ndefined by\n\n\[ \left( {b, t}\right) \mapsto {\exp }_{p}\left( {tN}\right) \]\n\nwhere \(... | Yes |
Theorem 5.2.10. If \( {G}_{0} \) is a compact subgroup of the diffeomorphism group of a compact manifold (possibly with boundary), then there is a neighborhood \( \mathcal{V} \) of \( {G}_{0} \) in the \( {C}^{\infty } \) topology on the diffeomorphism group such that every compact subgroup \( G \) of the diffeomorphis... | The proofs of these results are obtained by extracting suitable portions of the proof of Theorem 5.2.2. | No |
Theorem 5.3.10. Let \( \Omega = \{ \rho < 0\} \subseteq {\mathbb{C}}^{2} \) be smoothly bounded and \( P \in \partial \Omega \) . The point \( P \) is of finite geometric type \( m \geq 2 \) if and only if it is of finite analytic type \( m \) . | Proof. We may assume that \( P = 0 \) . Write \( \rho \) in the form\n\n\[ \rho \left( z\right) = 2\operatorname{Re}{z}_{2} + f\left( {z}_{1}\right) + \mathcal{O}\left( {\left| {{z}_{1}{z}_{2}}\right| + {\left| {z}_{2}\right| }^{2}}\right) . \]\n\nWe do this of course by examining the Taylor expansion of \( \rho \) and... | Yes |
Proposition 5.3.16. Let \( \Omega \subseteq {\mathbb{C}}^{n} \) be a bounded domain with compact automorphism group in the \( {C}^{k} \) topology, \( k > 0 \) an integer. Let \( \alpha \) be a multi-index such that \( \left| \alpha \right| \leq k \) . Then there is a positive, finite constant \( {K}_{\alpha } \) such t... | Proof of Proposition 5.3.16: Suppose to the contrary that, for some fixed multi-index \( \alpha \), there is no bound \( {K}_{\alpha } \) . Then there are a sequence \( {\varphi }_{j} \) of automorphisms of \( \Omega \) and points \( {P}_{j} \in \Omega \) such that\n\n\[ \left| {\frac{{\partial }^{\alpha }}{\partial {z... | Yes |
Proposition 5.3.17. Let \( k \) be a positive integer. Let \( \Omega \) be a smoothly bounded domain on which\n\n\[ \left| {\frac{{\partial }^{\alpha }}{\partial {z}^{\alpha }}\varphi \left( z\right) }\right| \leq {K}_{\alpha } \]\n\n(5.3.17.1)\n\nfor all \( \varphi \in \operatorname{Aut}\left( \Omega \right) \), all \... | Proof of the Proposition: From (5.3.17.1), there is a constant \( {K}_{{\Omega }_{1}} \) so that\n\n\[ \left| {\nabla {\varphi }_{j}\left( z\right) }\right| \leq {K}_{{\Omega }_{1}} \]\n\nfor all \( \varphi \in \operatorname{Aut}\left( \Omega \right) \), all \( j \), and all \( z \in \Omega \) . Let \( \epsilon > 0 \) ... | Yes |
Corollary 5.3.19. Let \( \Omega \subseteq {\mathbb{C}}^{n} \) be a smoothly bounded domain on which automorphisms satisfy uniform bounds on derivatives as in (5.3.17.1). Let \( {\varphi }_{j} \in \operatorname{Aut}\left( \Omega \right) \) be a sequence of automorphisms that converges uniformly on compact sets to a limi... | Proof. This is a special case of the preceding result. | No |
