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Theorem 5.1. Let \( f\left( z\right) \) be an analytic function on \( D, f ≢ 0 \) . Then \( f \in N \) if and only if \( \log \left| {f\left( z\right) }\right| \) has least harmonic majorant the Poisson integral of a finite measure on \( \partial D \) . | Proof. If\n\n\[ \log \left| {f\left( z\right) }\right| \leq \int {P}_{z}\left( \theta \right) {d\mu }\left( \theta \right) \]\n\nfor some finite measure, then\n\n\[ {\log }^{ + }\left| {f\left( z\right) }\right| \leq \int {P}_{z}\left( \theta \right) d{\mu }^{ + }\left( \theta \right) \]\n\nwhere \( {\mu }^{ + } \) is ... | Yes |
Lemma 5.2. Let \( f\left( z\right) \in N, f ≢ 0 \) . Let \( B\left( z\right) \) be the Blaschke product formed from the zeros of \( f\left( z\right) \) . Then \( B\left( z\right) \) converges, and \( g\left( z\right) = f\left( z\right) /B\left( z\right) \) is in \( N \) . Moreover, \( \log \left| {g\left( z\right) }\ri... | Proof. Since \( \log \left| {f\left( z\right) }\right| \) has a harmonic majorant, we know from Section 2 that \( B\left( z\right) \) converges. Let \( u\left( z\right) \) be the least harmonic majorant of \( \log \left| {f\left( z\right) }\right| \).\n\nThen since \( \left| {B\left( z\right) }\right| \leq 1 \), it is ... | Yes |
Theorem 5.3. Let \( f\left( z\right) \in N, f ≢ 0 \) . Then \( f\left( z\right) \) has a nontangential limit \( f\left( {e}^{i\theta }\right) \) almost everywhere, and\n\n\[ \log \left| {f\left( {e}^{i\theta }\right) }\right| \in {L}^{1}\left( {d\theta }\right) . \] | Proof. Let \( g\left( z\right) = f\left( z\right) /B\left( z\right) \), where \( B\left( z\right) \) is the Blaschke product with the same zeros as \( f\left( z\right) \) . We knew from Theorem 5.1 and Lemma 5.2 that\n\n\[ \log \left| {g\left( z\right) }\right| = \int {P}_{z}\left( \theta \right) {d\mu }\left( \theta \... | Yes |
Theorem 5.4. Let \( f\left( z\right) \in N, f ≢ 0 \) . Then the following are equivalent.\n\n(a) \( f\left( z\right) \in {N}^{ + } \)\n\n(b) The least harmonic majorant of \( {\log }^{ + }\left| {f\left( \theta \right) }\right| \) is\n\n\[ \int {\log }^{ + }\left| {f\left( \theta \right) }\right| {P}_{z}\left( \theta \... | Proof. We have already proved that (c) and (d) are equivalent.\n\nIf \( f\left( z\right) \in N \), then \( {\log }^{ + }\left| {f\left( z\right) }\right| \) has least harmonic majorant\n\n\[ U\left( z\right) = \int {P}_{z}\left( \theta \right) {dv}\left( \theta \right) \]\n\nwhere the positive measure \( v \) is the we... | Yes |
Theorem 5.5. Let \( f\left( z\right) \in N, f ≢ 0 \) . Then\n\n\[ f\left( z\right) = {CB}\left( z\right) F\left( z\right) {S}_{1}\left( z\right) /{S}_{2}\left( z\right) ,\;\left| C\right| = 1, \]\n\nwhere \( B\left( z\right) \) is a Blaschke product, \( F\left( z\right) \) is an outer function, and \( {S}_{1}\left( z\r... | Proof. We have already derived the factorization (5.10). There can be no difficulties about the uniqueness of the factors because \( B\left( z\right) \) is determined by the zeros of \( f\left( z\right) \), and as \( \left| {B\left( {e}^{i\theta }\right) }\right| = \left| {{S}_{1}\left( {e}^{i\theta }\right) }\right| =... | Yes |
Corollary 5.6. Let \( f\left( z\right) \in N, f ≢ 0 \) . Then in (5.10) the singular factor \( {S}_{2} \equiv 1 \) if and only if \( f\left( z\right) \in {N}^{ + } \) . | Proof. \( {S}_{2}\left( z\right) \equiv 1 \) if and only if\n\n\[ \n{d\mu } = \log \left| {f\left( \theta \right) }\right| \left( {{d\theta }/{2\pi }}\right) - d{\mu }_{1}, \n\]\n\nwith \( d{\mu }_{1} \geq 0 \), and this holds if and only if \( f\left( z\right) \in {N}^{ + } \) . | Yes |
Theorem 6.1. Let \( B\left( z\right) \) be a Blaschke product with zeros \( \left\{ {z}_{n}\right\} \), and let \( E \subset \partial D \) be the set of accumulation points of \( \left\{ {z}_{n}\right\} \) . Then \( B\left( z\right) \) extends to be analytic on the complement of \[ E \cup \left\{ {1/{\bar{z}}_{n} : n =... | Proof. Let \( \left\{ {{B}_{n}\left( z\right) }\right\} \) be the finite Blaschke products converging to \( B\left( z\right) \) . Then \[ {B}_{n}\left( {1/\bar{z}}\right) = 1/\overline{{B}_{n}\left( z\right) } \] by reflection, and \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{B}_{n}\left( z\right) \) exists on \(... | Yes |
Theorem 6.2. Let \( S\left( z\right) \) be the singular function determined by the measure \( \mu \) on \( \partial D \), and let \( E \subset \partial D \) be the closed support of \( \mu \) . Then \( S\left( z\right) \) extends analytically to \( \mathbb{C} \smallsetminus E \) . In particular \( S\left( z\right) \) i... | Proof. For any measure \( \mu \) on \( \partial D \), the function \n\[ \int \frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}{d\mu }\left( \theta \right) \nis analytic at all points not in the closed support \( E \) of \( \mu \) . Hence \( S\left( z\right) \) is analytic on \( \mathbb{C} \smallsetminus E \) .\n\nIf \( \mu... | Yes |
Theorem 6.3. Let \( f \in {H}^{p}, p > 0 \), and let \( \Gamma \) be an open arc on \( \partial D \) . If \( f\left( z\right) \) is analytic across \( \Gamma \), then its inner factor and its outer factor are analytic across \( \Gamma \) . If \( f\left( z\right) \) is continuous across \( \Gamma \), then its outer fact... | Proof. Write \( f = {BSF} \), where \( B \) is a Blaschke product, \( S \) is a singular function, and \( F \) is an outer function. We may suppose \( f ≢ 0 \) . If \( f \) is analytic or continuous across \( \Gamma \), then \( F \) is bounded on any compact subset of \( \Gamma \), because \( \left| F\right| = \left| f... | Yes |
Theorem 6.4 (Frostman). Let \( f\left( z\right) \) be a nonconstant inner function on the unit disc. Then for all \( \zeta ,\left| \zeta \right| < 1 \), except possibly for a set of capacity zero, the function\n\n\[ \n{f}_{\zeta }\left( z\right) = \frac{f\left( z\right) - \zeta }{1 - \bar{\zeta }f\left( z\right) }\n\]\... | Proof. Let \( K \) be a compact set of positive capacity and let \( \sigma \) be a positive mass on \( K \) such that \( {G}_{\sigma }\left( z\right) \) is bounded on \( D \) . We shall show\n\n\[ \n\sigma \left( \left\{ {\zeta \in K : {f}_{\zeta }\text{ is not a Blaschke product }}\right\} \right) = 0\n\]\n\nand that ... | Yes |
Corollary 6.5. The set of Blaschke products is uniformly dense in the set of inner functions. | Proof. If \( f\left( z\right) \) is an inner function and if \( \left| \zeta \right| \) is small, then\n\n\[ \n{\begin{Vmatrix}f - {f}_{\zeta }\end{Vmatrix}}_{\infty } < \varepsilon \n\]\n\nBy Frostman’s theorem \( {f}_{\zeta } \) is a Blaschke product for many small \( \zeta \) . | No |
Corollary 7.3. Let \( f\left( z\right) \in {H}^{2} \) . Then \( f\left( z\right) \) is an outer function if and only if \( \mathrm{P}f = \{ p\left( z\right) f\left( z\right) : p \in \mathrm{P}\} \) is dense in \( {H}^{2} \) . | Proof. Let \( M \) be the closure of \( \mathrm{P}f \) in \( {H}^{2} \) . Then \( M \) is invariant under the shift operator, so that \( M = G{H}^{2} \) for some inner function \( G\left( z\right) \) . Since \( f \in M \) , we have \( f = {Gh}, h \in {H}^{2} \) . So if \( f \) is an outer function, then \( G \) is cons... | Yes |
