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Theorem 5.4. Let \( G \) be a compact Lie group and \( T \) a connected Lie subgroup of G. Then \( T \) is a maximal torus if and only if \( \mathfrak{t} \) is a Cartan subalgebra. In particular, maximal tori and Cartan subalgebras exist. | Proof. Theorems 4.14 and 5.1 show that \( T \) is a maximal torus of \( G \) if and only \( \mathfrak{t} \) is a Cartan subalgebra of \( \mathfrak{g} \) . Since maximal Abelian subalgebras clearly exist, this also shows that maximal tori exist. | Yes |
Lemma 5.6. Let \( G \) be a compact Lie group and \( \left( {\pi, V}\right) \) a finite-dimensional representation of \( G \) .\n\n(a) There exists a \( G \) -invariant inner product, \( \left( {\cdot , \cdot }\right) \), on \( V \) and for any such \( G \) -invariant inner product on \( V,{d\pi X} \) is skew-Hermitian... | Proof. Part (b) is simply a special case of part (a), where \( \pi \) is the Adjoint representation on \( \mathfrak{g} \) . To prove part (a), recall that Theorem 2.15 provides the existence of a \( G \) - invariant inner product on \( V \) . If \( \left( {\cdot , \cdot }\right) \) is a \( G \) -invariant inner product... | Yes |
Lemma 5.7. Let \( G \) be a compact Lie group and \( \mathfrak{t} \) a Cartan subalgebra of \( \mathfrak{g} \). There exists \( X \in \mathfrak{t} \), so that \( \mathfrak{t} = {\mathfrak{z}}_{\mathfrak{g}}\left( X\right) \) where \( {\mathfrak{z}}_{\mathfrak{g}}\left( X\right) = \{ Y \in \mathfrak{g} \mid \left\lbrack... | Proof. By choosing a basis for \( \mathfrak{t} \) and using the fact that \( \mathfrak{t} \) is maximal Abelian in \( \mathfrak{g} \), there exist independent \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n},{X}_{i} \in \mathfrak{t} \), so that \( \mathfrak{t} = { \cap }_{i}\ker \left( {\operatorname{ad}{X}_{i}}\right) \). Be... | Yes |
Theorem 5.9. Let \( G \) be a compact Lie group and \( \mathfrak{t} \) a Cartan subalgebra. For \( X \in \mathfrak{g} \) , there exists \( g \in G \) so that \( \operatorname{Ad}\left( g\right) X \in \mathfrak{t} \) . | Proof. Let \( \left( {\cdot , \cdot }\right) \) be an Ad-invariant inner product on \( \mathfrak{g} \) . Using Lemma 5.7, write \( \mathfrak{t} = \) \( {\mathfrak{z}}_{\mathfrak{g}}\left( Y\right) \) for some \( Y \in \mathfrak{g} \) . It is necessary to find \( {g}_{0} \in G \) so that \( \left\lbrack {\operatorname{A... | Yes |
Corollary 5.10. (a) Let \( G \) be a compact Lie group with Lie algebra \( \mathfrak{g} \). Then \( \operatorname{Ad}\left( G\right) \) acts transitively on the set of Cartan subalgebras of \( G \). | Proof. For part (a), let \( {\mathfrak{t}}_{i} = {\mathfrak{z}}_{\mathfrak{g}}\left( {X}_{i}\right) ,{X}_{i} \in \mathfrak{g} \), be Cartan subalgebras. Using Theorem 5.9, there is a \( g \in G \) so that \( \operatorname{Ad}\left( g\right) {X}_{1} \in {\mathfrak{t}}_{2} \). Using the fact that \( \operatorname{Ad}\lef... | Yes |
Lemma 5.11. Let \( G \) be a compact connected Lie group. The kernel of the Adjoint map is the center of \( G \), i.e., \( \operatorname{Ad}\left( g\right) = I \) if and only if \( g \in Z\left( G\right) \), where \( Z\left( G\right) = \) \( \{ h \in G \mid {gh} = {hg}\} \) . | Proof. If \( g \in Z\left( G\right) \), then \( {c}_{g} \) is the identity, so that its differential, \( \operatorname{Ad}\left( g\right) \), is trivial as well. On the other hand, if \( \operatorname{Ad}\left( g\right) = I \), then \( {c}_{g}{e}^{X} = {e}^{\operatorname{Ad}\left( g\right) X} = {e}^{X} \) for \( X \in ... | Yes |
Corollary 5.13. Let \( G \) be a compact connected Lie group with maximal torus \( T \) . (a) Then \( {Z}_{G}\left( T\right) = T \), where \( {Z}_{G}\left( T\right) = \{ g \in G \mid {gt} = {tg} \) for \( t \in T\} \) . In particular, \( T \) is maximal Abelian. | Proof. Part (b) clearly follows from part (a). For part (a), obviously \( T \subseteq {Z}_{G}\left( T\right) \) . Conversely, let \( {g}_{0} \in {Z}_{G}\left( T\right) \) and consider the closed, therefore compact, connected Lie subgroup \( {Z}_{G}{\left( {g}_{0}\right) }^{0} \) . Using the Maximal Torus Theorem, write... | Yes |
Corollary 5.16. Let \( N \) be a connected Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) whose Lie algebra \( \mathfrak{n} \) lies in the set of strictly upper triangular matrices, i.e., if \( X \in \mathfrak{n} \), then \( {X}_{i, j} = 0 \) when \( i \geq j \) . Then the map \( \exp : \mathfrak{n} \rightarrow... | Proof. It is a simple exercise to see that \( \left\lbrack {{X}_{n},\ldots ,{X}_{3},{X}_{2},{X}_{1}}\right\rbrack = 0 \) for any strictly upper triangular \( X,{X}_{i} \in \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \) and that \( {e}^{X} \) is polynomial in \( X \) (Exercise 5.18). In particular, for \( X, Y \in \math... | No |
Theorem 5.18. Let \( G \) be a compact Lie group with Lie algebra \( \mathfrak{g} \) . Then \( \mathfrak{g} \) is reductive. If \( \mathfrak{z}\left( \mathfrak{g}\right) \) is the center of \( \mathfrak{g} \), i.e., \( \mathfrak{z}\left( \mathfrak{g}\right) = \{ X \in \mathfrak{g} \mid \left\lbrack {X,\mathfrak{g}}\rig... | Proof. Using Lemma 5.6, let \( \left( {\cdot , \cdot }\right) \) be an Ad-invariant inner product on \( \mathfrak{g} \), so that ad \( X, X \in \mathfrak{g} \), is skew-Hermitian. If \( \mathfrak{a} \) is an ideal of \( \mathfrak{g} \), then \( {\mathfrak{a}}^{ \bot } \) is also an ideal. It follows that \( \mathfrak{g... | Yes |
Theorem 5.22. (a) Let \( G \) be a compact connected Lie group. Then \( G = {G}^{\prime }Z{\left( G\right) }^{0} \) , \( Z\left( {G}^{\prime }\right) = {G}^{\prime } \cap Z\left( G\right) \) is a finite Abelian group, \( Z{\left( G\right) }^{0} \) is a torus, and | Proof. For part (a), first note that \( {G}^{\prime } \) is closed and therefore compact. It follows from the Maximal Torus Theorem that \( {G}^{\prime } = \exp {\mathfrak{g}}^{\prime } \) . Using the decomposition \( \mathfrak{g} = {\mathfrak{g}}^{\prime } \oplus \mathfrak{z}\left( \mathfrak{g}\right) \) and the fact ... | Yes |
Theorem 6.2. (a) Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) and \( \left( {\pi, V}\right) \) a finite-dimensional representation of \( G \) . Then \( \left( {{d\pi }, V}\right) \) is a representation of \( \mathfrak{g} \) satisfying \( {e}^{d\pi X} = \) \( \pi \left( {e}^{X}\right) \), wher... | Proof. Part (a) follows immediately from Theorem 4.8 by looking at the homomorphism \( \pi : G \rightarrow {GL}\left( V\right) \) and choosing a basis for \( V \) . | No |
Lemma 6.6. Let \( \mathfrak{g} \) be the Lie algebra of a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) and let \( \left( {\psi, V}\right) \) be a representation of \( \mathfrak{g} \) . Then \( V \) is irreducible under \( \mathfrak{g} \) if and only if it is irreducible under \( {\mathfrak{g}}_{\mathbb{C}} \)... | Proof. Simply observe that since a subspace \( W \subseteq V \) is a complex subspace, \( W \) is \( \psi \left( \mathfrak{g}\right) \) -invariant if and only if it is \( \psi \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) -invariant. | Yes |
Theorem 6.9. (a) Let \( G \) be a compact Lie group, \( \left( {\pi, V}\right) \) a finite-dimensional representation of \( G, T \) a maximal torus of \( G \), and \( V = {\bigoplus }_{\alpha \in \Delta \left( {V,{\mathrm{t}}_{\mathbb{C}}}\right) }{V}_{\alpha } \) the weight space decomposition. For each weight \( \alp... | Proof. Part (a) follows from the facts that \( {d\pi } \) is skew-Hermitian on \( \mathfrak{t} \) and is Hermitian on \( i\mathrm{t} \) . | Yes |