Theorem 5.4.1. Let \( \Omega \) be a smoothly bounded, finite-type domain in \( {\mathbb{C}}^{2} \) which has compact automorphism group in the compact-open topology. Let \( k \) an integer be sufficiently large. Then there is an \( \epsilon > 0 \) so that if \( {\Omega }^{\prime } \) is a smoothly bounded, finite-type... | Proof. The proof of this result has been indicated earlier in this chapter (see also [GRK3, Theorem 0.1]), so we only sketch the steps.\n\nStep 1: There is a Riemannian metric, smooth on \( \bar{\Omega } \), which is invariant under any automorphism of \( \Omega \) . We construct this metric simply by averaging the Euc... | No |
Theorem 5.4.2. Let \( \Omega \) be a smoothly bounded, finite-type domain in \( {\mathbb{C}}^{2} \) . Equip Aut \( \left( \Omega \right) \) with the \( {C}^{k} \) topology, some integer \( k \geq 0 \) . Assume that \( \Omega \) has compact automorphism group in the \( {C}^{k} \) topology. Then there is an \( \epsilon >... | Proof. The proof is just the same as that for the last theorem. The main point is to have a uniform bound for derivatives of automorphisms (Proposition 5.2.5), so that the smooth-to-the-boundary invariant metric can be constructed. | No |
Example 5.5.1. Let\n\n\[ \Omega = B\\left( {0,2}\\right) \\smallsetminus \\bar{B}\\left( {0,1}\\right) .\n\]\n\nThen \( \\Omega \) is a bounded domain, but it is not pseudoconvex. | Of course any automorphism of \( \\Omega \) continues analytically to \( B\\left( {0,2}\\right) \) . But it also must preserve \( {S}_{1} \\equiv \\{ z : \\left| z\\right| = 1\\} \) and \( {S}_{2} \\equiv \\{ z : \\left| z\\right| = 2\\} \) . It follows that \( \\operatorname{Aut}\\left( \\Omega \\right) = \) \( U\\lef... | No |
If we do not mandate that the domain \( \Omega \) have smooth boundary, then Theorem 5.2.3 need not be true. As a simple example, consider \[ \Omega = \left\{ {z \in {\mathbb{C}}^{n} : 0 < \left| z\right| < 1}\right\} . | Of course this \( \Omega \) is not pseudoconvex and does not have a smooth defining function (so does not have smooth boundary by our reckoning). The automorphism group of \( \Omega \) is \( U\left( n\right) \) . A \ | Yes |
Theorem 5.9.1. For every integer \( k > 0 \) there is a Reinhardt domain in \( {\mathbb{C}}^{2} \) with \( k \) -dimensional Bergman space. | Proof. We work in \( {\mathbb{C}}^{2} \) (although there are analogues of these results in any dimension greater than one). Consider the domains\n\n\[ \n{X}_{1} = \left\{ {\left( {z, w}\right) \in {\mathbb{C}}^{2} : \left| w\right| < 1/\left( {\left| z\right| \log \left| z\right| }\right) ,\left| z\right| > e}\right\} ... | Yes |
Lemma 5.9.2. The monomials in \( {A}^{2}\left( \Omega \right) \) are precisely\n\n\[ c{z}^{k}{w}^{k},\;k = 0,1,2,\ldots \]\n\nHere \( c \) is some complex constant. | Proof: We calculate\n\n\[ {\int }_{{X}_{1}}{\left| z\right| }^{2p}{\left| w\right| }^{2p}\mathrm{\;d}V = {\left( 2\pi \right) }^{2}{\int }_{{r}_{1} = e}^{\infty }{\int }_{{r}_{2} = 0}^{1/{r}_{1}\log {r}_{1}}{r}_{1}^{{2p} + 1}{r}_{2}^{{2q} + 1}\mathrm{\;d}{r}_{1}\mathrm{\;d}{r}_{2} \]\n\n\[ = {\left( 2\pi \right) }^{2}{... | Yes |
Lemma 5.9.3. For \( k = 0,1,2,\ldots \) we have\n\n\[ \n{z}^{p}{w}^{q} \in {A}^{2}\left( {\Omega }_{k}\right) \Leftrightarrow p = q < k.\n\] | Proof. We calculate, using the change of variables \( {r}_{1} + {r}_{2} = t,{r}_{1} - {r}_{2} = s \), that\n\n\[ \n{\begin{Vmatrix}{z}^{p}{w}^{q}\end{Vmatrix}}_{{A}^{2}\left( {B}_{m}\right) }^{2} = {\int }_{{B}_{m}}\left| {{z}^{2p}{w}^{2p}}\right| \mathrm{d}V\n\]\n\n\[ \n= {\left( 2\pi \right) }^{2}{\int }_{\begin{matr... | Yes |