Let \( f\left( z\right) \) be an outer function. Then there are functions \( \left\{ {f}_{n}\right\} \) in \( {H}^{\infty } \) such that\n\n(7.3)\n\n\[ \left| {{f}_{n}\left( z\right) f\left( z\right) }\right| \leq 1 \]\n\n(7.4)\n\n\[ {f}_{n}\left( \theta \right) f\left( \theta \right) \rightarrow 1\;\text{ almost every... | Proof. Let\n\n\[ {u}_{n}\left( \theta \right) = \min \left( {{A}_{n}, - \log \left| {f\left( \theta \right) }\right| }\right) ,\]\n\nwhere \( {A}_{n} \) is a large number to be determined later. Let \( {f}_{n} \) be the outer function with \( \log \left| {f}_{n}\right| = {u}_{n} \) and with \( {f}_{n}\left( 0\right) f\... | Yes |
Theorem 7.5. Let \( I \) be a nonzero ideal in \( {H}^{\infty } \) . If \( I \) is weak-star closed, then there is an inner function \( G \) such that\n\n\[ I = G{H}^{\infty } \]\n\nThe inner function \( G \) is unique except for a constant factor, and every set of the form (7.5) is a weak-star closed ideal in \( {H}^{... | Proof. It is clear that \( G \) is essentially unique and that (7.5) defines a weak-star closed ideal.\n\nNow let \( I \neq \{ 0\} \) be a weak-star closed ideal. Let \( M \) be the closure of \( I \) in \( {H}^{2} \) . We claim\n\n\[ M \cap {H}^{\infty } = I. \]\n\nSince \( M = G{H}^{2} \) for an inner function \( G \... | Yes |
Lemma 1.1. If \( f \in {L}^{1}\left( T\right) \), then \( \widetilde{u}\left( z\right) \) has nontangential limit \( \widetilde{f}\left( \theta \right) \) almost everywhere on \( T \) . | Proof. We may suppose \( f\left( \theta \right) \geq 0 \) . The analytic function \( g\left( z\right) = u\left( z\right) + i\widetilde{u}\left( z\right) \) then has nonnegative real part, and \( G\left( z\right) = g\left( z\right) /\left( {1 + g\left( z\right) }\right) \) is bounded. The function \( G\left( z\right) \)... | Yes |
Lemma 1.2. Let \( f\left( \theta \right) \in {L}^{1}\left( T\right) \) . For almost every \( \theta \) | Proof. Notice that for \( \theta \neq 0 \) ,\n\n\[ \mathop{\lim }\limits_{{r \rightarrow 1}}{Q}_{r}\left( \theta \right) = \mathop{\lim }\limits_{{r \rightarrow 1}}\frac{{2r}\sin \theta }{1 - {2r}\cos \theta + {r}^{2}} = \frac{\sin \theta }{1 - \cos \theta } = \cot \frac{\theta }{2} = {Q}_{1}\left( \theta \right) ,\]\n... | Yes |
Theorem 1.3. If \( f\left( \theta \right) \) is a Dini continuous function on \( T \), then \( \widetilde{f} \) exists at every point of \( T,\widetilde{f} \) is a continuous function, and\n\n(1.7)\n\n\[ \n{\omega }_{\widetilde{f}}\left( \delta \right) \leq C\left( {{\int }_{0}^{\delta }\frac{\omega \left( t\right) }{t... | Proof. If \( b\left( \theta \right) \) is a bounded function, then the convolution\n\n\[ \nb * f\left( \theta \right) = \frac{1}{2\pi }\int b\left( \varphi \right) f\left( {\theta - \varphi }\right) {d\varphi }\n\]\n\nsatisfies\n\n\[ \n{\omega }_{b * f}\left( \delta \right) \leq \parallel b{\parallel }_{\infty }{\omega... | Yes |
Corollary 1.4. Let \( I \) be an open arc on \( T \) . Let \( f\left( \theta \right) \in {L}^{1} \) and assume \( f \) is Dini continuous on the arc \( I \) . Then \( \widetilde{f}\left( \theta \right) \) is continuous at each point of \( I \) . | Proof. First note that if \( f\left( \theta \right) = 0 \) on \( I \), then by (1.1), \( \widetilde{f} \) is real analytic on \( I \) . If \( J \) is any compact subarc of \( I \), there is a function \( g\left( \theta \right) \) Dini continuous on \( T \) such that \( g = f \) on a neighborhood of \( J \) . By Theorem... | No |
Theorem 2.1. There is a constant \( {A}_{\alpha } \) depending only on \( \alpha \) such that\n\n\[ \left| \left\{ {\theta : {\left( \widetilde{f}\right) }^{ * }\left( \theta \right) > \lambda }\right\} \right| \leq \left( {{A}_{\alpha }/\lambda }\right) \parallel f{\parallel }_{1} \]\n\nif \( f \in {L}^{1}\left( T\rig... | Proof. If we can prove (2.1) for all positive \( f \in {L}^{1} \) with a constant \( {C}_{\alpha } \), then (2.1) holds in general with \( {A}_{\alpha } = 8{C}_{\alpha } \). So we assume \( f \geq 0 \). Then \( F\left( z\right) = \left( {{P}_{r} + i{Q}_{r}}\right) * f\left( \theta \right), z = r{e}^{i\theta } \), is an... | Yes |
Lemma 2.2. If \( \mu \) is a finite measure on \( \mathbb{R} \) with Poisson integral \( u\left( z\right) \), then\n\n\[ \int {d\mu } = \mathop{\lim }\limits_{{y \rightarrow \infty }}\int \frac{{y}^{2}}{{t}^{2} + {y}^{2}}{d\mu }\left( t\right) = \mathop{\lim }\limits_{{y \rightarrow \infty }}{\pi yu}\left( {iy}\right) ... | The lemma is elementary. | No |
Theorem 2.3. If \( 1 < p < \infty \), there is a constant \( {A}_{p} \) such that\n\n\[ \parallel \widetilde{f}{\parallel }_{p} \leq {A}_{p}\parallel f{\parallel }_{p} \]\n\nif \( f \in {L}^{p}\left( \mathbb{R}\right) \) or \( f \in {L}^{p}\left( T\right) \) . | From the interpolation we see that\n\n(2.3)\n\n\[ {A}_{p} \sim A/\left( {p - 1}\right) ,\;p \rightarrow 1, \]\n\n(2.4)\n\n\[ {A}_{p} \sim {Ap},\;p \rightarrow \infty . \]\n\nThese estimates are sharp, except for the choice of the constant \( A \) . By duality, (2.3) is sharp if (2.4) is sharp. Let \( f\left( t\right) =... | Yes |
Theorem 2.4. Let \( F\left( z\right) \) be an analytic function on the disc D. If \( \operatorname{Re}F\left( z\right) > 0 \) , then \( F \in {H}^{p} \) for all \( p < 1 \), and \( \parallel F{\parallel }_{{H}^{p}} \leq {C}_{p}\left| {F\left( 0\right) }\right| \) . | Proof. Write \( F = \left| F\right| {e}^{i\varphi } \) where \( \left| \varphi \right| < \pi /2 \) . Then \( {F}^{p} \) is analytic on \( D \) and\n\n\[ {F}^{p} = {\left| F\right| }^{p}\left( {\cos {p\varphi } + i\sin {p\varphi }}\right) . \]\n\nSince \( p < 1 \), this means\n\n\[ {\left| F\right| }^{p} \leq {C}_{p}\op... | Yes |
Corollary 2.5. If \( F\left( z\right) \) is analytic on \( D \) and if \( \left| {\arg F\left( z\right) }\right| \leq \lambda \leq \pi \), then \( F \in \) \( {H}^{p} \) for all \( p < \pi /{2\lambda } \) | Proof. Use Theorem 2.4 on \( {F}^{\pi /{2\lambda }} \) | Yes |
Corollary 2.6. If \( f\left( \theta \right) \in {L}^{\infty }\left( T\right) \) and if \( \parallel f{\parallel }_{\infty } \leq 1 \), then for \( p < \pi /2 \)\n\n\[ \frac{1}{2\pi }\int {e}^{p\left| \widetilde{f}\right| }{d\theta } < {C}_{p} \]\n\nIf \( f\left( \theta \right) \) is continuous on \( T \), then\n\n\[ \f... | Proof. If \( f \in {L}^{\infty },\parallel f{\parallel }_{\infty } \leq 1 \), then\n\n\[ F = \exp \left( {\pm \frac{1}{2}{\pi i}\left( {f + i\widetilde{f}}\right) }\right) \]\n\nmaps the disc into the right half plane. Hence \( F \in {H}^{p} \) for all \( p < 1 \) . If \( f\left( \theta \right) \) is continuous, there ... | No |
Theorem 2.7. Let \( E \subset T \) be a measurable set with measure \( \left| E\right| \) and let \( f = {\chi }_{E} \) . Then the distribution function\n\n\[ \n\left| {\{ \theta : }\right| \widetilde{f}\left( \theta \right) \left| { > \lambda \} }\right| \n\]\n\ndepends only on \( \left| E\right| \) . | Proof. Let \( F\left( z\right) = \left( {{P}_{r} + i{Q}_{r}}\right) * f\left( \theta \right) ,\;z = r{e}^{i\theta } \) . Then \( 0 < \operatorname{Re}F\left( z\right) < \) 1, Re \( F\left( {e}^{i\theta }\right) = {\chi }_{E}\left( \theta \right) \) almost everywhere, and \( F\left( 0\right) = \left| E\right| /{2\pi } \... | Yes |