Theorem 6.11. (a) Let \( G \) be a compact Lie group, \( \left( {\pi, V}\right) \) a finite-dimensional representation of \( G \), and \( \mathfrak{t} \) a Cartan subalgebra of \( \mathfrak{g} \). For \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) and \( \beta \in \Delta \left( V\right) \), \( {d\pi ... | Proof. For part (a), let \( H \in {\mathfrak{t}}_{\mathbb{C}},{X}_{\alpha } \in {\mathfrak{g}}_{\alpha } \), and \( {v}_{\beta } \in {V}_{\beta } \) and calculate\n\n\[ \n{d\pi }\left( H\right) {d\pi }\left( {X}_{\alpha }\right) {v}_{\beta } = \left( {{d\pi }\left( {X}_{\alpha }\right) {d\pi }\left( H\right) + \left\lb... | Yes |
Lemma 6.14. Let \( G \) be a compact Lie group and \( \mathfrak{t} \) be a Cartan subalgebra of \( \mathfrak{g} \). (a) If \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \), then \( - \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) and \( {\mathfrak{g}}_{-\alpha } = \theta {\mathfrak{g}}_... | Proof. Let \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \cup \{ 0\} \). Recalling that \( \theta \) is an involution, it suffices to show \( \theta {\mathfrak{g}}_{\alpha } \subseteq {\mathfrak{g}}_{-\alpha } \). Write \( Z \in {\mathfrak{g}}_{\alpha } \) uniquely as \( Z = X + {iY} \) for \( X, Y \in... | Yes |
Theorem 6.16. Let \( \mathfrak{g} \) be the Lie algebra of a compact Lie group \( G \) .\n\n(a) For \( X, Y \in \mathfrak{g}, B\left( {X, Y}\right) = \operatorname{tr}\left( {\operatorname{ad}X \circ \operatorname{ad}Y}\right) \) on \( \mathfrak{g} \) . | Proof. Part (a) is elementary. | No |
Lemma 6.19. Let \( G \) be a compact Lie group, \( \mathfrak{t} \) a Cartan subalgebra of \( \mathfrak{g} \), and \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) . (a) Then \( \alpha \left( {h}_{\alpha }\right) = 2 \) . | Proof. For part (a) simply use the definitions \[ \alpha \left( {h}_{\alpha }\right) = \frac{{2\alpha }\left( {u}_{\alpha }\right) }{B\left( {{u}_{\alpha },{u}_{\alpha }}\right) } = \frac{{2B}\left( {{u}_{\alpha },{u}_{\alpha }}\right) }{B\left( {{u}_{\alpha },{u}_{\alpha }}\right) } = 2. \] | Yes |
Theorem 6.20. Let \( G \) be a compact Lie group, \( \mathfrak{t} \) a Cartan subalgebra of \( \mathfrak{g} \), and \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) . Fix a nonzero \( {E}_{\alpha } \in {\mathfrak{g}}_{\alpha } \) and let \( {F}_{\alpha } = - \theta {E}_{\alpha } \) . Using Lemma 6.19, ... | Proof. For part (a), Lemma 6.19 and the definitions show that \( \left\lbrack {{H}_{\alpha },{E}_{\alpha }}\right\rbrack = 2{E}_{\alpha } \) , \( \left\lbrack {{H}_{\alpha },{F}_{\alpha }}\right\rbrack = - 2{F}_{\alpha } \), and \( \left\lbrack {{E}_{\alpha },{F}_{\alpha }}\right\rbrack = {H}_{\alpha } \) . Since these... | Yes |
Lemma 6.25. Let \( G \) be a compact connected Lie group with Cartan subalgebra \( \mathfrak{t} \) . For \( H \in \mathfrak{t},\exp H \in Z\left( G\right) \) if and only if \( \alpha \left( H\right) \in {2\pi i}\mathbb{Z} \) for all \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) . | Proof. Let \( g = \exp H \) and recall from Lemma 5.11 that \( g \in Z\left( G\right) \) if and only if \( \operatorname{Ad}\left( g\right) X = X \) for all \( X \in \mathfrak{g} \) . Now for \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \cup \{ 0\} \) and \( X \in {\mathfrak{g}}_{\alpha },\operatornam... | No |
Theorem 6.27. Let \( G \) be a compact Lie group with a maximal torus \( T \). (a) \( R \subseteq A \subseteq P \). (b) Given \( \lambda \in {\left( i\mathfrak{t}\right) }^{ * },\lambda \in A \) if and only if there exists \( {\xi }_{\lambda } \in \chi \left( T\right) \) satisfying \[ {\xi }_{\lambda }\left( {\exp H}\r... | Proof. Let \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) and suppose \( H \in \mathfrak{t} \) with \( \exp H = e \) . Lemma 6.25 shows that \( \alpha \left( H\right) \in {2\pi i}\mathbb{Z} \), so that \( R \subseteq A \) . Next choose a standard \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) -tri... | Yes |
Theorem 6.29. Let \( G \) be a compact Lie group with a semisimple Lie algebra \( \mathfrak{g} \) and let \( T \) be a maximal torus of \( G \) with corresponding Cartan subalgebra \( \mathfrak{t} \). (a) \( {R}^{ * } = {P}^{ \vee } \). (b) \( {P}^{ * } = {R}^{ \vee } \). (c) \( {A}^{ * } = \ker \mathcal{E} \). (d) \( ... | Proof. The equalities \( {R}^{ * } = {P}^{ \vee },{\left( {R}^{ \vee }\right) }^{ * } = P \), and \( {\left( \ker \mathcal{E}\right) }^{ * } = A \) follow immediately from the definitions. This proves parts (a), (b), and (c) (Exercise 6.24). Part (d) follows from Theorem 6.27 (Exercise 6.24). | No |
Theorem 6.30. Let \( G \) be a connected compact Lie group with a semisimple Lie algebra and maximal torus \( T \). (a) \( Z\left( G\right) \cong {P}^{ \vee }/\ker \mathcal{E} \cong A/R \) . | Proof (part (a) only). By Theorem 5.1, Corollary 5.13, and Lemma 6.25, the exponential map induces an isomorphism \[ Z\left( G\right) \cong \{ H \in \mathfrak{t} \mid \alpha \left( H\right) \in {2\pi i}\mathbb{Z}\text{ for }\alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \} /\{ H \in \mathfrak{t} \mid \exp ... | No |
Lemma 6.32. Let \( G \) be a compact connected Lie group, \( T \) a maximal torus of \( G \) , \( \widetilde{G} \) the simply connected covering of of \( G,\pi : \widetilde{G} \rightarrow G \) the associated covering homomorphism, and \( \widetilde{T} = {\left\lbrack {\pi }^{-1}\left( T\right) \right\rbrack }^{0} \) .\... | Proof. The proof of this lemma is a straightforward generalization of the proof of the Maximal Torus theorem, Theorem 5.12 (Exercise 6.26). | No |
Corollary 6.33. Let \( G \) be a compact connected Lie group with semisimple Lie algebra \( \mathfrak{g}, T \) a maximal torus of \( G,\widetilde{G} \) the simply connected covering of of \( G,\pi : \widetilde{G} \rightarrow G \) the associated covering homomorphism, and \( \widetilde{T} = {\left\lbrack {\pi }^{-1}\lef... | Proof. For part (a), observe that \( \widetilde{G} = \mathop{\bigcup }\limits_{{\widetilde{g} \in \widetilde{G}}}\left( {{c}_{\widetilde{g}}\widetilde{T}}\right) \) by Lemma 6.32. Thus \( \widetilde{G} \) is the continuous image of the compact set \( \widetilde{G}/Z\left( \widetilde{G}\right) \times \widetilde{T} \cong... | Yes |
Theorem 6.36. Let \( G \) be a compact connected Lie group with a maximal torus \( T \). (a) The action of \( W \) on it and on \( {\left( i\mathfrak{t}\right) }^{ * } \) is faithful, i.e., a Weyl group element acts trivially if and only it is the identity element. | Proof. For part (a), suppose \( w \in N \) acts trivially on \( \mathfrak{t} \) via Ad. Since \( \exp \mathfrak{t} = T \) and since \( {c}_{w} \circ \exp = \exp \circ \operatorname{Ad}\left( w\right) \), this implies that \( w \in {Z}_{G}\left( T\right) \) . However, Corollary 5.13 shows that \( {Z}_{G}\left( T\right) ... | Yes |
Lemma 6.42. Let \( G \) be compact Lie group with a Cartan subalgebra \( \mathfrak{t} \). (a) There is a one-to-one correspondence between \[ \{ \text{ systems of simple roots }\} \leftrightarrow \left\{ {\text{ Weyl chambers of }{\left( i\mathfrak{t}\right) }^{ * }}\right\} . | Proof. Suppose \( \Pi = \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{l}}\right\} \) is a simple system. From Equation 6.39, recall that \( \rho \in {\left( i\mathfrak{t}\right) }^{ * } \) satisfies \( B\left( {\rho ,{\alpha }_{j}}\right) = \frac{\parallel \alpha {\parallel }^{2}}{2} > 0 \), so that \( \rho \in C\left( \Pi... | Yes |