Theorem 5.9.4. Let \( \Omega \) be a domain in \( \mathbb{C} \). Then the dimension of \( {A}^{2}\left( \Omega \right) \) is either 0 or \( \infty \). | Proof: After a Möbius transformation, we may assume that \( \Omega \) contains the point \( \infty \). Let \( f \in {A}^{2}\left( \Omega \right) \) and suppose that \( f \) is not identically zero. There are now two cases:\n\nCase 1: The function \( f \) is rational. In this case we note that\n\n\[{\int }_{\mathbb{C}}{... | Yes |
Theorem 5.10.1. The form \( K\left( {z,\bar{z}}\right) \) is invariant under the group of holomorphic transformations of \( M \) . | Proof of the Theorem: Let \( \Phi \) be any one-to-one, onto, holomorphic transformation of \( M \) to itself. If \( {h}_{0},{h}_{1},{h}_{2},\ldots \) is a complete orthonormal basis for \( {A}^{2}\left( M\right) \), then so is \( {\Phi }^{ * }{h}_{0},{\Phi }^{ * }{h}_{1},{\Phi }^{ * }{h}_{2},\ldots \) a complete ortho... | No |
Theorem 5.10.3. We have that\n\n\[ K\left( {z,\bar{z}}\right) = \mathop{\max }\limits_{{\langle f, f\rangle = 1}}f\left( z\right) \land \bar{f}\left( z\right) . \]\n\nIf \( K\left( {z,\bar{z}}\right) \neq 0 \), then an \( n \) -form \( f \in {A}^{2}\left( M\right) \) satisfying the above identity is unique up to a cons... | Proof. Fix a point \( z \in M \) and let \( {A}^{2}{\left( M\right) }^{\prime } \) be the set of \( n \) -forms \( {f}^{\prime } \in {A}^{2}\left( M\right) \) which vanish at \( z \). If \( {A}^{2}{\left( M\right) }^{\prime } = {A}^{2}\left( M\right) \), then our result is trivial.\n\nSuppose instead that \( {A}^{2}{\l... | Yes |
Theorem 5.10.6. If \( {M}^{\prime } \) is a domain in \( M \) and if \( M \smallsetminus {M}^{\prime } \) is an analytic subvariety of \( M \) with complex dimension \( \leq n - 1 \), then\n\n\[ \n{K}_{{M}^{\prime }}\left( {z,\bar{z}}\right) = {K}_{M}\left( {z,\bar{z}}\right) \;\text{ for all }z\text{ in }{M}^{\prime }... | Proof. Let \( f \) be a square-integrable holomorphic \( n \) -form on \( {M}^{\prime } \) . We shall show that \( f \) can be continued analytically to \( M \) . Let \( {z}^{0} \) be any nonsingular point of the variety \( M \smallsetminus {M}^{\prime } \) . Let \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \) be local coordinat... | Yes |
Theorem 5.10.7. Let \( M \) and \( {M}^{\prime } \) be complex manifolds of complex dimensions \( n \) and \( {n}^{\prime } \), respectively. Then\n\n\[ \n{K}_{M \times {M}^{\prime }} = {\left( -1\right) }^{n{n}^{\prime }}{K}_{M} \land {K}_{{M}^{\prime }}.\n\] | Proof. In the displayed formula in the theorem, the projections from \( M \times {M}^{\prime } \) onto \( M \) and onto \( {M}^{\prime } \) are omitted. We can think of \( {K}_{M} \) and \( {K}_{{M}^{\prime }} \) as forms on \( M \times {M}^{\prime } \) in a natural manner.\n\nLet \( {z}^{0} \in M \) and \( {z}^{0}, \i... | Yes |