Lemma 3.3. If \( 0 < p \leq 2 \), and if \( {u}^{ * } \in {L}^{p} \), then\n\n\[{\left( \int {\left| u\left( x + iy\right) \right| }^{2}dx\right) }^{1/2} \leq 2{y}^{1/2 - 1/p}{\left( \int {\left| {u}^{ * }\right| }^{p}dt\right) }^{1/p},\;y > 0.\] | Proof. Fix \( y > 0 \) . Then\n\n\[{\left| u\left( x + iy\right) \right| }^{p} \leq \mathop{\inf }\limits_{{\left| {t - x}\right| < y}}{\left| {u}^{ * }\left( t\right) \right| }^{p} \leq \left( {1/{2y}}\right) {\int }_{x - y}^{x + y}{\left| {u}^{ * }\left( t\right) \right| }^{p}{dt}\]\n\nand so\n\n\[\mathop{\sup }\limi... | Yes |
Corollary 3.4. If \( 0 < p < \infty \) and if \( u\left( z\right) \) is a real-valued harmonic function, then \( u\left( z\right) = \operatorname{Re}f\left( z\right), f \in {H}^{p} \), if and only if \( {u}^{ * } \in {L}^{p} \). There are constants \( {c}_{1} \) and \( {c}_{2} \), depending only on \( p \), such that\n... | Proof. The inequality \( {c}_{1}{\begin{Vmatrix}{u}^{ * }\end{Vmatrix}}_{p} \leq \parallel f{\parallel }_{p} \) was proved in Chapter II. The other inequality is immediate from the theorem. | No |
Corollary 3.5. If \( 0 < p < \infty \) and if \( u\left( z\right) \) is a harmonic function such that \( {u}^{ * } \in {L}^{p} \), then there is a conjugate function \( v \) such that \( {v}^{ * } \in {L}^{P} \) and\n\n\[{\begin{Vmatrix}{v}^{ * }\end{Vmatrix}}_{p}^{p} \leq {C}_{p}{\begin{Vmatrix}{u}^{ * }\end{Vmatrix}}... | Proof. By Corollary 3.4, \( f = u + {iv} \) is in \( {H}^{p} \) with \( \parallel f{\parallel }_{p}^{p} \leq {c}_{2}^{p}{\begin{Vmatrix}{u}^{ * }\end{Vmatrix}}_{p}^{p} \) . The maximal theorem then shows that \( {f}^{ * } \in {L}^{p} \) with \( {\begin{Vmatrix}{f}^{ * }\end{Vmatrix}}_{p}^{p} \leq A{c}_{2}^{p}{\begin{Vm... | Yes |
Theorem 3.6. If \( 0 < p < \infty \) and if \( u\left( z\right) \) is harmonic on \( \mathrm{H} \), then\n\n\[ \int {\left| {u}^{ * }\right| }^{p}{dt} \leq {c}_{p}\int {\left| {u}^{ + }\right| }^{p}{dt} \]\n\nwhere \( {c}_{p} \) is a constant depending only on \( p \) . | The proof of Theorem 3.6 rests on a remarkable inequality due to Hardy and Littlewood. | No |
Lemma 3.7. If \( u\left( z\right) \) is harmonic on the disc \( \Delta \left( {{z}_{0}, R}\right) \) and if \( 0 < p < \infty \) , then\n\n\[ \left| {u\left( {z}_{0}\right) }\right| \leq {K}_{p}{\left( \frac{1}{\pi {R}^{2}}{\iint }_{\Delta \left( {{z}_{0}, R}\right) }{\left| u\left( z\right) \right| }^{p}dxdy\right) }^... | Proof. We only have to treat the case \( p < 1 \) . We may change variables and assume \( {z}_{0} = 0, R = 1 \) . Write\n\n\[ {m}_{q}\left( r\right) = {\left( \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| u\left( r{e}^{i\theta }\right) \right| }^{q}d\theta \right) }^{1/q}. \]\n\nWe can assume that\n\n\[ 2{\int }_{0}^{1}{m}... | Yes |
Lemma 1.1. If \( {x}^{ * } \in {X}^{ * } \), then\n\n\( \left( {1.1}^{\prime }\right) \)\n\n\( \sup \left\{ {\left| \left\langle {{x}^{ * }, y}\right\rangle \right| : y \in Y,\parallel y\parallel \leq 1}\right\} = \inf \left\{ {\begin{Vmatrix}{{x}^{ * } - k}\end{Vmatrix} : k \in {Y}^{ \bot }}\right\} . \) | Proof. The left side of \( \left( {1.1}^{\prime }\right) \) is the norm of the restriction of \( {x}^{ * } \) to the subspace \( Y \) and the right-hand side of \( \left( {1.1}^{\prime }\right) \) is the norm of the coset \( {x}^{ * } + {Y}^{ \bot } \) in \( {X}^{ * }/{Y}^{ \bot } \) . By (1.1) these quantities are equ... | No |
Theorem 1.2. Let \( 1 \leq p < \infty \), and let \( f \in {L}^{p}, \notin {H}^{p} \) . Then the distance from f to \( {H}^{p} \) is\n\n\[ \operatorname{dist}\left( {f,{H}^{p}}\right) = \mathop{\inf }\limits_{{g \in {H}^{p}}}\parallel f - g{\parallel }_{p} \]\n\n\[ = \sup \left\{ {\left| {\int {fF}\frac{d\theta }{2\pi ... | Proof. The identity (1.3) follows from \( \left( {1.2}^{\prime }\right) \) . Let \( {g}_{n} \in {H}^{p} \) be such that \( \parallel f - \) \( {\left. {g}_{n}\right.\parallel }_{p} \rightarrow \operatorname{dist}\left( {f,{H}^{p}}\right) \) . The Poisson integrals of \( f - {g}_{n} \) are bounded on any compact subset ... | Yes |
Theorem 1.3. If \( f \in {L}^{\infty } \), then the distance from f to \( {H}^{\infty } \) is\n\n\[ \operatorname{dist}\left( {f,{H}^{\infty }}\right) = \mathop{\inf }\limits_{{g \in {H}^{\infty }}}\parallel f - g{\parallel }_{\infty } = \sup \left\{ {\left| {\int {fF}\frac{d\theta }{2\pi }}\right| : F \in {H}_{0}^{1},... | Proof. The dual expression for the distance follows from \( \left( {1.1}^{\prime }\right) \) . Just as in the proof of Theorem 1.2, a normal families argument shows there is a best approximating function \( g \in {H}^{\infty } \) . If a dual extremal function \( F \in {H}_{0}^{1} \) exists, then\n\n\[ \operatorname{dis... | Yes |
Example 1.4. Let \( f\left( 0\right) = {e}^{-{2i\theta }} \) . Taking \( {F}_{0}\left( \theta \right) = {e}^{2i\theta } \) we see that | \[ \frac{1}{2\pi }\int f{F}_{0}{d\theta } = 1 \] Hence \( \operatorname{dist}\left( {f,{H}^{\infty }}\right) = 1 \) and \( {F}_{0} \) is a dual extremal function. However \[ {F}_{\alpha }\left( z\right) = \frac{z\left( {z + \alpha }\right) \left( {1 + \bar{\alpha }z}\right) }{1 + {\left| \alpha \right| }^{2}},\;\left| ... | Yes |
Lemma 1.6. If \( f \in C \), then\n\n\[ \operatorname{dist}\left( {f,{H}^{\infty }}\right) = \operatorname{dist}\left( {f,{A}_{o}}\right) = \mathop{\inf }\limits_{{g \in {A}_{o}}}\parallel f - g{\parallel }_{\infty }.\] | Proof. There exists \( g \in {H}^{\infty } \) such that \( \parallel f - g{\parallel }_{\infty } = \operatorname{dist}\left( {f,{H}^{\infty }}\right) \) . Let \( {f}_{r} = \) \( f * {P}_{r} \) be the Poisson integral of \( f \) and let \( {g}_{r} = g * {P}_{r} \) . Since \( {\begin{Vmatrix}{P}_{r}\end{Vmatrix}}_{1} = 1... | Yes |
Theorem 1.7. If \( f \in {H}^{\infty } + C \), then there exists \( F \in {H}_{0}^{1},\parallel F{\parallel }_{1} = 1 \), such that\n\n(1.8)\n\n\[ \n\frac{1}{2\pi }\int {fFd\theta } = \operatorname{dist}\left( {f,{H}^{\infty }}\right) \n\]\n\nand there exists unique \( g \in {H}^{\infty } \) such that \( \parallel f - ... | Proof. Write \( f = g + h, g \in {H}^{\infty }, h \in C \) . Then \( \operatorname{dist}\left( {f,{H}^{\infty }}\right) = \operatorname{dist}\left( {h,{H}^{\infty }}\right) \) and we can assume that \( f \) is continuous. By Theorem 1.3 there are \( {F}_{n} \in \) \( {H}_{0}^{1},{\begin{Vmatrix}{F}_{n}\end{Vmatrix}}_{1... | Yes |
Corollary 1.9. Let \( {z}_{n + 1} \in D \) be distinct from \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \) . Assume (1.9) has a solution \( f \in {H}^{\infty } \) with \( \parallel f{\parallel }_{\infty } \leq 1 \) . Among such solutions let \( {f}_{0} \) he one for which \( \left| {f\left( {z}_{n + 1}\right) }\right| \) is lar... | Corollary 1.9 is a simple consequence of Corollary 1.8. By normal families there exists an extremal function \( {f}_{0},{\begin{Vmatrix}{f}_{0}\end{Vmatrix}}_{\infty } \leq 1 \) . Let \( {w}_{n + 1} = {f}_{0}\left( {z}_{n + 1}\right) \) . If \( f \in \) \( {H}^{\infty } \) interpolates (1.9) and if also\n\n\[ f\left( {... | Yes |
Lemma 1.10. If \( G \in {H}^{1} \) is real almost everywhere on an arc \( I \subset T \), then \( G \) extends analytically across \( I \) . | Proof. On \( \left| z\right| > 1 \) define \( G\left( z\right) = \overline{G\left( {1/z}\right) } \) . This is an \( {H}^{1} \) function on \( \left| z\right| > 1 \) with nontangential limits \( G\left( \theta \right) \) at almost every point of \( I \) . Let \( \zeta \in I \) and center at \( \zeta \) a disc \( \Delta... | Yes |