Theorem 7.3. Let \( G \) be a connected compact Lie group and \( V \) an irreducible representation of \( G \) .\n\n(a) \( V \) has a unique highest weight, \( {\lambda }_{0} \) .\n\n(b) The highest weight \( {\lambda }_{0} \) is dominant and analytically integral, i.e., \( {\lambda }_{0} \in A\left( T\right) \) .\n\n(... | Proof. Existence of a highest weight \( {\lambda }_{0} \) follows from the finite dimensionality of \( V \) and Theorem 6.11. Let \( {v}_{0} \) be a highest weight vector for \( {\lambda }_{0} \) and inductively define \( {V}_{n} = {V}_{n - 1} + {\mathfrak{n}}^{ - }{V}_{n - 1} \) where \( {V}_{0} = \mathbb{C}{v}_{0} \)... | Yes |
Lemma 7.5. Let \( G \) be connected. If \( V\left( \lambda \right) \) is an irreducible representation of \( G \), then \( V{\left( \lambda \right) }^{ * } \cong V\left( {-{w}_{0}\lambda }\right) \), where \( {w}_{0} \in W\left( {\Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) }\right) \) is the unique element mapping... | Proof. Since \( V\left( \lambda \right) \) is irreducible, the character theory of Theorems 3.5 and 3.7 show that \( V{\left( \lambda \right) }^{ * } \) is irreducible. It therefore suffices to show that the highest weight of \( V{\left( \lambda \right) }^{ * } \) is \( - {w}_{0}\lambda \) .\n\nFix a \( G \) -invariant... | Yes |
Theorem 7.7. Let \( G \) be a compact connected Lie group.\n\n(a) \( {\mathfrak{g}}^{\text{reg }} \) is open dense in \( \mathfrak{g} \), | Proof. Let \( l \) be the dimension of a Cartan subalgebra and \( n = \dim \mathfrak{g} \) . Any element \( X \in \mathfrak{g} \) lies in at least one Cartan subalgebra, so that \( \dim \left( {\ker \left( {\operatorname{ad}\left( X\right) }\right) }\right) \geq l \) . Thus\n\n\[ \det \left( {\operatorname{ad}\left( X\... | Yes |
Lemma 7.9. Let \( G \) be a compact connected Lie group and \( T \) a maximal torus. Then \( \mathfrak{g} = \mathfrak{t} \oplus \left( {\mathfrak{g} \cap {\bigoplus }_{\alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) }{\mathfrak{g}}_{\alpha }}\right) \) and there exists an open neighborhood \( {U}_{\mathfrak... | Proof. The decomposition \( \mathfrak{g} = \mathfrak{t} \oplus \left( {\mathfrak{g} \cap {\bigoplus }_{\alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) }{\mathfrak{g}}_{\alpha }}\right) \) follows from Theorem 6.20. In fact, \( \left( {\mathfrak{g} \cap {\bigoplus }_{\alpha \in \Delta \left( {\mathfrak{g}}_{... | Yes |
Lemma 7.10. Let \( G \) be a compact connected Lie group and \( T \) a maximal torus. Choose \( {U}_{G} \subseteq G \) as in Lemma 7.9. For \( g \in G \) and \( t \in T \), let \( \phi : {U}_{G}T \rightarrow G \) be given by\n\n\[ \phi = {l}_{g{t}^{-1}{g}^{-1}} \circ \psi \circ \left( {{l}_{gT} \times {l}_{t}}\right) \... | Proof. Calculate\n\n\[ {d\phi }\left( H\right) = {\left. \frac{d}{ds}\phi \left( {e}^{sH}\right) \right| }_{s = 0} = {\left. \frac{d}{ds}g{e}^{sH}{g}^{-1}\right| }_{s = 0} = \operatorname{Ad}\left( g\right) H \]\n\n\[ {d\phi }\left( X\right) = {\left. \frac{d}{ds}\phi \left( {e}^{sX}\right) \right| }_{s = 0} = {\left. ... | Yes |
Theorem 7.11. Let \( G \) be a compact connected Lie group and \( T \) a maximal torus. The map\n\n\[ \psi : G/T \times {T}^{\text{reg }} \rightarrow {G}^{\text{reg }}\text{given by} \]\n\n\[ \psi \left( {{gT}, t}\right) = {gt}{g}^{-1} \] \nis a surjective, \( \left| {W\left( G\right) }\right| \) -to-one local diffeomo... | Proof. For \( g \in G \) and \( t \in {T}^{\mathrm{{reg}}} \), Lemma 7.10 and Theorem 7.7 show that \( \psi \) is a surjective local diffeomorphism at \( \left( {{gT}, t}\right) \) . Moreover if \( w \in N\left( T\right) \), then\n\n(7.12)\n\n\[ \psi \left( {g{w}^{-1}T,{wt}{w}^{-1}}\right) = \psi \left( {{gT}, t}\right... | Yes |
Possibly replacing \( {\omega }_{T} \) by \( - {\omega }_{T} \) (which does not change integration), there exists a \( G \) -invariant form \( \widetilde{{\omega }_{T}} \in \mathop{\bigwedge }\limits_{l}^{ * }\left( G\right) \), so that\n\n\[ \n{\omega }_{T} = {\iota }^{ * }\widetilde{{\omega }_{T}} \n\]\n\nand\n\n\[ \... | Proof. Clearly the restriction map \( {\left. {\iota }^{ * }\right| }_{e} : {\mathfrak{g}}^{ * } \rightarrow {\mathfrak{t}}^{ * } \) is surjective. Choose any \( {\left( \widetilde{{\omega }_{T}}\right) }_{e} \in \) \( \mathop{\bigwedge }\limits_{l}^{ * }{\left( G\right) }_{e} \), so \( {\iota }^{ * }{\left( \widetilde... | Yes |
Lemma 7.19. Let \( G \) be a compact connected Lie group with maximal torus \( T \). (a) If \( f : \mathrm{t} \rightarrow \mathbb{C} \) descends to \( T \) and is \( W \) -invariant, then \( f : T \rightarrow \mathbb{C} \) is \( W \) -invariant. (b) Restriction of domain establishes a bijection between the continuous c... | Proof. For part (a), recall that the identification of \( W\left( G\right) \) with \( W\left( {\Delta {\left( {\mathfrak{g}}_{\mathbb{C}}\right) }^{ \vee }}\right) \) from Theorem 6.43 via the Ad-action of Equation 6.35. It follows that when \( f \) descends to \( T \) and is \( W \) -invariant, then \( f\left( {{c}_{w... | Yes |
Lemma 7.20. Let \( G \) be a compact Lie group with a maximal torus \( T \).\n\n(a) \( \rho = \frac{1}{2}\mathop{\sum }\limits_{{\alpha \in {\Delta }^{ + }\left( {\mathfrak{g}}_{\mathbb{C}}\right) }}\alpha \) .\n\n(b) For \( w \in W\left( {\Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) }\right) ,{w\rho } - \rho \in R... | Proof. For part (a), write \( \Pi \left( {\mathfrak{g}}_{\mathbb{C}}\right) = \left\{ {{\alpha }_{1},\ldots {\alpha }_{l}}\right\} \) and let \( {\rho }^{\prime } = \frac{1}{2}\mathop{\sum }\limits_{{\alpha \in {\Delta }^{ + }\left( {\mathfrak{g}}_{\mathbb{C}}\right) }}\alpha \) (c.f. Exercise 6.34). By the definitions... | Yes |
Lemma 7.22. Let \( G \) be a compact Lie group with a maximal torus \( T \). (a) The function \( \Delta \) is skew-symmetric on \( \mathfrak{t} \). (b) The function \( \Delta \) descends to \( T \) if and only if the function \( H \rightarrow {e}^{-\rho \left( H\right) } \) descends to \( T \). (c) The function \( {\le... | Proof. For part (a), it suffices to show that \( \Delta \circ {r}_{{h}_{\alpha }} = - \Delta \) for \( \alpha \in {\Delta }^{ + }\left( {\mathfrak{g}}_{\mathbb{C}}\right) \). This follows from three observations. The first is that composition with \( {r}_{{h}_{\alpha }} \) maps \( \left( {{e}^{\alpha /2} - {e}^{-\alpha... | Yes |
Lemma 7.26. Let \( G \) be a compact connected Lie group with a maximal torus \( T \) . Fix an analytically integral weight \( \lambda \in A\left( T\right) \) . The function \( {\Theta }_{\lambda } \) descends to a smooth \( W \) -invariant function on \( {T}^{\mathrm{{reg}}} \) . In turn, this function, still denoted ... | Proof. The first expression for \( {\Theta }_{\lambda } \) shows that it is symmetric since the numerator and denominator are skew-symmetric. The second expression for \( {\Theta }_{\lambda } \) shows it descends to a function on \( {T}^{\text{reg }} \) since the numerator and denominator both descend to \( T \) and th... | No |
Theorem 7.30 (Weyl Denominator Formula). Let \( G \) be a compact connected Lie group with a maximal torus \( T \) . Then\n\n\[ \n\Delta \left( H\right) = \mathop{\sum }\limits_{{w \in W\left( {\Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) }\right) }}\det \left( w\right) {e}^{\left( {w\rho }\right) \left( H\right) }... | Proof. Simply take the trivial representation \( V\left( 0\right) = \mathbb{C} \) with \( {\chi }_{0}\left( g\right) = 1 \) and apply the Weyl Character Formula to \( g = {e}^{H} \) for \( H \in \Xi \) . The formula extends to all \( \mathfrak{t} \) by continuity. | No |