Theorem 5.10.8. The quadratic form \( \mathrm{d}{s}^{2} \) is positive semidefinite and invariant under the holomorphic automorphisms of \( M \) . | Proof. Let \( z \) be any point of \( M \) and let \( {z}_{1},\ldots ,{z}_{n} \) be local coordinates around \( z \) . Let\n\n\[ \n{h}_{j} = {h}_{j}^{ * }\mathrm{\;d}{z}_{1} \land \cdots \land \mathrm{d}{z}_{n},\;j = 0,1,2,\ldots ,\n\]\n\nbe an orthonormal basis for \( {A}^{2}\left( M\right) \) such that\n\n\[ \n{h}_{0... | Yes |
Theorem 5.10.11. Let \( M \) be a complex manifold satisfying \( {A1} \) and \( {A2} \) . Then the automorphism group Aut \( \left( M\right) \) is a (real) Lie group. Furthermore, the isotropy group of \( \operatorname{Aut}\left( M\right) \) at each point of \( M \) is compact. | Now let us specialize down to the case that \( M \) is a bounded domain in \( {\mathbb{C}}^{n} \) . As we usually do, call it \( \Omega \) . Fix a point \( P \in \Omega \) . Then the mapping\n\n\[ \operatorname{Aut}\left( \Omega \right) \rightarrow {\mathbb{C}}^{n} \times {\mathbb{C}}^{{n}^{2}} \]\n\n\[ \varphi \mapsto... | Yes |
Theorem 6.1.1. Any function holomorphic on a neighborhood of \( \bar{\Omega } \) actually continues analytically to \( {D}^{2}\left( {0,1}\right) \equiv D \times D \) . Thus \( \bar{\Omega } \) cannot have a neighborhood basis of pseudoconvex domains. Instead, it has a nontrivial Nebenhülle. | Proof: Let \( U \) be a neighborhood of \( \bar{\Omega } \) . For \( \left| {z}_{1}\right| < 1 \), the analytic discs\n\n\[ \zeta \mapsto \left( {{z}_{1},\zeta \cdot \left| {z}_{1}\right| }\right) \]\n\nhave boundary lying in \( U \) . But, for \( \left| {z}_{1}\right| \) sufficiently small, the entire disc lies in \( ... | Yes |
Theorem 6.1.5. Let \( {\Omega }_{j} \subset {\mathbb{C}}^{n} \) be smoothly bounded, Levi pseudoconvex domains. Suppose that one of the two domains satisfies Condition \( R \) . If \( \Phi : {\Omega }_{1} \rightarrow {\Omega }_{2} \) is a biholomorphic mapping, then \( \Phi \) extends to be a \( {C}^{\infty } \) diffeo... | This result established the centrality of Condition \( R \) . The techniques of proof are so natural and accessible that it seems that Condition \( R \) is certainly the \ | No |
Proposition 6.2.2. There exists no defining function \( \widetilde{\rho } \) for \( \mathcal{W} \) that is plurisubharmonic on the entire boundary. | Proof: Suppose that such a defining \( \widetilde{\rho } \) exist. Then, there exists a smooth positive function \( h \) such that \( \widetilde{\rho } = {h\rho } \) . A direct calculation shows that the complex Hessian for \( \widetilde{\rho } \) at a point \( z \in \mathcal{A} \) acting on \( v = \left( {{v}_{1},{v}_... | Yes |
Theorem 6.2.4. Let \( \\Omega \) be a smoothly bounded pseudoconvex domain, and let \( P \) denotes its Bergman projection. Let \( \\widetilde{\\rho } \) be a smooth defining function for \( \\Omega \) such that \( - {\\left( -\\widetilde{\\rho }\\right) }^{\\delta } \) is strictly plurisubharmonic. Then there exists \... | \[ P : {W}^{s}\\left( \\Omega \\right) \\rightarrow {W}^{s}\\left( \\Omega \\right) \] is continuous for all \( 0 \\leq s < {s}_{0} \) . | Yes |