Lemma 2.3. Let \( G = u + {iv} \in {H}^{1} \) and let \( I \) be an arc on \( T \) such that almost everywhere on I\n\n\[ u > 0,\;\left| v\right| \leq {\alpha u},\]\n\nwhere \( \alpha > 0 \) . Let \( J \) be a relatively compact subarc of \( I \) and let \( V \) be the domain \( \left\{ {r{e}^{i\theta } : {r}_{0} < r <... | Proof. Recall from Corollary 2.5 in Chapter III that an \( {H}^{1} \) function whose boundary values lie in the sector \( S = \{ x > 0,\left| y\right| < {\alpha x}\} \) is in \( {H}^{p} \) if arctan \( \alpha < \pi /{2p} \) . Enlarging \( J \), we may assume \( G \) has a finite radial limit at the endpoints of \( J \)... | Yes |
Lemma 2.4. Let \( f\left( \theta \right) \in C \), let \( g\left( \theta \right) \in {H}^{\infty } \) and let \( F\left( \theta \right) \in {H}_{0}^{1} \) be functions such that (2.1) holds. Then\n\n(a) \( F \in {H}^{p} \) for all \( p < \infty \), and\n\n(b) if \( \tau \in \left\lbrack {0,{2\pi }}\right\rbrack \) and ... | Proof. To prove (a) let \( p < \infty \) and let \( \varepsilon > 0 \) satisfy \( \arctan \left( {\varepsilon /\left( {1 - \varepsilon }\right) }\right) < \) \( \pi /{2p} \) . Choose \( \delta \) so that \( \left| {f\left( \theta \right) - f\left( \tau \right) }\right| < \varepsilon \) when \( \left| {\theta - \tau }\r... | Yes |
Lemma 2.5. Let \( \delta > 0 \) . Let\n\n\[ f\left( {e}^{it}\right) = \left\{ \begin{matrix} \omega \left( t\right) , & 0 \leq t \leq \delta \\ 0, & - \delta \leq t < 0 \end{matrix}\right. \]\n\nExtend \( f\left( {e}^{it}\right) \) to be smooth on \( \delta < \left| t\right| < \pi \) and continuous on \( \left\lbrack {... | Proof. Since \( f \) is real, \( f \notin {H}^{\infty } \) and \( \parallel f - g{\parallel }_{\infty } > 0 \) . We may suppose \( \parallel f - g{\parallel }_{\infty } = 1 \), so that \( \left| {f - g}\right| = 1 \) on \( T \) . We must prove \( \operatorname{Re}g\left( 1\right) = 0 \) . Since \( \left| {f - g}\right|... | Yes |
Theorem 3.2 (Helson-Szegö). The subspaces \( \overline{\mathrm{F}} \) and \( \overline{\mathrm{G}} \) are at positive angle if and only if\n\n(3.2)\n\n\[{\mu }_{\mathrm{s}} = 0\]\n\nand\n\n(3.3)\n\n\[ \log w = u + \widetilde{v} \]\n\nwhere \( u \in {L}^{\infty }, v \in {L}^{\infty } \) and \( \parallel v{\parallel }_{\... | Proof. First let us show that (3.2) is necessary. Suppose \( d{\mu }_{\mathrm{s}} > 0 \) . By Exercise 2, Chapter III, there are \( {p}_{n} \in \mathrm{F} \) such that \( \left| {p}_{n}\right| \leq 1,{p}_{n} \rightarrow 1 \) almost everywhere \( d{\mu }_{\mathrm{s}} \), and \( {p}_{n} \rightarrow 0 \) almost everywhere... | No |
Lemma 3.3. If \( \psi \) is a real measurable function, then\n\n\[ \mathop{\inf }\limits_{{g \in {H}^{\infty }}}\begin{Vmatrix}{{e}^{-{i\psi }} - g}\end{Vmatrix} < 1 \]\n\nif and only if there are \( \varepsilon > 0 \) and \( h \in {H}^{\infty } \) such that\n\n(3.4)\n\n\[ \left| h\right| \geq \varepsilon \;\text{ a.e.... | Proof. Notice that if \( g \in {H}^{\infty } \) and \( {\begin{Vmatrix}{e}^{-{i\psi }} - g\end{Vmatrix}}_{\infty } < 1 \) then \( g \) satisfies (3.4) and (3.5). On the other hand, if (3.4) and (3.5) hold for \( h \), then for a small \( \lambda > 0,{\begin{Vmatrix}{e}^{-{i\psi }} - \lambda h\end{Vmatrix}}_{\infty } < ... | Yes |
Theorem 3.4. Let \( \mu \) be a positive finite measure on the circle. Then there is a constant \( K \) such that\n\n\[ \int {\left| \widetilde{p}\right| }^{2}{d\mu } \leq {K}^{2}\int \left| {p}^{2}\right| {d\mu } \]\n\nfor all trigonometric polynomials if and only if \( \mu \) is absolutely continuous, \( {d\mu } = {w... | Proof. We show the conjugation operator is bounded in \( {L}^{2}\left( \mu \right) \) if and only if the subspaces \( \overline{\mathrm{F}} \) and \( \overline{\mathrm{G}} \) are at positive angle in \( {L}^{2}\left( \mu \right) \) . With Theorem 3.2 that will prove the result.\n\nLet \( T \) be the operator defined on... | Yes |
Theorem 4.1. (Nevanlinna). If there are two distinct functions of unit norm in \( {H}^{\infty } \) that do the interpolation (4.1), then there is an inner function that also satisfies (4.1). | If \( {f}_{0} \) fulfills (4.1) and if \( \begin{Vmatrix}{f}_{0}\end{Vmatrix} < 1 \), then the two function hypothesis is trivially satisfied. Indeed, let \( B\left( z\right) \) be the Blaschke product with zeros \( \left\{ {z}_{j}\right\} \) . Then for some \( s > 0 \) and for some \( t < 0,{f}_{0} + {sB} \) and \( {f... | No |
Let \( \lim {z}_{j} = 1 \) . Let \( I \) be an open are on \( \mathrm{T} \) containing \( z = 1 \) . Let \( f \in {\left( {H}^{\infty }\right) }^{-1} \) have \( \parallel f\parallel = 1,\left| f\right| = 1 \) on \( I \) but \( \left| f\right| ≢ 1 \) . Let \( {w}_{j} = f\left( {z}_{j}\right) \) . Then every other interp... | So assume \( g \in {H}^{\infty } \) and \( \parallel f - {Gg}\parallel \leq 1 \) . Then\n\n\[ \left| {1 - {Bg}/f}\right| \leq 1 \]\n\nalmost everywhere on \( I \), so that\n\n\[ \operatorname{Re}{Bg}/f \geq 0 \]\n\nalmost everywhere on \( I \) . By Exercise 14, Chapter II, the inner factor of \( {Bg}/f \) is analytic a... | Yes |
Theorem 4.4. If \( \\begin{Vmatrix}{{h}_{0} + {H}^{\\infty }}\\end{Vmatrix} < 1 \), there is \( h \\in {h}_{0} + {H}^{\\infty } \) and there is \( F \\in \) \( {H}^{1}, F \\neq 0 \) such that\n\n(4.7)\n\n\[ h = \\bar{F}/\\left| F\\right| \]\n\nalmost everywhere. | Proof. Let \( h \) and \( \\left\{ {F}_{n}\\right\} \) be as in the above discussion. We claim the sequence \( \\left\{ {F}_{n}\\right\} \) has a subsequence converging weakly in \( {L}^{1} \) . If \( F \) is any weak limit point, then by (4.5) \( \\int {hFd\\theta }/{2\\pi } = 1 \) . Since \( \\parallel h\\parallel \\... | Yes |
Lemma 4.5. If \( \left\{ {E}_{k}\right\} \) is a sequence of measurable subsets of \( T \) such that \( \left| {E}_{k}\right| \rightarrow 0 \), then there is a sequence \( \left\{ {g}_{k}\right\} \) of functions in \( {H}^{\infty } \) such that\n\n(i) \( \mathop{\sup }\limits_{{E}_{k}}\left| {g}_{k}\right| \rightarrow ... | Proof of Lemma 4.5. Choose \( {A}_{k} \rightarrow \infty \) so slowly that \( {A}_{k}\left| {E}_{k}\right| \rightarrow 0 \) . Let \( {f}_{k} \) be the Poisson integral of\n\n\[ {A}_{k}{\chi }_{{E}_{k}} + i{A}_{k}{\widetilde{\chi }}_{{E}_{k}} \]\n\nThen \( {f}_{k} \) is an analytic function in \( D \) and \( {f}_{k} \) ... | Yes |
Lemma 4.6. If \( E \subset T \) is a set of positive measure, then there is \( g \in {H}^{\infty } \) such that \( g\left( 0\right) = 1 \) and such that \( g \) is real and negative on \( T \smallsetminus E \) . | Proof of Lemma 4.6. Let \( G \) be the outer function such that \( \left| G\right| = e \) on \( E,\left| G\right| = 1 \) on \( T \smallsetminus E \) . Then \( G\left( 0\right) = \exp \left| E\right| > 1 \) and \( G \) has values in the annulus \( \{ 1 < \left| w\right| < e\} \) . The function\n\n\[ \varphi \left( w\rig... | Yes |
Theorem 5.2. Let \( h \in {L}^{\infty },\left| h\right| = 1 \) almost everywhere, and assume \( {S}_{h} \) contains at least two distinct functions. Let \( {z}_{0} \in D \) . Then \( \left\{ {F\left( {z}_{0}\right) : F \in {S}_{h}}\right\} \) contains a disc centered at the origin. | Proof. When \( \left| {z}_{0}\right| \leq 1,\left| {z}_{1}\right| \leq 1 \) ,\n\n\[ \n\frac{\left( {z - {z}_{1}}\right) \left( {1 - {\bar{z}}_{1}z}\right) }{\left( {z - {z}_{0}}\right) \left( {1 - {\bar{z}}_{0}z}\right) } = \frac{\left( {z - {z}_{1}}\right) z\left( {\bar{z} - {\bar{z}}_{1}}\right) }{\left( {z - {z}_{0}... | Yes |