Lemma 7.35. Let \( G \) be a compact connected Lie group with maximal torus \( T \) . Let \( {G}^{\text{sing }} = G \smallsetminus {G}^{\text{reg }} \) . Then \( {G}^{\text{sing }} \) is a closed subset with \( \operatorname{codim}{G}^{\text{sing }} \geq 3 \) in \( G \) . | Proof. It follows from Theorem 7.7 that \( {G}^{\text{sing }} \) is closed and the map \( \psi : G/T \times \) \( {T}^{\text{sing }} \rightarrow {G}^{\text{sing }} \) is surjective. Moreover \( t \in {T}^{\text{sing }} \) if and only if there exists \( \alpha \in \) \( {\Delta }^{ + }\left( {\mathfrak{g}}_{\mathbb{C}}\... | No |
Lemma 7.37. Let \( G \) be a compact connected Lie group with maximal torus \( T \) and fix a base \( {t}_{0} = {e}^{{H}_{0}} \in {T}^{\text{reg }} \) with \( {H}_{0} \in \mathfrak{t} \) . (a) Any continuous loop \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow {G}^{\text{reg }} \) with \( \gamma \left( 0\right)... | Proof. Using the Maximal Torus Theorem, write \( \gamma \left( s\right) = {c}_{{g}_{s}}\tau \left( s\right) \) with \( \tau \left( s\right) \in {T}^{\text{reg }} \) , \( \tau \left( 0\right) = {t}_{0} \), and \( {g}_{0} = e \) . In fact, since \( \psi : G/T \times {T}^{\mathrm{{reg}}} \rightarrow {G}^{\mathrm{{reg}}} \... | Yes |
Lemma 7.38. Let \( G \) be a compact connected Lie group with maximal torus \( T \) . (a) Each homotopy class in \( G \) with base \( e \) can be represented by a loop of the form \[ \gamma \left( s\right) = {e}^{s{X}_{\gamma }} \] for some \( {X}_{\gamma } \in {2\pi i}\ker \mathcal{E} \), i.e., for some \( {X}_{\gamma... | Proof. Lemma 7.37 shows that each homotopy class in \( G \) with base \( {t}_{0} \) can be represented by a curve of the form \( \gamma \left( s\right) = {c}_{{g}_{s}}{e}^{H\left( s\right) } \) with \( H\left( 1\right) = \operatorname{Ad}{\left( {g}_{1}\right) }^{-1}{H}_{0} + {X}_{\gamma } \) for some \( {X}_{\gamma } ... | No |
Lemma 7.40. Let \( G \) be a compact connected Lie group with maximal torus \( T \) . (a) The affine Weyl group is generated by the reflections across the hyperplanes \( {\alpha }^{-1}\left( {2\pi in}\right) \) for \( \alpha \in \Delta \left( {\mathfrak{g}}_{\mathbb{C}}\right) \) and \( n \in \mathbb{Z} \) . (b) The af... | Proof. Recall that \( {h}_{\alpha } \in {R}^{ \vee } \) and notice the reflection across the hyperplane \( {\alpha }^{-1}\left( {2\pi in}\right) \) is given by \( {r}_{{h}_{\alpha }, n}\left( H\right) = {r}_{{h}_{\alpha }}H + {2\pi i}{h}_{\alpha } \) (Exercise 7.25). Since the Weyl group is generated by the reflections... | No |
Theorem 7.41. Let \( G \) be a connected compact Lie group with semisimple Lie algebra and maximal torus \( T \) . Then \( {\pi }_{1}\left( G\right) \cong \ker \mathcal{E}/{R}^{ \vee } \cong P/A\left( T\right) \) . | Proof. By Lemma 7.38, it suffices to show that the loop \( \gamma \left( s\right) = {e}^{s{X}_{\gamma }},{X}_{\gamma } \in \) \( {2\pi i}\ker \mathcal{E} \), is trivial if and only if \( {X}_{\gamma } \in {2\pi i}{R}^{ \vee } \) . For this, first consider the standard \( \mathfrak{{su}}\left( 2\right) \) -triple corres... | Yes |
Theorem 7.44. Let \( G \) be a Lie group and \( H \) a closed subgroup of \( G \) . There is a bijection between equivalence classes of homogenous vector bundles \( \mathcal{V} \) on \( G/H \) and representations of \( H \) . | Proof. The correspondence maps \( \mathcal{V} \) to \( {\mathcal{V}}_{eH} \) . By definition \( {\mathcal{V}}_{eH} \) is a representation of \( H \) . Conversely, given a representation \( V \) of \( H \), the vector bundle \( G{ \times }_{H}V \) inverts the correspondence. | Yes |
Theorem 7.46. Let \( G \) be a Lie group, \( H \) a closed subgroup of \( G \), and \( V \) a representation of \( H \) . There is a linear \( G \) -intertwining bijection between \( \Gamma \left( {G/H, G{ \times }_{H}V}\right) \) and \( {\operatorname{Ind}}_{H}^{G}\left( V\right) \) . | Proof. Identify \( {\left( G{ \times }_{H}V\right) }_{eH} \) with \( V \) by mapping \( \left( {h, v}\right) \in {\left( G{ \times }_{H}V\right) }_{eH} \) to \( {h}^{-1}v \in \) \( V \) . Given \( s \in \Gamma \left( {G/H, G{ \times }_{H}V}\right) \), let \( {f}_{s} \in {\operatorname{Ind}}_{H}^{G}\left( V\right) \) be... | No |
Theorem 7.47 (Frobenius Reciprocity). Let \( G \) be a Lie group and \( H \) a closed subgroup of \( G \). If \( V \) is a representation of \( H \) and a \( W \) is a representation of \( G \), then as vector spaces\n\n\[{\operatorname{Hom}}_{G}\left( {W,{\operatorname{Ind}}_{H}^{G}\left( V\right) }\right) \cong {\ope... | Proof. Map \( T \in {\operatorname{Hom}}_{G}\left( {W,{\operatorname{Ind}}_{H}^{G}\left( V\right) }\right) \) to \( {S}_{T} \in {\operatorname{Hom}}_{H}\left( {{\left. W\right| }_{H}, V}\right) \) by \( {S}_{T}\left( w\right) = \) \( \left( {T\left( w\right) }\right) \left( e\right) \) for \( w \in W \) and map \( S \i... | No |
Lemma 7.49. Let \( G \) be a compact connected Lie group with maximal torus \( T \). (a) The map \( \exp : {\mathfrak{n}}^{ + } \rightarrow N \) is a bijection. | Proof. Since \( T \) consists of commuting unitary matrices, we may assume \( T \) is contained in the set of diagonal matrices of \( {GL}\left( {n,\mathbb{C}}\right) \). By using the Weyl group of \( {GL}\left( {n,\mathbb{C}}\right) \), we may further assume \( {u}_{\rho } = \operatorname{diag}\left( {{c}_{1},\ldots ,... | Yes |
Theorem 7.50. Let \( G \) be a compact connected Lie group with maximal torus \( T \) . The inclusion \( G \hookrightarrow {G}_{\mathbb{C}} \) induces a diffeomorphism | Proof. Recall that \( \mathfrak{g} = \left\{ {X + {\theta X} \mid X \in {\mathfrak{g}}_{\mathbb{C}}}\right\} \), so that \( \mathfrak{g}/\mathrm{t} \) and \( {\mathfrak{g}}_{\mathbb{C}}/\mathfrak{b} \) are both spanned by the projections of \( \left\{ {{X}_{\alpha } + \theta {X}_{\alpha } \mid {X}_{\alpha } \in {\mathf... | Yes |
Lemma 7.53. Let \( G \) be a compact connected Lie group with maximal torus \( T \) and \( \lambda \in A\left( T\right) \) . Then \( \Gamma \left( {G/T,{L}_{\lambda }}\right) \cong \Gamma \left( {{G}_{\mathbb{C}}/B,{L}_{\lambda }^{\mathbb{C}}}\right) \) and \( {\operatorname{Ind}}_{T}^{G}\left( {\xi }_{\lambda }\right)... | Proof. Since the map \( G \rightarrow {G}_{\mathbb{C}}/B \) induces an isomorphism \( G/T \cong {G}_{\mathbb{C}}/B \), any \( h \in {G}_{\mathbb{C}} \) can be written as \( h = {gb} \) for \( g \in G \) and \( b \in B \) . Moreover, if \( h = {g}^{\prime }{b}^{\prime } \) , \( {g}^{\prime } \in G \) and \( {b}^{\prime ... | No |
Theorem 7.61 (Bott-Borel-Weil Theorem). Let \( G \) be a compact connected Lie group and \( \lambda \in A\left( T\right) \) . If \( \lambda + \rho \) lies on a Weyl chamber wall, then \( {H}^{p}\left( {G/T,{L}_{\lambda }}\right) = \{ 0\} \) for all p. Otherwise, | \[ {H}^{p}\left( {G/T,{L}_{\lambda }}\right) \cong \left\{ \begin{matrix} V\left( {w\left( {\lambda + \rho }\right) - \rho }\right) \text{ for }p = \left| \left\{ {\alpha \in {\Delta }^{ + }\left( {\mathfrak{g}}_{\mathbb{C}}\right) \mid B\left( {\lambda + \rho ,\alpha }\right) < 0}\right\} \right| \\ \{ 0\} \text{ else... | Yes |