Theorem 6.5.1. Let \( \Omega \subseteq {\mathbb{C}}^{n} \) be a smoothly bounded, pseudoconvex domain with \( n \geq 2 \) . Assume that there is a complex variety \( V \), of complex dimension at least 1, in \( \partial \Omega \) . Then \[ {K}_{\Omega }\left( {z, w}\right) \notin {C}^{\infty }\left( {\bar{\Omega } \tim... | Proof: Let \( p \in V \) be a regular point. Let \( {n}_{p} \) be the unit outward normal vector at \( p \) . Then there are small numbers \( \delta ,{\varepsilon }_{0} > 0 \) such that \( w - \varepsilon {n}_{p} \in \Omega \) for all \( w \in \) \( \partial \Omega \cap B\left( {p,\delta }\right) \) and all \( 0 < \var... | Yes |
Theorem 6.7.1. Let \( \Omega \subseteq {\mathbb{C}}^{n} \) be a bounded domain of holomorphy with defining function given by\n\n\[ \rho \left( z\right) = \left\{ \begin{array}{r} - {\operatorname{dist}}_{\partial \Omega }\left( z\right) \text{ if }z \in \Omega \\ {\operatorname{dist}}_{\partial \Omega }\left( z\right) ... | The proof will be broken up into a sequence of lemmas and will occupy most of the rest of this section. At the end we shall comment on the Fefferman's asymptotic expansion and how it trumps the work of Bergman, Hörmander, and Diederich. We note that, by the main theorem of [HOR1], the hypothesis of closedness of the \(... | No |
Lemma 6.7.2. If \( \Omega \) is as in the theorem and \( {\Omega }^{\prime } \subseteq \Omega \) is another domain, then\n\n\[ \left| {{K}_{{\Omega }^{\prime }}\left( {z, z}\right) }\right| \geq \left| {{K}_{\Omega }\left( {z, z}\right) }\right| . \] | Proof: This is obvious from the characterization\n\n\[ \mid {K}_{\Omega }\left( {z, z}\right) = \mathop{\sup }\limits_{\substack{{u \in {A}^{2}\Omega } \\ {\parallel u{\parallel }_{{A}^{2}} = 1} }}\frac{{\left| u\left( z\right) \right| }^{2}}{\parallel u{\parallel }_{{A}^{2}}^{2}}. \] | No |
Lemma 6.7.3. Let \( \Omega \) be a bounded, pseudoconvex domain. Let \( P \in \partial \Omega \) and suppose that, for some neighborhood \( U \) of \( P \), there is a holomorphic function \( {u}_{0} \) on \( {\Omega }^{\prime } \equiv \Omega \cap U \) such that \( \left| {u}_{0}\right| \leq 1 \) in \( {\Omega }^{\prim... | Proof of Lemma 6.7.3. Let \( \chi \) be a \( {C}^{\infty } \) function with compact support in \( U \) which is identically equal to 1 on \( {U}_{0} \) . Assume that \( 0 \leq \chi \leq 1 \) everywhere. If \( {u}^{\prime } \in {L}_{\left( 0,0\right) }^{2}\left( {\Omega }^{\prime }\right) \) and is holomorphic there, th... | Yes |
Lemma 6.7.6. Let \( {a}_{jk}\left( {j, k = 1,\ldots, n}\right) \) be a positive definite Hermitian symmetric matrix. Set\n\n\[ \n{F}_{0} = \left\{ {z \in {\mathbb{C}}^{n} : \operatorname{Im}{z}_{n} > \mathop{\sum }\limits_{{j, k = 1}}^{n}{a}_{jk}{z}_{j}{\bar{z}}_{k}}\right\} .\n\]\n\nThen\n\n\[ \n\left| {{K}_{{F}_{0}}\... | Proof: By a unitary transformation of the variables \( {z}_{1},{z}_{2},\ldots ,{z}_{n - 1} \), we may reduce the matrix \( {\left( {a}_{jk}\right) }_{j, k = 1}^{n - 1} \) to diagonal form; and the statement of the theorem remains invariant. Assuming this reduction to have been made, we can introduce new variables\n\n\[... | Yes |