Lemma 5.4. There exists an outer function \( F \in {H}^{1},\parallel F{\parallel }_{1} = 1 \), such that\n\n\[ \n{h}_{0} = F/\left| F\right| \n\]\n\nalmost everywhere. | Proof. We know \( \left| {h}_{0}\right| = 1 \) almost everywhere and we know there is \( g \in \) \( {H}^{\infty }, g ≢ 0 \), such that \( {\begin{Vmatrix}{h}_{0} - g\end{Vmatrix}}_{\infty } \leq 1 \) . Then \( \left| {1 - {\bar{h}}_{0}g}\right| \leq 1 \) almost everywhere. Let \( \alpha = \arg {\bar{h}}_{0}g \) . Then... | Yes |
If \( w\left( z\right) \in {H}^{\infty },\parallel w{\parallel }_{\infty } \leq 1 \), then \( g\left( z\right) = \frac{F\left( z\right) \left( {1 - \chi \left( z\right) }\right) \left( {1 - w\left( z\right) }\right) }{1 - \chi \left( z\right) w\left( z\right) } \) is in \( {H}^{\infty } \) and \( {\begin{Vmatrix}\frac{... | Proof. Note that \( \operatorname{Re}\frac{1 + \chi \left( z\right) }{1 - \chi \left( z\right) } = \int {P}_{z}\left( \theta \right) \left| {F\left( \theta \right) }\right| \frac{d\theta }{2\pi } > 0. \) Let \( \left| {w\left( z\right) }\right| < 1 \) and set \( \varphi \left( z\right) = \frac{1 + \chi \left( z\right) ... | Yes |
Lemma 5.6. The function \( F \in {H}^{1},\parallel F{\parallel }_{1} = 1 \), such that\n\n\[ \n{h}_{0} = F/\left| F\right| \n\]\n\nis unique. | Proof. Lemma 5.4 shows there exists at least one such \( F \) . What we must show is that \( {\mathrm{S}}_{{h}_{0}} \) does not contain two functions. But if \( {\mathrm{S}}_{{h}_{0}} \) contains two functions, then by Theorem 5.2, there is \( {F}_{1} \in {S}_{{h}_{0}} \) with \( \operatorname{Re}{F}_{1}\left( 0\right)... | Yes |
Lemma 6.2. If \( {\mathrm{E}}_{n} \neq \varnothing \), then when \( \left| z\right| = 1,{\Delta }_{n}\left( z\right) = \bar{D},{\rho }_{n}\left( z\right) = 1 \), and | Proof. The conditions (6.13) follow from the assertion that \( {\Delta }_{n}\left( z\right) = \bar{D} \), by the well-known characterization of the coefficients of the linear fractional transformations\n\n\[ \zeta \rightarrow \frac{A + {B\zeta }}{C + {D\zeta }} \]\n\nthat map \( \bar{D} \) onto itself. That \( {\Delta ... | Yes |
Lemma 6.3. \( {C}_{n}\left( z\right) \) has no zeros in \( \left| z\right| \leq 1 \) . | Proof. If \( {C}_{n}\left( z\right) = 0 \), then by taking \( \zeta = 0 \) in (6.11), we see that \( {A}_{n}\left( z\right) = 0 \) . Then\n\n\[ \n{A}_{n}\left( z\right) {D}_{n}\left( z\right) - {B}_{n}\left( z\right) {C}_{n}\left( z\right) = 0 \n\] \n\nand by (6.12), \( z = {z}_{j} \) for some \( j = 1,2,\ldots, n \) .... | Yes |
Assume \( {\mathrm{E}}_{\infty } = \bigcap {\mathrm{E}}_{n} \) contains two functions with different values at \( z = 0 \) . Then there are \( {n}_{j} \rightarrow \infty \) such that the limits\n\n\[ P\left( z\right) = \mathop{\lim }\limits_{j}{P}_{{n}_{j}}\left( z\right) \]\n\n\[ Q\left( z\right) = \mathop{\lim }\limi... | Proof. The hypothesis that \( \bigcap {\mathrm{E}}_{n} \) contains two functions with different values at \( z = 0 \) implies that\n\n\[ \mathop{\lim }\limits_{n}{\rho }_{n}\left( 0\right) > 0 \]\n\n(This limit exists because \( {\rho }_{n + 1} \leq {\rho }_{n} \) since \( {\mathrm{E}}_{n + 1} \subset {\mathrm{E}}_{n} ... | Yes |
Theorem 1.1. Every complex homomorphism of \( A \) is a continuous linear functional with norm at most one, | Proof. Because \( m \) is linear, we only have to prove that \( \parallel m\parallel \leq 1 \) . If \( m \) is unbounded, or if \( \parallel m\parallel > 1 \), then there is \( f \in A \) such that \( \parallel f\parallel < 1 \) but such that \( m\left( f\right) = 1 \) . By (1.1) the series\n\n\[ \mathop{\sum }\limits_... | Yes |
Theorem 1.3. The Gelfand transform is an isometry if and only if\n\n\[ \begin{Vmatrix}{f}^{2}\end{Vmatrix} = \parallel f{\parallel }^{2} \]\n\nfor all \( f \in A \) . | Proof. Since \( \parallel \widehat{f}\parallel \) is a supremum, \( \begin{Vmatrix}{\widehat{f}}^{2}\end{Vmatrix} = \parallel \widehat{f}{\parallel }^{2} \) and (1.5) holds for any uniform algebra.\n\nNow assume (1.5). By Theorem 1.1, we have \( \parallel \widehat{f}\parallel \leq \parallel f\parallel \) . To complete ... | Yes |
Suppose \( A \) is any algebra of continuous complex functions on a compact Hausdorff space \( Y \) . If \( A \) has the uniform norm, \( \parallel f\parallel = \mathop{\sup }\limits_{{y \in Y}}\left| {f\left( y\right) }\right| \) and if \( A \) is complete, then \( A \) is a uniform algebra. If \( A \) contains the co... | This is the generic example, because any uniform algebra \( A \) is clearly a uniform algebra on its spectrum \( Y = {\mathfrak{M}}_{A} \) . If \( A = C\left( Y\right) \), then \( {\mathfrak{M}}_{A} = Y \) . (See Exercise 5.) | No |
The Gelfand transform of \( {l}^{\infty } \) is \( C\left( {\beta \mathbb{N}}\right) \). | To see this, note that if \( x \in {l}^{\infty } \) is real, that is, if \( {x}_{n} \in \mathbb{R} \) for all \( n \), then \( \widehat{x}\left( m\right) \) is real on \( {\mathfrak{M}}_{{l}^{\infty }} = \beta \mathbb{N} \), because then \( {\left( x - \lambda \right) }^{-1} \in {l}^{\infty } \) whenever \( \operatorna... | No |
Theorem 1.4. Let \( Y \) be a compact Hausdorff space and let \( \tau : \mathbb{N} \rightarrow Y \) be a continuous mapping. Then the mapping \( \tau \) has a unique continuous extension \( \widetilde{\tau } : \beta \mathbb{N} \rightarrow Y \) . | Proof. The mapping\n\n\[ T : C\left( Y\right) \rightarrow {l}^{\infty } \]\n\ndefined by \( {Tf}\left( n\right) = f \circ \tau \left( n\right) \), is a homomorphism from \( C\left( Y\right) \) into \( {l}^{\infty } \) . Because \( T \) is continuous, the adjoint mapping \( {T}^{ * } : {\left( {l}^{\infty }\right) }^{ *... | No |
The space \( {L}^{\infty } \) of essentially bounded, measurable functions on the unit circle is a uniform algebra when it is given the pointwise multiplication and the essential supremum norm | Under the Gelfand transform, \( {L}^{\infty } \) is isomorphic to \( C\left( X\right) \), the algebra of continuous complex functions on \( X \) . This has the same proof as the corresponding result on \( {l}^{\infty } \) . If \( f \in {L}^{\infty } \) is real, then \( {\left( f - \lambda \right) }^{-1} \in {L}^{\infty... | No |
Theorem 1.5. There is a smallest closed boundary \( {K}_{0} \), which is contained in every boundary \( K \) . | Proof. Let \( {K}_{0} \) be the intersection of all boundaries. We must show \( {K}_{0} \) is a boundary for \( A \) . | No |
Lemma 1.6. Let \( {f}_{1},{f}_{2},\ldots ,{f}_{n} \in A \) and set\n\n\[ U = \left\{ {m : \left| {{f}_{j}\left( m\right) }\right| < 1, j = 1,\ldots, n}\right\} .\n\]\n\nThen either \( U \cap K \neq \varnothing \) for every boundary \( K \), or else \( K \smallsetminus U \) is a boundary for every boundary \( K \) . | Proof of Lemma 1.6. We suppose that \( K \) is a boundary but that \( K \smallsetminus U \) is not a boundary, and we show that \( U \) intersects every boundary for \( A \) . By hypothesis there is \( f \in A \) such that \( \parallel f\parallel = 1 \) but such that \( \mathop{\sup }\limits_{{K \smallsetminus U}}\left... | Yes |