Lemma 1.1. If \( f\left( z\right) \in \mathrm{B} \), and if \( f\left( 0\right) = 0 \), then\n\n\[ \left| {f\left( z\right) }\right| \leq \left| z\right| ,\;z \neq 0, \]\n\n(1.1)\n\n\[ \left| {{f}^{\prime }\left( 0\right) }\right| \leq 1 \]\n\nEquality holds in (1.1) at some point \( z \) if and only if \( f\left( z\ri... | The proof consists in observing that the analytic function \( g\left( z\right) = f\left( z\right) /z \) satisfies \( \left| g\right| \leq 1 \) by virtue of the maximum principle. | No |
Lemma 1.2. If \( f\left( z\right) \in \mathrm{B} \), then\n\n(1.2)\n\n\[ \frac{\left| f\left( z\right) - f\left( {z}_{0}\right) \right| }{\left| 1 - \overline{f\left( {z}_{0}\right) }f\left( z\right) \right| } \leq \left| \frac{z - {z}_{0}}{1 - {\bar{z}}_{0}z}\right| ,\;z \neq {z}_{0}, \]\n\nand\n\n(1.3)\n\n\[ \frac{\l... | The proof is the same as the proof of Schwarz’s lemma if we regard \( \tau \left( z\right) \) as the independent variable and\n\n\[ \frac{f\left( z\right) - f\left( {z}_{0}\right) }{1 - \overline{f\left( {z}_{0}\right) }f\left( z\right) } \]\n\nas the analytic function. Letting \( z \) tend to \( {z}_{0} \) in (1.2) gi... | Yes |
Corollary 1.3. If \( f\left( z\right) \in \mathrm{B} \), then\n\n(1.9)\n\n\[ \left| {f\left( z\right) }\right| \leq \frac{\left| {f\left( 0\right) }\right| + \left| z\right| }{1 + \left| {f\left( 0\right) }\right| \left| z\right| }.\] | Proof. By Lemma 1.2, \( \rho \left( {f\left( z\right), f\left( 0\right) }\right) \leq \left| z\right| \), so that \( f\left( z\right) \in \overline{K\left( {f\left( 0\right) ,\left| z\right| }\right) } \) . The bound on \( \left| {f\left( z\right) }\right| \) then follows from (1.8). Equality can hold in (1.9) only if ... | Yes |
Lemma 1.4. For any three points \( {z}_{0},{z}_{1},{z}_{2} \) in \( D \) , \n\n1.10) \( \frac{\rho \left( {{z}_{0},{z}_{2}}\right) - \rho \left( {{z}_{2},{z}_{1}}\right) }{1 - \rho \left( {{z}_{0},{z}_{2}}\right) \rho \left( {{z}_{2},{z}_{1}}\right) } \leq \rho \left( {{z}_{0},{z}_{1}}\right) \leq \frac{\rho \left( {{z... | Proof. We can suppose \( {z}_{2} = 0 \) because \( \rho \) is invariant. Then (1.10) becomes \n\n(1.11) \n\n\[ \frac{\left| {z}_{0}\right| - \left| {z}_{1}\right| }{1 - \left| {z}_{0}\right| \left| {z}_{1}\right| } \leq \left| \frac{{z}_{1} - {z}_{0}}{1 - {\bar{z}}_{0}{z}_{1}}\right| \leq \frac{\left| {z}_{0}\right| + ... | Yes |
Theorem 2.1 (Carathéodory). If \( f\left( z\right) \in \mathrm{B} \), then there is a sequence \( \left\{ {B}_{k}\right\} \) of finite Blaschke products that converges to \( f\left( z\right) \) pointwise on \( D \) . | Proof. Write\n\n\[ f\left( z\right) = {c}_{0} + {c}_{1}z + \cdots . \]\n\nBy induction, we shall find a Blaschke product of degree at most \( n \) whose first \( n \) coefficients match those of \( f \) ;\n\n\[ {B}_{n} = {c}_{0} + {c}_{1}z + \cdots + {c}_{n - 1}{z}^{n - 1} + {d}_{n}{z}^{n} + \cdots . \]\n\nThat will pr... | Yes |
There exists \( f \in \mathrm{B} \) satisfying the interpolation (2.1) if and only if the quadratic form\n\n\[ \n{Q}_{n}\left( {{t}_{1},\ldots ,{t}_{n}}\right) = \mathop{\sum }\limits_{{j, k = 1}}^{n}\frac{1 - {w}_{j}{\bar{w}}_{k}}{1 - {z}_{j}{\bar{z}}_{k}}{t}_{j}{\bar{t}}_{k} \]\n\nis nonnegative, \( {Q}_{n} \geq 0 \)... | Proof. We use induction on \( n \) . The case \( n = 1 \) holds because the Möbius transformations act transitively on \( D \) . Assume \( n > 1 \) . Suppose (2.1) holds. Then clearly \( \left| {w}_{n}\right| \leq 1 \), and if \( \left| {w}_{n}\right| = 1 \), then the interpolating function is the constant \( {w}_{n} \... | Yes |
Corollary 2.3. Suppose \( {Q}_{n} \geq 0 \) . Then (2.1) has a unique solution \( f\left( z\right) \in \mathrm{B} \) if and only if \( \det \left( {Q}_{n}\right) = 0 \) . If \( \det \left( {Q}_{n}\right) = 0 \) and \( m < n \) is the rank of \( {Q}_{n} \), then the interpolating function is a Blaschke product of degree... | Proof. If \( \left| {w}_{n}\right| = 1 \) the whole thing is very trivial because then \( {Q}_{n} = 0 \) , \( m = 0 \), and \( {B}_{n} = {w}_{n} \) . So we may assume \( \left| {w}_{n}\right| < 1 \) . We may then suppose \( {z}_{n} = {w}_{n} = 0 \), because by (2.4), \( {Q}_{n} \) and \( {Q}_{n}^{\prime } \) have the s... | Yes |
Corollary 2.4. Suppose \( {Q}_{n} \geq 0 \) and \( \det \left( {Q}_{n}\right) > 0 \) . Let \( z \in D, z \neq {z}_{j}, j = \) \( 1,2,\ldots, n \) . The set of values\n\n\[ \nW = \left\{ {f\left( z\right) : f \in \mathrm{B}, f\left( {z}_{j}\right) = {w}_{j},1 \leq j \leq n}\right\} \n\]\n\nis a nondegenerate closed disc... | Proof. We may again suppose \( {z}_{n} = {w}_{n} = 0 \) . Then \( \det \left( {\widetilde{Q}}_{n - 1}\right) > 0 \) by (2.7). By induction,\n\n\[ \n\widetilde{W} = \left\{ {g\left( z\right) : g \in \mathrm{B}, g\left( {z}_{j}\right) = {w}_{j}/{z}_{j},1 \leq j \leq n - 1}\right\} \n\]\n\nis a closed disc contained in \(... | Yes |
Theorem 3.1. (a) If \( 1 \leq p < \infty \) and if \( f\left( x\right) \in {L}^{p} \), then\n\n\[ \n{\begin{Vmatrix}{P}_{y} * f - f\end{Vmatrix}}_{p} \rightarrow 0\;\left( {y \rightarrow 0}\right) .\n\] | Proof. Let \( f \in {L}^{p},1 \leq p \leq \infty \) . When \( p = \infty \) we suppose in addition that \( f \) is uniformly continuous. Then\n\n\[ \n{P}_{y} * f\left( x\right) - f\left( x\right) = \int {P}_{y}\left( t\right) \left( {f\left( {x - t}\right) - f\left( x\right) }\right) {dt}.\n\]\n\nMinkowski's inequality... | Yes |
Corollary 3.2. Assume \( f\left( x\right) \) is bounded and uniformly continuous, and let\n\n\[ u\left( {x, y}\right) = \left\{ \begin{array}{ll} \left( {{P}_{y} * f}\right) \left( x\right) , & y > 0, \\ f\left( x\right) , & y = 0. \end{array}\right. \]\n\nThen \( u\left( {x, y}\right) \) is harmonic on \( \mathrm{H} \... | This corollary follows from (d). We also need the local version of the corollary. | No |
Lemma 3.3. Assume \( f\left( x\right) \in {L}^{p},1 \leq p \leq \infty \), and assume \( f \) is continuous at \( {x}_{0} \). Let \( u\left( {x, y}\right) = {P}_{y} * f\left( x\right) \). Then\n\n\[ \mathop{\lim }\limits_{{\left( {x, y}\right) \rightarrow {x}_{0}}}u\left( {x, y}\right) = f\left( {x}_{0}\right) \] | Proof. We have\n\n\[ \left| {u\left( {x, y}\right) - f\left( {x}_{0}\right) }\right| \leq {\int }_{\left| t\right| < \delta }{P}_{y}\left( t\right) \left| {f\left( {x - t}\right) - f\left( {x}_{0}\right) }\right| {dt} + {\int }_{\left| t\right| \geq \delta }.\n\nWith \( \delta \) small and \( \left| {x - {x}_{0}}\right... | Yes |
Lemma 3.4. If \( u\left( z\right) \) is harmonic on \( \mathrm{H} \) and bounded and continuous on H then\n\n\[ u\left( z\right) = \int {P}_{y}\left( {x - t}\right) u\left( t\right) {dt} \] | Proof. The lemma is not a trivial consequence of the definition of \( {P}_{z}\left( t\right) \) , because \( u\left( z\right) \) may not be continuous at \( \infty \) . But let\n\n\[ U\left( z\right) = u\left( z\right) - \int {P}_{y}\left( {x - t}\right) u\left( t\right) {dt}. \]\n\nThen \( U\left( z\right) \) is harmo... | Yes |
Theorem 3.5. Let \( u\left( z\right) \) be a harmonic function on the upper half plane \( \mathrm{H} \) . Then\n\n(a) If \( 1 < p \leq \infty, u \) is the Poisson integral of a function in \( {L}^{p} \) if and only if\n\n(3.7)\n\n\[ \mathop{\sup }\limits_{y}\int \parallel u\left( {x + y}\right) {\parallel }_{{L}^{p}\le... | Proof. We have already noted that (3.7) and (3.8) are necessary conditions because of Minkowski’s inequality. Suppose \( u\left( z\right) \) satisfies (3.7) or (3.8). Then we have the estimate\n\n(3.9)\n\n\[ \left| {u\left( z\right) }\right| \leq {\left( \frac{2}{\pi y}\right) }^{1/p}\mathop{\sup }\limits_{{\eta > 0}}\... | Yes |