Proposition 6.7.7. Let \( p \in \partial \Omega \) be a strictly pseudoconvex point. Then there is a neighborhood \( V \) of \( p \) with the following property: For any point \( Q \in V \) there is a biholomorphic mapping \( {\zeta }_{Q} \) sending \( V \) to a neighborhood of the origin, sending \( Q \) to the point ... | Proof: Let \( q \) be the point of \( \partial \Omega \) that is closest to \( Q \) . Then \( Q - q \) is normal to \( \partial \Omega \) . After a suitable translation and rotation of \( {\mathbb{C}}^{n} \), we may suppose that \( q = 0 \) and that\n\n---\n\nthe tangent plane to \( \partial \Omega \) at \( q \) is \( ... | Yes |
Example 6.7.11. Let \( \Omega \) be the connected component of\n\n\[ \left\{ {\left( {{z}_{1},{z}_{2}}\right) \in {\mathbb{C}}^{2} : {\left| {z}_{1}\right| }^{2} + {\left| {z}_{2}\right| }^{2} - c{\left| {z}_{2}\right| }^{8} < 1}\right\} \]\n\ncontaining the origin, where \( c \) is small and positive. Note that this \... | To do so, we apply the reproducing property of \( {K}_{\Omega } \) to the anti-holomorphic function \( F\left( w\right) = {K}_{B}\left( {{w}^{0}, w}\right) = {c}_{1}{\left( 1 - \tau {\bar{w}}_{1}\right) }^{-3} \) to obtain\n\n\[ {c}_{1}{\left( 1 - {\tau }^{2}\right) }^{-3} = F\left( {w}^{0}\right) \]\n\n\[ = {\int }_{\... | Yes |
Lemma 6.9.3 (Hopf). Let \( \Omega \subset \subset {\mathbb{R}}^{N} \) have \( {C}^{2} \) boundary. Let \( u \in C\left( \bar{\Omega }\right) \) with \( u \) harmonic and nonconstant on \( \Omega \) . Let \( P \in \bar{\Omega } \) and assume that \( u \) takes a local minimum at \( P \) . Then\n\n\[ \frac{\partial u}{\p... | Proof: Suppose without loss of generality that \( u > 0 \) on \( \Omega \) near \( P \) and that \( u\left( P\right) = \) 0 . Let \( {B}_{R} \) be a ball that is internally tangent to \( \bar{\Omega } \) at \( P \) . We may assume that the center of this ball is at the origin and that \( P \) has coordinates \( \left( ... | Yes |
Lemma 6.13.3. Let \( p \geq 2 + 1/k \) . Any function \( g \in {L}^{p}\left( \Omega \right) \cap \mathcal{O}\left( \Omega \right) \) which is independent of the variable \( w \) extends to a holomorphic function of \( z \) on the disc \( \left| z\right| < 4 \) . | Proof of Lemma 6.13.3. This is the trickiest of the three lemmas.\n\nFirst observe that if \( p = \infty \), then the result follows from the Riemann removable singularities theorem. So assume that \( p < \infty \), and let \( h \in {L}^{p}\left( \Omega \right) \cap \mathcal{O}\left( \Omega \right) \) be independent of... | Yes |
Theorem 6.13.4. Let \( p \geq 2 + 1/k \) . Then the space \( P\left( {{C}_{c}^{\infty }\left( \Omega \right) }\right) \) is not a subset of \( {L}^{p}\left( \Omega \right) \) . | Proof: Lemma 6.13.1 tells us that it suffices to show that \( {L}^{p}\left( \Omega \right) \cap \mathcal{O}\left( \Omega \right) \) is not dense in \( {L}^{2}\left( \Omega \right) \cap \mathcal{O}\left( \Omega \right) \) when \( p \geq 2 + 1/k \) . By Lemma 6.13.2, it thus suffices to show that the function \( 1/z \) c... | Yes |