Example 5. \( {H}^{\infty } \) is a uniform algebra with pointwise multiplication and with the supremum norm | \[ \parallel f\parallel = \mathop{\sup }\limits_{{z \in D}}\left| {f\left( z\right) }\right| \] We shall always write \( \mathfrak{M} \) for the maximal ideal space of \( {H}^{\infty } \). For each point \( \zeta \in D \) there exists \( {m}_{\zeta } \in \mathfrak{M} \) such that \( {m}_{\zeta }\left( z\right) = \zeta ... | Yes |
Theorem 1.7. The Šilov boundary of \( {H}^{\infty } \) is \( X = {\mathfrak{M}}_{{L}^{\infty }} \) . | Every inner function has unit modulus on \( X \), because it has unit modulus when viewed as an element of \( {L}^{\infty } \) . So the singular function \( S\left( z\right) = \exp ((z + 1)/(z - 1)) satisfies \( \left| S\right| = 1 \) on \( X \) . Now \( S \notin {\left( {H}^{\infty }\right) }^{-1} \), but \( S \) has ... | No |
Theorem 1.8. The open disc \( D \) is dense in \( \mathfrak{M} \) if and only if the following condition holds: if \( {f}_{1},\ldots ,{f}_{n} \in {H}^{\infty } \) and if\n\n(1.6)\n\n\[ \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {{f}_{j}\left( z\right) }\right| \geq \delta > 0 \]\n\nfor all \( z \in D \), then ther... | Proof. Suppose \( D \) is dense in \( \mathfrak{M} \) . Then by continuity we have\n\n\[ \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {{f}_{j}\left( m\right) }\right| \geq \delta \]\n\nfor all \( m \in \mathfrak{M} \), so that \( \left\{ {{f}_{1},\ldots ,{f}_{n}}\right\} \) is contained in no proper ideal of \( {H}^... | Yes |
Theorem 2.1 (Douglas-Rudin). Suppose \( U \) is a unimodular function in \( {L}^{\infty },\left| {U\left( {e}^{i\theta }\right) }\right| = 1 \) almost everywhere. For any \( \varepsilon > 0 \) there exist inner functions \( {u}_{1},{u}_{2} \) in \( {H}^{\infty } \) such that\n\n\[ \begin{Vmatrix}{U - {u}_{1}/{u}_{2}}\e... | Proof of Theorem 2.1. Let \( E \) be a measurable subset of \( T \) with \( \left| E\right| > 0 \) . We may assume\n\n\[ U = \alpha {\chi }_{E} + \beta {\chi }_{T \smallsetminus E} \]\n\nwhere \( \left| \alpha \right| = \left| \beta \right| = 1,\alpha \neq \beta \), because the finite products of functions of this form... | Yes |
Theorem 2.3 (A. Bernard). Let \( A \) be a uniform algebra on a compact Haus-dorffspace \( Y \) and let\n\n\[ U = \{ u \in A : \left| u\right| = 1\\text{ on }Y\}\]\n\nbe the set of unimodular functions in A. If \( \\mathrm{U} \) generates \( A \), then the unit ball of \( A \) is the norm closed convex hull of \( \\mat... | Proof. Let \( f \in A,\\parallel f\\parallel < 1 \) . We can suppose \( f = \\mathop{\\sum }\\limits_{{j = 1}}^{n}{\\lambda }_{j}{u}_{j},{u}_{j} \in \\mathrm{U},{\\lambda }_{j} \in \) \( \\mathbb{C} \), because functions of this form are dense in ball \( \\left( A\\right) \) . Write \( u = \\mathop{\\prod }\\limits_{{j... | Yes |
Corollary 2.4. The unit ball of the disc algebra \( {A}_{0} \) is the closed convex hull of the set of finite Blaschke products. | Proof. The inner functions in \( {A}_{0} \) are the finite Blaschke products. They generate \( {A}_{0} \) because the polynomials are dense in \( {A}_{0} \) . | No |
Corollary 2.6. The unit ball of \( {H}^{\infty } \) is the norm closed convex hull of the set of Blaschke products. | The corollary follows immediately from Theorem 2.5, from Theorem 2.3, and from Frostman's theorem. Notice that the Carathéodory result for pointwise convergence (Chapter I, Theorem 2.1) is much easier than Corollary 2.6. | No |
Theorem 3.1. The mapping \( L : \mathrm{D} \rightarrow \mathfrak{M} \) has the following three properties:\n\n(a) \( L\left( \mathrm{D}\right) \subset {\mathfrak{M}}_{1} \), the fiber of \( \mathfrak{M} \) over \( z = 1 \) ;\n\n(b) \( L \) is an analytic mapping, that is, \( f \circ L\left( \zeta \right) \) is analytic... | Proof. Take \( f\left( z\right) = z \) in (3.2). Then\n\n\[ \widehat{z}\left( {L\left( \zeta \right) }\right) = \mathop{\lim }\limits_{j}\left( \frac{\zeta + {z}_{{n}_{j}}}{1 + {\bar{z}}_{{n}_{j}}\zeta }\right) = 1 \]\n\nsince \( \lim {z}_{n} = 1 \), and (a) holds. Property (b) follows from (3.2) and from the fact that... | Yes |
Theorem 4.1. Let \( A \) be a uniform algebra on a compact Hausdorff space \( Y \) and let \( m \in {\mathfrak{M}}_{A} \). Then\n\n\[ \parallel m\parallel = \mathop{\sup }\limits_{{\parallel f\parallel \leq 1}}\left| {m\left( f\right) }\right| = 1 \]\n\nThere exists a positive Borel measure \( \mu \) on \( Y \) such th... | Proof. We have \( \parallel m\parallel \leq 1 \) by Theorem 1.1. But since \( A \) is a uniform algebra we also have \( \parallel 1\parallel = 1 \), so that \( \parallel m\parallel \geq \left| {m\left( 1\right) }\right| = 1 \). Hence \( \parallel m\parallel = 1 \). By the Hahn-Banach theorem, the linear functional \( m... | Yes |
Theorem 4.2. Suppose \( A \) is a logmodular algebra on \( Y \) . Then each \( m \in {\mathfrak{M}}_{A} \) has unique representing measure on \( Y \) . | Proof. Suppose \( {\mu }_{1} \) and \( {\mu }_{2} \) are representing measures on \( Y \) for \( m \in {\mathfrak{M}}_{A} \) . Let \( f \in {A}^{-1} \) . Then since \( {\mu }_{1} \) and \( {\mu }_{2} \) are probability measures, we have\n\n\[ \left| {m\left( f\right) }\right| \leq \int \left| f\right| d{\mu }_{1},\;\le... | Yes |
Theorem 4.3. Let \( 0 < p \leq \infty \) . Then the correspondence\n\n\[{\chi }_{E} \rightarrow {\chi }_{\widetilde{E}}\]\nextends to a unique positive isometric linear operator \( S \) from \( {L}^{p}\left( {T,{d\theta }/{2\pi }}\right) \) onto \( {L}^{p}\left( {X,{\mu }_{0}}\right) \) . | Proof. For \( f = \sum {\alpha }_{j}{\chi }_{{E}_{j}} \) a simple function, we define \( {Sf} = \sum {\alpha }_{j}{\chi }_{{\widetilde{E}}_{j}} \) . This is the only possible definition of a linear extension of the mapping \( {\chi }_{E} \rightarrow {\chi }_{\widetilde{E}} \) . By (4.2), \( \parallel {Sf}{\parallel }_{... | Yes |
Lemma 4.5. Assume \( m \in {\mathfrak{M}}_{A} \) has unique representing measure \( \mu \) on \( Y \) . Let \( E \subset Y \) be an \( {F}_{\sigma } \) set such that \( \mu \left( E\right) = 0 \) . Then there are \( {f}_{n} \in A \) such that\n\n(i) \( \begin{Vmatrix}{f}_{n}\end{Vmatrix} \leq 1 \) ,\n\n(ii) \( {f}_{n}\... | Proof. We need a preliminary observation. If \( u \in {C}_{\mathbb{R}}\left( Y\right) \) then\n\n(4.3) \( \sup \{ \operatorname{Re}m\left( f\right) : f \in A,\operatorname{Re}f < u\} = \inf \{ \operatorname{Re}m\left( f\right) : f \in A,\operatorname{Re}f > u\} \)\n\n\[ = \int {ud\mu } \]\n\nTo establish (4.3), conside... | Yes |
Lemma 5.1. Let \( v \) be a finite complex Borel measure on \( X \) . Assume \( v \) is singular with respect to \( {\mu }_{0} \) . For any \( \varepsilon > 0 \) there is a peak set \( P \subset X \) such that \( {\mu }_{0}\left( P\right) = 0 \) but such that \( \left| v\right| \left( {X \smallsetminus P}\right) < \var... | Proof. By hypothesis, the total variation \( \left| v\right| \) is singular to \( {\mu }_{0} \) and there is a compact set \( K \subset X \) such that\n\n\[ \n{\mu }_{0}\left( K\right) = 0\;\text{ and }\;\left| v\right| \left( {X \smallsetminus K}\right) < \varepsilon .\n\]\n\nBecause the open-closed sets form a base f... | Yes |