Lemma 4.1. If \( \left( {X,\mu }\right) \) is a measure space, if \( f\left( x\right) \) is measurable, and if \( 0 < p < \infty \), then\n\n\[ \int {\left| f\right| }^{p}{d\mu } = {\int }_{0}^{\infty }p{\lambda }^{p - 1}m\left( \lambda \right) {d\lambda }. \] | Proof. We may assume \( f \) vanishes except on a set of \( \sigma \) -finite measure, because otherwise both sides of (4.1) are infinite. Then Fubini's theorem shows that both sides of (4.1) equal the product measure of the ordinate set \( \left\{ {\left( {x,\lambda }\right) : 0 < \lambda < {\left| f\left( x\right) \r... | Yes |
Theorem 4.2. For \( \alpha > 0 \) and \( t \in \mathbb{R} \), let \( {\Gamma }_{\alpha }\left( t\right) \) be the cone in \( \mathrm{H} \) with vertex \( t \) and angle 2 arctan \( \alpha \), as shown in Figure I.3,\n\n\[ \n{\Gamma }_{\alpha }\left( t\right) = \{ \left( {x, y}\right) : \left| {x - t}\right| < {\alpha y... | Proof. We may assume \( t = 0 \) . Let us first consider the points \( \left( {0, y}\right) \) on the axis of the cone \( {\Gamma }_{\alpha }\left( 0\right) \) . Then\n\n\[ \nu\left( {0, y}\right) = \int {P}_{y}\left( s\right) f\left( s\right) {ds} \]\n\nand the kernel \( {P}_{y}\left( s\right) \) is a positive even fu... | Yes |
Lemma 4.4. Let \( \mu \) be a positive Borel measure on \( \mathbb{R} \) and let \( \left\{ {{I}_{1},\ldots ,{I}_{n}}\right\} \) be a finite family of open intervals in \( \mathbb{R} \). There is a subfamily \( \left\{ {{J}_{1},\ldots ,{J}_{m}}\right\} \) such that the \( {J}_{i} \) are pairwise disjoint and such that\... | Proof. By induction \( \left\{ {{I}_{1},\ldots ,{I}_{n}}\right\} \) can be replaced by a subfamily of intervals such that no interval \( {I}_{j} \) is contained in the union of the others and such that the refined family has the same union as the original family. Write the \( {I}_{j} \) in the refined family as \( \lef... | Yes |
Theorem 4.5. Let \( \\left( {X,\\mu }\\right) \) and \( \\left( {Y, v}\\right) \) be measure spaces, and let \( 1 < {p}_{1} \\leq \\infty \) . Suppose \( T \) is a mapping from \( {L}^{1}\\left( {X,\\mu }\\right) + {L}^{{p}_{1}}\\left( {X,\\mu }\\right) \) to v-measurable functions such that\n\n(i)\n\n\[\\left| {T\\lef... | Proof of Theorem 4.5. Fix \( f \\in {L}^{p},1 < p < {p}_{1} \), and, \( \\lambda > 0 \) . Let\n\n\[{E}_{\\lambda } = \\{ y : \\left| {{Tf}\\left( y\\right) }\\right| > \\lambda \\} .\n\]\nWe are going to get a tight grip on \( v\\left( {E}_{\\lambda }\\right) \) and then use Lemma 4.1 to estimate \( \\parallel {Tf}{\\p... | Yes |
Theorem 5.1. Let \( u\left( z\right) \) be harmonic on \( \mathrm{H} \) and let \( 1 \leq p < \infty \) . Assume\n\n\[ \mathop{\sup }\limits_{y}\int {\left| u\left( x + iy\right) \right| }^{p}{dx} < \infty . \]\n\nIf \( p > 1 \), then \( {u}^{ * }\left( t\right) \in {L}^{p} \), and\n\n(5.1)\n\n\[ {\begin{Vmatrix}{u}^{ ... | Proof. Let \( p > 1 \) . Then \( u\left( z\right) \) is the Poisson integral of a function \( f\left( t\right) \in \) \( {L}^{p}\left( \mathbb{R}\right) \), and\n\n\[ \parallel f{\parallel }_{p} \leq \mathop{\sup }\limits_{y}{\left( \int {\left| u\left( x + iy\right) \right| }^{p}dx\right) }^{1/p}. \]\n\nTheorem 4.2 sa... | Yes |
Corollary 5.2. If \( u\left( z\right) \) is harmonic on \( \mathrm{H} \) and if \( p > 1 \), then\n\n\[ \int \mathop{\sup }\limits_{y}{\left| u\left( x + iy\right) \right| }^{p}{dx} \leq {B}_{p}\mathop{\sup }\limits_{y}\int {\left| u\left( x + iy\right) \right| }^{p}{dx}. \] | Note that Corollary 5.2 is false at \( p = 1 \) . Take \( u\left( {x, y}\right) = {P}_{y} * f\left( x\right), f \in \) \( {L}^{1}, f > 0 \) . Then \( \mathop{\sup }\limits_{y}\left| {u\left( {x, y}\right) }\right| \geq {Mf}\left( x\right) \) and \( {Mf} \notin {L}^{1} \) . | No |
Lemma 5.4. If \( v \) is a finite singular measure on \( \mathbb{R} \), then \( \left( {{P}_{y} * v}\right) \left( x\right) \) converges nontangentially to zero almost everywhere. | Proof. We may assume \( v \geq 0 \) . Because \( v \) is singular, we have\n\n(5.4)\n\n\[ \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{v\left( \left( {t - h, t + h}\right) \right) }{2h} = 0 \] \n\nfor Lebesgue almost all \( t \) . Indeed, if (5.4) were not true, there would be a compact set \( K \) such that \( \left... | Yes |
Lemma 5.5. Let \( \sigma \) be a positive measure on \( \mathrm{H} \), and let \( \alpha > 0 \) . Then \( \sigma \) is a Carleson measure if and only if there exists \( A = A\left( \alpha \right) \) such that\n\n\[ \sigma \left( \left\{ {\left| {u\left( z\right) }\right| > \lambda }\right\} \right) \leq A\left| \left\{... | Proof. We take \( \alpha = 1 \) . The proof for a different \( \alpha \) is similar. Assume \( \sigma \) is a Carleson measure. The open set \( \left\{ {t : {u}^{ * }\left( t\right) > \lambda }\right\} \) is the union of a disjoint sequence of open intervals \( \left\{ {I}_{j}\right\} \), with centers \( c\left( {I}_{j... | Yes |
Theorem 5.6. (Carleson). Let \( f \in {L}^{p}\left( \mathbb{R}\right) \) and let \( u\left( z\right) \) denote the Poisson integral of \( f \) . If \( \sigma \) is a positive measure on the upper half plane, then the following are equivalent.\n\n(a) \( \sigma \) is a Carleson measure.\n\n(b) For \( 1 < p < \infty \), a... | Proof. If (a) holds, then by (5.6) and Theorem 5.1, (c) and (d) hold. Clearly, (c) implies (b), and if (b) holds for some \( p \), then the closed graph theorem for Banach spaces shows that (c) holds for the same value of \( p \) .\n\nNow suppose that (d) holds or that (c) holds for some \( p,1 < p < \infty \) . As in ... | Yes |
Lemma 6.1. (Jensen’s Inequality). Let \( \\left( {X,\\mu }\\right) \) be a measure space such that \( \\mu \) is a probability measure, \( \\mu \\left( X\\right) = 1 \) . Let \( v \\in {L}^{1}\\left( \\mu \\right) \) be a real function, and let \( \\varphi \\left( t\\right) \) be a convex function on \( \\mathbb{R} \) ... | Proof. The convexity of \( \\varphi \) means that \( \\varphi \\left( t\\right) \) is the supremum of the linear functions lying below \( \\varphi \) :\n\n\[ \n\\varphi \\left( {t}_{0}\\right) = \\sup \\left\\{ {a{t}_{0} + b : {at} + b \\leq \\varphi \\left( t\\right), t \\in \\mathbb{R}}\\right\\} .\n\]\n\nWhenever \(... | Yes |
Theorem 6.2. Let \( v\left( z\right) \) be a subharmonic function on \( \Omega \), and let \( \varphi \left( t\right) \) be an increasing convex function on \( \lbrack - \infty ,\infty ) \), continuous at \( t = - \infty \) . Then \( \varphi \circ v \) is a subharmonic function on \( \Omega \) . | Proof. Since every convex function is continuous on \( \mathbb{R},\varphi \) is continuous on \( \lbrack - \infty ,\infty ) \) . It follows immediately that \( \varphi \circ v \) is upper semicontinuous. If \( {z}_{0} \in \Omega \) and if \( r < r\left( {z}_{0}\right) \), then because \( \varphi \) is increasing\n\n\[ ... | Yes |
Corollary 6.4. If \( \Omega \) is a connected open set and if \( v\left( z\right) \) is a subharmonic function on \( \Omega \) such that \( v\left( z\right) ≢ - \infty \), then whenever \( \overline{\Delta \left( {{z}_{0}, r}\right) } \subset \Omega \), \[ \frac{1}{2\pi }\int v\left( {{z}_{0} + r{e}^{i\theta }}\right) ... | Proof. Let \( {u}_{n}\left( z\right) \) be continuous functions decreasing to \( v\left( z\right) \) on \( \partial \Delta \left( {{z}_{0}, r}\right) \), and let \( {U}_{n}\left( z\right) \) denote the harmonic extension of \( {u}_{n} \) to \( \Delta \left( {{z}_{0}, r}\right) \) . If \[ \frac{1}{2\pi }\int v\left( {{z... | Yes |