Theorem 6.13.7. With \( X, Y, M \) as above, let \( L \) be any partial differential operator on \( M \) of the form \( L = - {x}^{2} - {Y}^{2} + a \), where \( a \in {C}^{\infty }\left( M\right) \) and\n\n\[ \parallel u{\parallel }^{2} \leq C\langle {Lu}, u\rangle \]\n\n(6.13.7.1)\n\nfor all \( u \in {C}^{2}\left( M\r... | We note that our hypotheses, particularly inequality (6.13.7.1), imply that \( L \) has a well-defined inverse \( {L}^{-1} \) which is a bounded linear operator on \( {L}^{2}\left( M\right) \) . | No |
Theorem 6.13.8. Let \( X, Y, M, L \) be as above. Then \( L \) has the following global properties:\n\n(a) There is a positive number \( {s}_{0} \) such that, for every \( 0 < s < {s}_{0},{L}^{-1} \) preserves \( {W}^{s}\left( M\right) \) .\n\n(b) For each \( s > {s}_{0},{L}^{-1} \) fails to map \( {C}^{\infty }\left( ... | The proof of Theorem 6.13.8.1 breaks into two parts. The first part consists of proving the a priori inequality (17). The second part, following ideas of Barrett in [BAR2], shows that, for any \( s \geq {s}_{0} \), the operator \( L \) cannot be exactly regular on \( {W}^{s}\left( M\right) \) . We refer the reader to [... | No |
Proposition 6.14.2. Let \( 1 < p < \infty \) . We have the estimate\n\n\[{\int }_{0}^{\infty }{\left| Hf\left( x\right) \right| }^{p}{dx} \leq {C}_{p}{\int }_{0}^{\infty }\left| {f\left( x\right) }\right| \mathrm{d}x.\] | Proof: Let \( \epsilon > 0 \) and consider the integral\n\n\[{\int }_{0}^{\infty }\frac{{\tau }^{-\epsilon }}{x + \tau }\mathrm{d}\tau\]\n\n(6.14.2.1)\n\n\nThe substitution \( \tau = {x\mu } \) gives that\n\n\[\text{(6.14.2.1)} = {x}^{-\epsilon } \cdot {\int }_{0}^{\infty }\frac{{\mu }^{-\epsilon }}{1 + \mu }\mathrm{d}... | Yes |
Proposition 6.14.3. Let \( L\left( {x, t}\right) \) be a nonnegative kernel for \( \left( {x, t}\right) \in {\mathbb{R}}^{N} \times {\mathbb{R}}_{ + } \) which satisfies:\n\n1. \( L\left( {{\lambda x},{\lambda t}}\right) = {\lambda }^{-\left( {N + 1}\right) } \cdot L\left( {x, t}\right) \; \) for all \( \lambda > 0 \),... | Proof: It is enough to prove (6.14.3.1) with \( K \) replaced by \( L\left( {x - y, t + u}\right) \) and we do so. By homogeneity and change of variables in the integral, \n\n\[ {\int }_{{\mathbb{R}}^{N}}L\left( {x, t + u}\right) \mathrm{d}x = {\left( t + u\right) }^{-\left( {N + 1}\right) }{\int }_{{\mathbb{R}}^{N}}L\... | Yes |
Proposition 7.2.5. Let \( \Omega \) be a bounded domain in \( {\mathbb{C}}^{n} \) with a \( {C}^{2} \) smooth, strictly pseudoconvex boundary point \( p \) . Let \( {B}^{n} \) be the unit open ball in \( {\mathbb{C}}^{n} \) . Let \( \eta \) be a positive real number satisfying \( 0 < \eta < 1 \) . Then, for every \( \e... | Proof. Assume to the contrary that there exist holomorphic mappings \( {\varphi }_{j} : {B}^{n} \rightarrow \Omega \) satisfying the following two conditions:\n\n(a) \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{\varphi }_{j}\left( 0\right) = p \) .\n\n(b) \( \exists \epsilon > 0 \) for which there exists a sequen... | Yes |
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