Lemma 1.1. Let \( \varphi \in \mathrm{{BMO}} \) and let \( I \) and \( J \) be bounded intervals.\n\n(a) If \( I \subset J \) and \( \left| J\right| > 2\left| I\right| \), then\n\n\[ \left| {{\varphi }_{I} - {\varphi }_{J}}\right| \leq c\log \left( {\left| J\right| /\left| I\right| }\right) \parallel \varphi {\parallel... | Proof. For part (a) let\n\n\[ I = {I}_{1} \subset {I}_{2} \subset \cdots \subset {I}_{n} = J \]\n\nwhere \( \left| {I}_{k + 1}\right| \leq 2\left| {I}_{k}\right| \) and where \( n \leq c\log \left( {\left| J\right| /\left| I\right| }\right) \) . Then (1.3) gives\n\n\[ \left| {{\varphi }_{I} - {\varphi }_{J}}\right| \le... | Yes |
Theorem 1.2. Let \( \varphi \in {L}_{\text{loc }}^{1} \) . Then \( \varphi \in \mathrm{{BMO}} \) if and only if\n\n(1.4)\n\n\[ \int \frac{\left| \varphi \left( t\right) \right| }{1 + {t}^{2}}{dt} < \infty \]\n\nand\n\n(1.5)\n\n\[ \mathop{\sup }\limits_{{\operatorname{Im}z > 0}}\int \left| {\varphi \left( t\right) - \va... | Proof. Suppose \( \varphi \) satisfies (1.4) and (1.5). Let \( I \) be a bounded interval, and let \( z = x + {iy} \), where \( x \) is the center of \( I \) and \( y = \frac{1}{2}\left| I\right| \) . Then\n\n\[ \frac{{\chi }_{I}\left( t\right) }{\left| I\right| } \leq \pi {P}_{y}\left( {x - t}\right) \]\n\nand by (1.5... | Yes |
Corollary 1.3. Under the conformal mapping \[ z = i\frac{1 - w}{1 + w},\;\left| w\right| < 1 \] BMO ( \( \mathbb{R} \) ) and \( \operatorname{BMO}\left( T\right) \) are transformed into each other. The norms of \( \varphi \in \) \( \operatorname{BMO}\left( \mathbb{R}\right) \) and its image \( \psi \in \operatorname{BM... | Condition (1.6) also says that \( \operatorname{BMO}\left( T\right) \) has an equivalent norm invariant under Möbius transformations. | No |
Corollary 1.4. Let \( \psi \in {L}^{1}\left( T\right) \) and let \( \tau \left( z\right) \) be a Möbius transformation. Then \( \psi \in \mathrm{{BMO}}\left( T\right) \) if and only if \( \psi \circ \tau \in \mathrm{{BMO}}\left( T\right) \) . There is a constant \( C \) independent of \( \tau \) such that \[ \parallel ... | Proof. Let \[ \parallel \psi {\parallel }_{ * }^{\prime } = \mathop{\sup }\limits_{{z \in D}}\int \left| {\psi - \psi \left( z\right) }\right| {P}_{z}\left( \theta \right) {d\theta }/{2\pi }. \] By (1.6) and (1.7), \( \parallel \psi {\parallel }_{ * }^{\prime } \) is an equivalent norm on \( \operatorname{BMO}\left( T\... | Yes |
Theorem 1.5. If \( \varphi \in {L}^{\infty } \), then the conjugate function \( \widetilde{\varphi } \) is in BMO, and\n\n\[ \parallel \widetilde{\varphi }{\parallel }_{ * } \leq C\parallel \varphi {\parallel }_{\infty } \]\n\nfor some universal constant \( C \) . | Proof. Because of Corollary 1.3 it makes no difference whether we prove the theorem on the line or on the circle. On the circle the proof is quite transparent if we use Corollary 1.4. Let \( \tau \) be any Möbius transformation. The normalization \( \widetilde{\psi }\left( 0\right) = 0 \) in the definition of the conju... | Yes |
Theorem 1.6. If \( \sigma \) is a finite measure on the upper half plane and if \( \left| \sigma \right| \) is a Carleson measure, then \( {S\sigma } \in \mathrm{{BMO}} \) and\n\n\[ \n\parallel {S\sigma }{\parallel }_{ * } \leq {CN}\left( \sigma \right)\n\]\n\nwhere \( N\left( \sigma \right) \) is the constant in (1.10... | Proof. The argument is very similar to the real-variables proof of Theorem 1.5. Writing \( \sigma = {\sigma }^{ + } - {\sigma }^{ - } \), where \( \left| \sigma \right| = {\sigma }^{ + } + {\sigma }^{ - },{\sigma }^{ + } \geq 0,{\sigma }^{ - } \geq 0 \), we can suppose \( \sigma \geq 0 \) . Let \( {I}_{0} \) be a fixed... | Yes |
Lemma 2.2. Let \( I \) be a bounded interval, let \( u \in {L}^{1}\left( I\right) \) and let\n\n\[ \alpha > \frac{1}{\left| I\right| }{\int }_{I}\left| u\right| {dt} \]\n\nThen there is a finite or infinite sequence \( \left\{ {I}_{j}\right\} \) of pairwise disjoint open subintervals of \( I \) such that\n\n(2.2)\n\n\[... | Proof. We may assume \( I = \left( {0,1}\right) \) . Partition \( I \) into two intervals \( {\omega }_{0} = \left( {0,\frac{1}{2}}\right) \) and \( {\omega }_{1} = \left( {\frac{1}{2},1}\right) \) . For each interval \( \omega \) there are two cases.\n\n\[ \text{Case(i):}\frac{1}{\left| \omega \right| }{\int }_{\omega... | Yes |
Let \( \varphi \in {L}_{\mathrm{{Ioc}}}^{1}\left( \mathbb{R}\right) \) . If\n\n\[ \mathop{\sup }\limits_{I}\frac{1}{\left| I\right| }{\int }_{I}\left| {\varphi - {\varphi }_{I}}\right| {dt} = \parallel \varphi {\parallel }_{ * } < \infty ,\]\n\nthen for any finite \( p > 1 \),\n\n\[ \mathop{\sup }\limits_{I}{\left( \fr... | By hypothesis, \( \varphi \in \mathrm{{BMO}} \) and we have (2.1) at our disposal. Fix \( I \) and write\n\n\[ m\left( \lambda \right) = \frac{\left| \left\{ t \in I : \left| \varphi \left( t\right) - {\varphi }_{I}\right| > \lambda \right\} \right| }{\left| I\right| }\]\n\nfor the distribution function of \( \left| {\... | Yes |
Corollary 2.4. Let \( \varphi \in {L}_{\mathrm{{loc}}}^{1}\left( \mathbb{R}\right) \) . Then \( \varphi \in \mathrm{{BMO}} \) if and only if\n\n\[ \int \frac{{\left| \varphi \left( t\right) \right| }^{2}}{1 + {t}^{2}}{dt} < \infty \]\n\nand\n\n(2.7)\n\n\[ \mathop{\sup }\limits_{{\operatorname{Im}z > 0}}\int {\left| \va... | Proof. If the Poisson integral of \( {\left| \varphi \right| }^{2} \) converges, and if (2.7) holds, then Hölder’s inequality and Theorem 1.2 show that \( \varphi \in \) BMO with\n\n\[ \parallel \varphi {\parallel }_{ * } \leq {c}_{2}{B}_{2}^{1/2} \]\n\nThe proof of the converse resembles the proof of Theorem 1.2. Supp... | Yes |
Corollary 2.5. If \( \varphi \in \mathrm{{BMO}} \), then the conjugate function \( \widetilde{\varphi } \in \mathrm{{BMO}} \), and\n\n\[ \n{c}_{1}\parallel \varphi {\parallel }_{ * } \leq \parallel \widetilde{\varphi }{\parallel }_{ * } \leq {c}_{2}\parallel \varphi {\parallel }_{ * }\n\]\n\nfor some constants \( {c}_{... | Proof. By Corollary 2.4 it is enough to prove\n\n\[ \n\int {\left| \widetilde{\varphi } - \widetilde{\varphi }\left( z\right) \right| }^{2}{P}_{z}\left( t\right) {dt} = \int {\left| \varphi - \varphi \left( z\right) \right| }^{2}{P}_{z}\left( t\right) {dt}\n\]\n\nfor any fixed \( z \in \mathrm{H} \) . This reduces to t... | Yes |
Lemma 3.1. If \( g\left( {e}^{it}\right) \in {L}^{1}\left( T\right) \), and if \( g\left( 0\right) = \left( {1/{2\pi }}\right) \int g\left( \theta \right) {d\theta } \) is its mean value, then\n\n\[ \frac{1}{\pi }{\iint }_{D}{\left| \nabla g\left( z\right) \right| }^{2}\log \frac{1}{\left| z\right| }{dxdy} = \frac{1}{2... | Proof. We may assume that \( g\left( 0\right) = 0 \) . Notice that\n\n\[ 2{\left| \nabla g\left( z\right) \right| }^{2} = \Delta \left( {\left| g\left( z\right) \right| }^{2}\right) .\n\nFor \( r < 1 \), Green’s theorem, with \( u\left( z\right) = {\left| g\left( z\right) \right| }^{2} \) and \( v\left( z\right) = \log... | Yes |
Lemma 3.2. If \( g\left( {e}^{i\theta }\right) \in {L}^{1}\left( T\right) \), then\n\n(3.2) \( \frac{1}{\pi }{\iint }_{D}{\left| \nabla g\left( z\right) \right| }^{2}\left( {1 - {\left| z\right| }^{2}}\right) {dxdy} \leq \frac{1}{\pi }\int {\left| g\left( {e}^{i\theta }\right) - g\left( 0\right) \right| }^{2}{d\theta }... | Proof. The leftmost inequality in (3.2) follows from (3.1) and the simple fact that \( 1 - {\left| z\right| }^{2} \leq 2\log \left( {1/\left| z\right| }\right) ,\left| z\right| < 1 \) . To prove the other inequality, suppose that the integral on the right is finite and normalize \( g\left( z\right) \) so that\n\n\[ \fr... | Yes |