Theorem 6.5. Let \( v\left( z\right) \) be a subharmonic function in the unit disc D. Assume \( v\left( z\right) ≢ - \infty \) . For \( 0 < r < 1 \), let\n\n\[ \n{v}_{r}\left( z\right) = \left\{ \begin{array}{ll} v\left( z\right) , & \left| z\right| \leq r, \\ \frac{1}{2\pi }\int {P}_{z/r}\left( \theta \right) v\left( ... | Proof. By Corollary 6.4 and by Section 3 we know \( {v}_{r}\left( z\right) \) is finite and harmonic on \( \Delta \left( {0, r}\right) = \{ \left| z\right| < r\} \) . To see that \( {v}_{r}\left( z\right) \) is upper semicontinuous at a point \( {z}_{0} \in \partial \Delta \left( {0, r}\right) \) we must show\n\n\[ \nv... | Yes |
Theorem 6.7. Let \( v\left( z\right) \) be a subharmonic function in the unit disc D. Then \( v \) has a harmonic majorant if and only if \[ \mathop{\sup }\limits_{r}\frac{1}{2\pi }\int v\left( {r{e}^{i\theta }}\right) {d\theta } = \mathop{\sup }\limits_{r}{v}_{r}\left( 0\right) < \infty . \] | Proof. If \( \mathop{\sup }\limits_{r}{v}_{r}\left( 0\right) \) is finite, then by Harnack’s theorem the functions \( {v}_{r}\left( z\right) \) increase to a finite harmonic function \( u\left( z\right) \) on \( D \) . Since \( v\left( z\right) \leq {v}_{r}\left( z\right), u\left( z\right) \) is a harmonic majorant of ... | Yes |
Theorem 6.8. Let \( v\left( z\right) \) be a subharmonic function in the upper half plane H . If\n\n\[ \n\mathop{\sup }\limits_{y}\int \left| {v\left( {x + {iy}}\right) }\right| {dx} = M < \infty ,\n\]\n\nthen \( v\left( z\right) \) has a harmonic majorant in \( \mathrm{H} \) of the form\n\n\[ \nu\left( z\right) = \int... | Proof. The inequality\n\n(6.2)\n\n\[ \nv\left( z\right) \leq \frac{2}{\pi y}\mathop{\sup }\limits_{\eta }\int \left| {v\left( {\xi + {i\eta }}\right) }\right| {d\xi },\;z = x + {iy},\;y > 0, \]\n\n\nis proved in the same way that the similar inequality (3.9) was proved to begin the proof of Theorem 3.5.\n\nFix \( {y}_{... | Yes |
Lemma 1.1. If \( 0 < p < \infty \) and if \( f\left( z\right) \in {H}^{p}\left( {dt}\right) \), then the subharmonic function \( {\left| f\left( z\right) \right| }^{p} \) has a harmonic majorant \( u\left( z\right) \) in \( \mathrm{H} \) and\n\n\[ u\left( i\right) \leq \left( {1/\pi }\right) \parallel f{\parallel }_{{H... | Proof. This follows from Theorem 6.8 of Chapter I. | No |
Lemma 1.2. If \( 0 < p < \infty \) and if \( f\left( z\right) \) is an analytic function in the upper half plane such that the subharmonic function \( {\left| f\left( z\right) \right| }^{p} \) has a harmonic majorant, then\n\n\[ F\left( z\right) = \frac{{\pi }^{-1/p}}{{\left( z + i\right) }^{2/p}}f\left( z\right) \]\n\... | Proof. Let \( u\left( z\right) \) be the least harmonic majorant of \( {\left| f\left( z\right) \right| }^{p} \) . The positive harmonic function \( u\left( z\right) \) has the form\n\n(1.2)\n\n\[ u\left( z\right) = {cy} + \int {P}_{y}\left( {x - t}\right) {d\mu }\left( t\right) ,\]\n\nwhere \( c \geq 0 \) and where \(... | Yes |
Theorem 1.3. For \( 0 < p \leq \infty ,{H}^{p} \) is complete. | Proof. We can assume \( p < \infty \) . We give the proof in the upper half plane; the reasoning for the disc is very similar. The key inequality\n\n(1.3)\n\n\[ \left| {f\left( {x + {iy}}\right) }\right| \leq {\left( 2/\pi y\right) }^{1/p}\parallel f{\parallel }_{{H}^{p}},\;y > 0, \]\n\nfollows from (6.2) of Chapter I.... | Yes |
Theorem 2.1. Let \( f\left( z\right) \) be an analytic function on the disc, \( f ≢ 0 \), and let \( \left\{ {z}_{n}\right\} \) be the zeros of \( f\left( z\right) \) . If \( \log \left| {f\left( z\right) }\right| \) has a harmonic majorant, then\n\n\[ \sum \left( {1 - \left| {z}_{n}\right| }\right) < \infty \]\n\nIf \... | Proof. Replacing \( f\left( z\right) \) by \( f\left( z\right) /{z}^{N} \) if necessary, we can assume \( f\left( 0\right) \neq 0 \) . Then by Theorem I.6.7,\n\n\[ \mathop{\sup }\limits_{r}\int \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| {d\theta }/{2\pi } = u\left( 0\right) ,\]\n\nwhere \( u \) is the least... | Yes |
Theorem 2.2. Let \( \left\{ {z}_{n}\right\} \) be a sequence of points in \( D \) such that\n\n\[ \sum \left( {1 - \left| {z}_{n}\right| }\right) < \infty \]\n\nLet \( m \) be the number of \( {z}_{n} \) equal to 0 . Then the Blaschke product\n\n(2.2)\n\n\[ B\left( z\right) = {z}^{m}\mathop{\prod }\limits_{{\left| {z}_... | Proof. We can suppose \( \left| {z}_{n}\right| > 0 \) for all \( n \) . Let\n\n\[ {b}_{n}\left( z\right) = \frac{-{\bar{z}}_{n}}{\left| {z}_{n}\right| }\frac{z - {z}_{n}}{1 - {\bar{z}}_{n}z}. \]\n\nThen the product \( \prod {b}_{n} \) converges on \( D \) to an analytic function having \( \left\{ {z}_{n}\right\} \) for... | Yes |
Theorem 2.3 (F. Riesz). Let \( 0 < p < \infty \) . Let \( f\left( z\right) \in {H}^{p}\left( D\right), f ≢ 0 \), let \( \left\{ {z}_{n}\right\} \) be the zeros of \( f\left( z\right) \), and let \( B\left( z\right) \) be the Blaschke product with zeros \( \left\{ {z}_{n}\right\} \) . Then \( g\left( z\right) = f\left( ... | Proof. It was noted above that \( B\left( z\right) \) converges when \( f \in {H}^{p} \) . Let \( {B}_{n} \) be the finite Blaschke product with zeros \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \), and let \( {g}_{n} = f/{B}_{n} \) . Fix \( r < 1 \) . Then by Theorem I.6.6, \[ \int {\left| {g}_{n}\left( r{e}^{i\theta }\right) ... | Yes |
Theorem 2.4. Let \( f\left( z\right) \in {H}^{\infty }\left( D\right) \parallel f{\parallel }_{\infty } \leq 1 \) . Then the following are equivalent.\n\n(a) \( f\left( z\right) = {\lambda B}\left( z\right) \), where \( \lambda \) is constant, \( \left| \lambda \right| = 1 \), and \( B\left( z\right) \) is a Blaschke p... | Proof. Theorem I.6.7 shows that (b) and (c) are equivalent.\n\nSuppose \( f\left( z\right) \) is the Blaschke product with zeros \( \left\{ {z}_{n}\right\} \), and let \( \varepsilon > 0 \) . We may divide \( f\left( z\right) \) by a finite Blaschke product \( {B}_{n}\left( z\right) \) so that \( \left| {\left( {f/{B}_... | Yes |
Theorem 3.1. Let \( 0 < p < \infty \) and let \( f\left( z\right) \) be a function in \( {H}^{p}\left( {dt}\right) \) . Then for any \( \alpha > 0 \), the nontangential maximal function\n\n\[ \n{f}^{ * }\left( t\right) = \mathop{\sup }\limits_{{z \in {\Gamma }_{\alpha }\left( t\right) }}\left| {f\left( z\right) }\right... | Proof. Except for the fact that the constant \( {A}_{\alpha } \) in (3.1) does not depend on \( p \) , the case \( p > 1 \) of the theorem is proved in Section 5 of Chapter I. To stretch \( p \) below 1 we use Theorem 2.3 above. Suppose \( f \in {H}^{p}, f ≢ 0 \) . Let \( B\left( z\right) \) be the Blaschke product for... | Yes |
Let \( 1 \leq p \leq \infty \) and let \( f\left( t\right) \in {L}^{p}\left( \mathbb{R}\right) \) . Then \( f\left( t\right) \) is almost everywhere the nontangential limit of an \( {H}^{p}\left( {dt}\right) \) function if and only if its Poisson integral\n\n\[ f\left( z\right) = {P}_{y} * f\left( x\right) \]\n\nis ana... | Proof. If \( f \in {L}^{p}\left( \mathbb{R}\right) \) and if \( f\left( z\right) = {P}_{y} * f\left( x\right) \) is analytic, then by Chapter I, \( f\left( z\right) \in {H}^{p}\left( {dt}\right) \) and \( f\left( z\right) \) converges nontangentially to \( f\left( t\right) \) .\n\nConversely, suppose \( f\left( z\right... | Yes |
Corollary 3.3. Let \( N \) be a positive integer. For \( 0 < p < \infty \), the class \( {\mathfrak{A}}_{N} \) is dense in \( {H}^{p}\left( {dt}\right) \) . For \( f\left( z\right) \in {H}^{\infty } \), there are functions \( {f}_{n}\left( z\right) \) in \( {\mathfrak{A}}_{N} \) such that \( {\begin{Vmatrix}{f}_{n}\end... | Proof. Were it not for the decay condition (ii), we could approximate \( f \) by the smooth functions \( f\left( {z + i/n}\right) \), which converge in \( {H}^{p} \) norm to \( f\left( t\right) \) if \( p < \infty \) and which converge boundedly pointwise almost everywhere to \( f\left( t\right) \) if \( p = \infty \) ... | Yes |
Let \( 0 < p < \infty \), let \( f\left( z\right) \) be an analytic function in the upper half plane and let\n\n\[ F\left( z\right) = \frac{{\pi }^{-1/p}}{{\left( z + i\right) }^{2/p}}f\left( z\right) .\n\]\n\nThen \( {\left| f\left( z\right) \right| }^{p} \) has a harmonic majorant if and only if \( F\left( z\right) \... | Proof. Let \( g\left( w\right) = f\left( z\right), z = i\left( {1 - w}\right) /\left( {1 + w}\right) \) . The corollary asserts that \( g \in {H}^{p}\left( D\right) \) if and only if \( F \in {H}^{p}\left( {dt}\right) \), and that \( \parallel g{\parallel }_{p} = \parallel F{\parallel }_{p} \) . If \( N > 2/p \) and if... | Yes |
Corollary 3.5. Let \( 0 < p < \infty \) and let \( u\left( z\right) \) be a real-valued harmonic function on the upper half plane \( \mathrm{H} \) . If \( u\left( z\right) \) is the real part of a function \( f\left( z\right) \in {H}^{p} \) , then\n\n\[ \n{u}^{ * }\left( t\right) = \mathop{\sup }\limits_{{{\Gamma }_{\a... | The proof from Theorem 3.1 is trivial. | No |
Theorem 3.6. If \( f\left( z\right) \in {H}^{1}\left( {dt}\right) \), then \( f\left( z\right) \) is the Poisson integral of its boundary values:\n\n\[ f\left( z\right) = \int {P}_{y}\left( {x - t}\right) f\left( t\right) {dt} \] | Proof. If \( f\left( z\right) \in {H}^{1} \), then (3.5) was already obtained in the proof of Corollary 3.2. Conversely, if \( f\left( z\right) = {P}_{y}, * \mu \left( x\right) \) is an analytic function, then by Minkowski’s integral inequality it is an \( {H}^{1} \) function and hence it is the Poisson integral of its... | Yes |
Lemma 3.7. Let \( f\left( z\right) \in {H}^{1} \) . Then the Fourier transform\n\n\[ \n\widehat{f}\left( s\right) = {\int }_{-\infty }^{\infty }f\left( t\right) {e}^{-{2\pi ist}}{dt} = 0 \n\]\n\nfor all \( s \leq 0 \) . | Proof. By the continuity of \( f \rightarrow \widehat{f} \), we may suppose \( \int \in {\mathfrak{A}}_{N} \) . Then for \( s \leq 0, F\left( z\right) = f\left( z\right) {e}^{-{2\pi isz}} \) is also in \( {\mathfrak{A}}_{N} \) . The result now follows from Cauchy’s theorem because\n\n\[ \n{\int }_{0}^{\pi }\left| {F\le... | Yes |
Theorem 3.8. Let \( {d\mu }\left( t\right) \) be a finite complex measure on \( \mathbb{R} \) such that either\n\n(a)\n\n\[ \int \frac{{d\mu }\left( t\right) }{t - z} = 0\;\text{ on }\;\operatorname{Im}z < 0 \]\n\n or \n\n(b)\n\n\[ \widehat{\mu }\left( s\right) = \int {e}^{-{2\pi ist}}{d\mu }\left( t\right) = 0\;\text{... | Proof. If (a) holds, then by (3.6) \( f\left( z\right) = {P}_{y} * \mu \left( x\right) \) is analytic and the result follows from Theorem 3.6.\n\nAssume (b) holds. The Poisson kernel \( {P}_{y}\left( t\right) \) has Fourier transform\n\n\[ \int {e}^{-{2\pi ist}}{P}_{y}\left( t\right) {dt} = {e}^{-{2\pi }\left| s\right|... | Yes |
Theorem 3.9 (Carleson). Let \( \sigma \) be a positive measure in the upper half plane. Then the following are equivalent:\n\n(a) \( \sigma \) is a Carleson measure: for some constant \( N\left( \sigma \right) \), \n\n\[ \n\sigma \left( Q\right) \leq N\left( \sigma \right) h \n\] \n\nfor all squares \n\n\[ \nQ = \left\... | Proof. That (a) implies (b) follows from (3.1) and Lemma I.5.5 just as in the proof of Theorem I.5.6.\n\nTrivially, (b) implies (c). On the other hand, if (c) holds for some fixed \( p < \infty \), then (b) holds for the same value \( p \). This follows from the closed graph theorem, which is valid here even when \( p ... | Yes |
Theorem 4.1. If \( 0 < p \leq \infty \) and if \( f\left( z\right) \in {H}^{p}\left( D\right), f ≢ 0 \), then\n\n\[ \n\frac{1}{2\pi }\int \log \left| {f\left( {e}^{i\theta }\right) }\right| {d\theta } > - \infty \n\]\n\nIf \( f\left( 0\right) ≢ 0 \), then\n\n(4.1)\n\n\[ \n\log \left| {f\left( 0\right) }\right| \leq \fr... | Proof. By Theorem I.6.7 and by the subharmonicity of \( \log \left| f\right| \) ,\n\n\[ \n\log \left| {f\left( z\right) }\right| \leq \mathop{\lim }\limits_{{r \rightarrow 1}}\frac{1}{2\pi }\int \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| {P}_{z}\left( \theta \right) {d\theta }. \n\]\n\nSince \( \log \left| ... | Yes |
Corollary 4.3. Let \( 0 < p, r \leq \infty \) . If \( f\left( z\right) \in {H}^{p} \) and if the boundary function \( f\left( t\right) \in {L}^{r} \), then \( f\left( z\right) \in {H}^{r} \) . | Proof. Applying Jensen’s inequality, with the convex function \( \varphi \left( s\right) = \) \( \exp \left( {rs}\right) \) and with the probability measure \( {P}_{y}\left( {x - t}\right) {dt} \), to (4.3) gives \[ {\left| f\left( z\right) \right| }^{r} \leq \int {\left| f\left( t\right) \right| }^{r}{P}_{y}\left( {x ... | Yes |
Theorem 4.4. Let \( h\left( t\right) \) be a nonzero nonnegative function in \( {L}^{p}\left( \mathbb{R}\right) \) . Then there is \( f\left( z\right) \in {H}^{p}\left( {dt}\right) \) such that \( \left| {f\left( t\right) }\right| = h\left( t\right) \) almost everywhere if and only if\n\n\[ \int \frac{\log h\left( t\ri... | Proof. It has already been proved that (4.4) is a necessary condition. To show (4.4) is sufficient, note that since \( \log h \leq \left( {1/p}\right) {\left| h\right| }^{p} \) ,(4.4) holds if and only if \( \log h \in {L}^{1}\left( {{dt}/\left( {1 + {t}^{2}}\right) }\right) \) . Let \( u\left( z\right) \) be the Poiss... | Yes |
Theorem 4.6. Let \( 0 < p \leq \infty \) and let \( f\left( z\right) \in {H}^{p}, f ≢ 0 \) . Then the following are equivalent.\n\n(a) \( f\left( z\right) \) is an outer function.\n\n(b) For each \( z \in \mathrm{H} \), equality holds in (4.3); that is,\n\n(4.6)\n\n\[ \log \left| {f\left( z\right) }\right| = \int \log ... | Proof. First, (e) is merely a rewording of the definition of an outer function in \( {H}^{p} \), because any function \( f\left( z\right) \) without zeros is an exponential, \( f = {e}^{u + {iv}}, u = \) \( \log \left| f\right| \) . Thus (a) and (e) are equivalent.\n\nBy definition, (a) implies (b). If (b) holds, then ... | Yes |
Corollary 4.7. If \( f\left( z\right) \in {H}^{p} \) and if for some \( r > 0,1/f\left( z\right) \in {H}^{r} \), then \( f\left( z\right) \) is an outer function. | This holds because \( f \) satisfies (4.6). | No |
Corollary 4.8. Let \( f\left( z\right) \in {H}^{p} \) . Either of the following two conditions imply that \( f\left( z\right) \) is an outer function.\n\n(a) Re \( f\left( z\right) \geq 0, z \in \mathrm{H} \) .\n\n(b) There exists a \( {C}^{1} \) are \( \Gamma \) terminating at 0 such that\n\n\[ f\left( \mathrm{H}\righ... | Proof. If (a) holds then \( f + \varepsilon \) is an outer function for any \( \varepsilon > 0 \), because \( {\left( f + \varepsilon \right) }^{-1} \in {H}^{\infty } \) . Now, since \( \operatorname{Re}f \leq 0 \),\n\n\[ \int \log \left| {f\left( t\right) + \varepsilon }\right| {P}_{y}\left( {x - t}\right) {dt} \right... | Yes |
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