Lemma 3.3. A positive measure \( \lambda \) on the disc is a Carleson measure if and only if\n\n(3.6)\n\n\[ \mathop{\sup }\limits_{{{z}_{0} \in D}}\int \frac{1 - {\left| {z}_{0}\right| }^{2}}{{\left| 1 - {\bar{z}}_{0}z\right| }^{2}}{d\lambda }\left( z\right) = M < \infty . \] | Proof. Suppose (3.6) holds, and let \( S \) be any sector, \( S = \{ 1 - h \leq r < 1, \mid \theta - \) \( \left. {\left. {\theta }_{0}\right| \; < h}\right\} \) . Since (3.6) with \( {z}_{0} = 0 \) shows that \( \lambda \left( D\right) \leq M \), we can suppose \( h < \frac{1}{4} \) .\n\nTake \( {z}_{0} = \left( {1 - ... | Yes |
Theorem 3.4. Let \( \varphi \in {L}^{1}\left( T\right) \), and let\n\n\[ d{\lambda }_{\varphi } = {\left| \nabla \varphi \left( z\right) \right| }^{2}\log \frac{1}{\left| z\right| }{dxdy} \]\n\nwhere \( \nabla \varphi \) is the gradient of the Poisson integral \( \varphi \left( z\right) \). Then \( \varphi \in \operato... | Proof. By Corollary 2.4 and by the inequalities (3.4), we know that \( \varphi \in \) \( \operatorname{BMO}\left( T\right) \) if and only if\n\n(3.9)\n\n\[ \mathop{\sup }\limits_{{{z}_{0} \in D}}{\iint }_{D}{\left| \nabla \varphi \left( z\right) \right| }^{2}\frac{\left( {1 - {\left| z\right| }^{2}}\right) \left( {1 - ... | Yes |
Lemma 4.1. If \( 1 \leq p < \infty \), if \( q = p/\left( {p - 1}\right) \), and if \( g \in {L}^{q} \), then \( g \in \) \( {\left( {H}^{p}\right) }^{ \bot } \), that is,\n\n\[\n\int {fgdt} = 0\n\]\n\nfor all \( f \in {H}^{p} \) if and only if \( g \in {H}^{q} \) . | Proof. If \( g \in {H}^{q} \), then by Hölder’s inequality \( {fg} \in {H}^{1} \), and (4.1) follows from Lemma II.3.7.\n\nOn the other hand if (4.1) holds then the Poisson integral of \( g \) is analytic on \( \mathrm{H} \) . Indeed, if \( z \in \mathrm{H} \) and if \( {z}_{0} \in \mathrm{H} \) is fixed, then\n\n\[\n{... | Yes |
Theorem 4.2. For \( 1 < p < \infty ,{\left( {H}^{p}\right) }^{ * } \) is isomorphic to \( {\bar{H}}^{q} \), the space of complex conjugates of functions in \( {H}^{q} \) . The isomorphism \( T : {\bar{H}}^{q} \rightarrow {\left( {H}^{p}\right) }^{ * } \) is defined by\n\n\[ \left( {Tg}\right) \left( f\right) = \int {fg... | Proof. Since \( 1 < q < \infty \), the Hilbert transform \( H \) is bounded on \( {L}^{q} \) . The operator\n\n\[ S\left( g\right) = g - {iHg} \]\n\n is then also bounded on \( {L}^{q} \) . The kernel of \( S \) is \( {H}^{q} \) and the range of \( S \) is \( {\bar{H}}^{q} \) . By the open mapping theorem, the induced ... | Yes |
Lemma 4.3. Let \( L \) be a continuous real linear functional on \( {H}_{\mathbb{R}}^{1} \) . Then there are \( {\varphi }_{1} \) and \( {\varphi }_{2} \) in \( {L}_{\mathbb{R}}^{\infty } \) such that\n\n\[ \n{\begin{Vmatrix}{\varphi }_{1}\end{Vmatrix}}_{\infty } \leq \parallel L\parallel ,\;{\begin{Vmatrix}{\varphi }_... | Proof. The space \( {H}_{\mathbb{R}}^{1} \) is a closed subspace of \( {L}_{\mathbb{R}}^{1} \oplus {L}_{\mathbb{R}}^{1} \) when this latter space is given the norm \( \parallel \left( {u, v}\right) \parallel = \parallel u{\parallel }_{1} + \parallel v{\parallel }_{1} \) . Extend \( L \) to a bounded real linear functio... | Yes |
Theorem 4.4 (C. Fefferman). The dual space of \( {H}_{\mathbb{R}}^{1} \) is BMO. More precisely, if \( \varphi \in \mathrm{{BMO}} \) is real valued and if \( L \) is defined by\n\n(4.8)\n\n\[ L\left( u\right) = \int {u\varphi dt},\;u \in \mathfrak{A}, \]\n\nthen \( \left| {L\left( u\right) }\right| \leq {C}_{1}\paralle... | Proof. By Lemma 4.3, each \( L \in {\left( {H}_{\mathbb{R}}^{1}\right) }^{ * } \) yields a unique \( \varphi \in \) BMO with \( \parallel \varphi {\parallel }_{ * } \leq C\parallel L\parallel \) such that (4.8) holds. The important thing that remains to be proved is that\n\n(4.9)\n\n\[ \left| {\int {u\varphi dt}}\right... | Yes |
Corollary 4.5. If \( \varphi \in {L}_{\mathrm{{loc}}}^{1} \), then \( \varphi \in \mathrm{{BMO}} \) if and only if\n\n(4.11)\n\n\[ \varphi = {\varphi }_{1} + H{\varphi }_{2} + \alpha \]\n\nwhere \( \alpha \) is a constant and where \( {\varphi }_{1} \) and \( {\varphi }_{2} \) are \( {L}^{\infty } \) functions. When \(... | Proof. This corollary is equivalent to the theorem. Theorem 1.5 tells us that every function of the form (4.11) is in BMO that\n\n(4.13)\n\n\[ \parallel \varphi {\parallel }_{ * } \leq C\left( {{\begin{Vmatrix}{\varphi }_{1}\end{Vmatrix}}_{\infty } + {\begin{Vmatrix}{\varphi }_{2}\end{Vmatrix}}_{\infty }}\right) \]\n\n... | Yes |
Corollary 4.6. Let \( f \in {L}^{\infty } \) . Then the distance\n\n\[ \operatorname{dist}\left( {f,{H}^{\infty }}\right) = \mathop{\inf }\limits_{{g \in {H}^{\infty }}}\parallel f - g{\parallel }_{\infty } \]\n\nsatisfies\n\n(4.14)\n\n\[ {C}_{1}\parallel f - {iHf}{\parallel }_{ * } \leq \operatorname{dist}\left( {f,{H... | Proof. By (4.2) and the Hahn-Banach theorem, we have\n\n\[ \operatorname{dist}\left( {f,{H}^{\infty }}\right) = \sup \left\{ {\left| {\int {fFdt}}\right| : F \in {H}^{1},\parallel F{\parallel }_{1} \leq 1}\right\} .\n\nBy density, we may suppose \( F = u + {iHu} \), where \( u \in \mathfrak{A} \) . Then,\n\n\[ \int {fF... | Yes |
Corollary 4.7. Let \( f \in {L}^{\infty } \) be real valued. Then the distances\n\n\[ \n\operatorname{dist}\left( {f,\operatorname{Re}{H}^{\infty }}\right) = \mathop{\inf }\limits_{{g \in {H}^{\infty }}}\parallel f - \operatorname{Re}g{\parallel }_{\infty }\n\]\n\nand\n\n\[ \n{\operatorname{dist}}_{ * }\left( {{Hf},{L}... | Proof. If \( g \in {H}^{\infty } \), then \( \operatorname{Im}g \in {L}^{\infty } \) and \( {Hf} - \operatorname{Im}g = H\left( {f - \operatorname{Re}g}\right) \), so that by Theorem 1.5,\n\n\[ \n\parallel {Hf} - \operatorname{Im}g{\parallel }_{ * } \leq {C}_{2}\parallel f - \operatorname{Re}g{\parallel }_{ * }.\n\]\n\... | Yes |
Lemma 6.3. If \( 1 < p < \infty \) and if \( \mu \) is a positive measure on \( \mathbb{R} \) such that\n\n\[ \int {\left| Mf\right| }^{p}{d\mu } \leq C\int {\left| f\right| }^{p}{d\mu } \]\n\nfor all \( f \), then \( \mu \) is absolutely continuous, \( {d\mu } = w\left( x\right) {dx} \), and \( w\left( x\right) \) sat... | Proof. Let \( E \) be a compact set with \( \left| E\right| = 0 \), let \( \varepsilon > 0 \), and let \( V \) be an open neighborhood of \( E \) with \( \mu \left( {V \smallsetminus E}\right) < \varepsilon \) . Then \( f = {\chi }_{V \smallsetminus E} \) has \( \int {\left| f\right| }^{p}{d\mu } < \varepsilon \) . On ... | Yes |
Lemma 6.4. If \( 1 < p < \infty \) and if \( \mu \) is a positive measure on \( \mathbb{R} \), finite on compact sets, such that for all \( f \)\n\n\[ \int {\left| Hf\right| }^{p}{d\mu } \leq C\int {\left| f\right| }^{p}{d\mu } \]\n\nthen \( {d\mu } = w\left( x\right) {dx} \), where \( w\left( x\right) \) satisfies the... | Proof. Testing the hypothesis of the lemma with \( f = {\chi }_{\left\lbrack 0,1\right\rbrack } \), we see that\n\n(6.2)\n\n\[ \int \frac{{d\mu }\left( x\right) }{{\left( 1 + \left| x\right| \right) }^{p}} < \infty \]\n\nLet \( g \in {L}^{q}\left( \mu \right), q = p/\left( {p - 1}\right) \), be real. Then by the hypoth... | Yes |
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