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Proposition 2.2 Suppose \( f \in {L}^{\infty } \) is supported on a set of finite measure. Then \( f \in {L}^{p} \) for all \( p < \infty \), and\n\n\[ \parallel f{\parallel }_{{L}^{p}} \rightarrow \parallel f{\parallel }_{{L}^{\infty }}\;\text{ as }p \rightarrow \infty . \] | Proof. Let \( E \) be a measurable subset of \( X \) with \( \mu \left( E\right) < \infty \), and so that \( f \) vanishes in the complement of \( E \) . If \( \mu \left( E\right) = 0 \), then \( \parallel f{\parallel }_{{L}^{\infty }} = \) \( \parallel f{\parallel }_{{L}^{p}} = 0 \) and there is nothing to prove. Othe... | Yes |
Proposition 3.1 A linear functional on a Banach space is continuous, if and only if it is bounded. | Proof. The key is to observe that \( \ell \) is continuous if and only if \( \ell \) is continuous at the origin.\n\nIndeed, if \( \ell \) is continuous, we choose \( \epsilon = 1 \) and \( g = 0 \) in the above definition so that \( \left| {\ell \left( f\right) }\right| \leq 1 \) whenever \( \parallel f\parallel \leq ... | Yes |
Theorem 3.2 The vector space \( {\mathcal{B}}^{ * } \) is a Banach space. | Proof. It is clear that \( \parallel \cdot \parallel \) defines a norm, so we only check that \( {\mathcal{B}}^{ * } \) is complete. Suppose that \( \left\{ {\ell }_{n}\right\} \) is a Cauchy sequence in \( {\mathcal{B}}^{ * } \) . Then, for each \( f \in \mathcal{B} \), the sequence \( \left\{ {{\ell }_{n}\left( f\rig... | Yes |
Theorem 4.1 Suppose \( 1 \leq p < \infty \), and \( 1/p + 1/q = 1 \) . Then, with \( \mathcal{B} = \) \( {L}^{p} \) we have\n\n\[{\mathcal{B}}^{ * } = {L}^{q}\]\n\nin the following sense: For every bounded linear functional \( \ell \) on \( {L}^{p} \) there is a unique \( g \in {L}^{q} \) so that\n\n\[ \ell \left( f\ri... | The proof of the theorem is based on two ideas. The first, as already seen, is Hölder's inequality; to which a converse is also needed. The second is the fact that a linear functional \( \ell \) on \( {L}^{p},1 \leq p < \infty \), leads naturally to a (signed) measure \( \nu \) . Because of the continuity of \( \ell \)... | No |
Lemma 4.2 Suppose \( 1 \leq p, q \leq \infty \), are conjugate exponents.\n\n(i) If \( g \in {L}^{q} \), then \( \parallel g{\parallel }_{{L}^{q}} = \mathop{\sup }\limits_{{\parallel f{\parallel }_{{L}^{p}} \leq 1}}\left| {\int {fg}}\right| \) . | Proof. We start with (i). If \( g = 0 \), there is nothing to prove, so we may assume that \( g \) is not 0 a.e., and hence \( \parallel g{\parallel }_{{L}^{q}} \neq 0 \) . By Hölder’s inequality, we have that\n\n\[ \parallel g{\parallel }_{{L}^{q}} \geq \mathop{\sup }\limits_{{\parallel f{\parallel }_{{L}^{p}} \leq 1}... | Yes |
Theorem 5.2 Suppose \( {V}_{0} \) is a linear subspace of \( V \), and that we are given a linear functional \( {\ell }_{0} \) on \( {V}_{0} \) that satisfies\n\n\[{\ell }_{0}\left( v\right) \leq p\left( v\right) ,\;\text{ for all }v \in {V}_{0}.\n\]\nThen \( {\ell }_{0} \) can be extended to a linear functional \( \el... | Proof. Suppose \( {V}_{0} \neq V \), and pick \( {v}_{1} \) a vector not in \( {V}_{0} \) . We will first extend \( {\ell }_{0} \) to the subspace \( {V}_{1} \) spanned by \( {V}_{0} \) and \( {v}_{1} \), as we did before. We can do this by defining a putative extension \( {\ell }_{1} \) of \( {\ell }_{0} \), defined o... | No |
Proposition 5.3 Suppose \( {f}_{0} \) is a given element of \( \mathcal{B} \) with \( \begin{Vmatrix}{f}_{0}\end{Vmatrix} = M \) . Then there exists a continuous linear functional \( \ell \) on \( \mathcal{B} \) so that \( \ell \left( {f}_{0}\right) = M \) and \( \parallel \ell {\parallel }_{{\mathcal{B}}^{ * }} = 1 \)... | Proof. Define \( {\ell }_{0} \) on the one-dimensional subspace \( {\left\{ \alpha {f}_{0}\right\} }_{\alpha \in \mathbb{R}} \) by \( {\ell }_{0}\left( {\alpha {f}_{0}}\right) = {\alpha M} \), for each \( \alpha \in \mathbb{R} \) . Note that if we set \( p\left( f\right) = \parallel f\parallel \) for every \( f \in \ma... | Yes |
Proposition 5.4 Let \( {\mathcal{B}}_{1},{\mathcal{B}}_{2} \) be a pair of Banach spaces and \( \mathcal{S} \subset {\mathcal{B}}_{1} \) a dense linear subspace of \( {\mathcal{B}}_{1} \). Suppose \( {T}_{0} \) is a linear transformation from \( \mathcal{S} \) to \( {\mathcal{B}}_{2} \) that satisfies \( {\begin{Vmatri... | Proof. If \( f \in {\mathcal{B}}_{1} \), let \( \left\{ {f}_{n}\right\} \) be a sequence in \( \mathcal{S} \) which converges to \( f \). Then since \( {\begin{Vmatrix}{T}_{0}\left( {f}_{n}\right) - {T}_{0}\left( {f}_{m}\right) \end{Vmatrix}}_{{\mathcal{B}}_{2}} \leq M{\begin{Vmatrix}{f}_{n} - {f}_{m}\end{Vmatrix}}_{{\... | Yes |
Theorem 5.5 The operator \( {T}^{ * } \) defined by (13) is a bounded linear transformation from \( {\mathcal{B}}_{2}^{ * } \) to \( {\mathcal{B}}_{1}^{ * } \) . Its norm \( \begin{Vmatrix}{T}^{ * }\end{Vmatrix} \) satisfies \( \parallel T\parallel = \begin{Vmatrix}{T}^{ * }\end{Vmatrix} \) . | Proof. First, if \( {\begin{Vmatrix}{f}_{1}\end{Vmatrix}}_{{\mathcal{B}}_{1}} \leq 1 \), we have that\n\n\[ \left| {{\ell }_{1}\left( {f}_{1}\right) }\right| = \left| {{\ell }_{2}\left( {T\left( {f}_{1}\right) }\right) }\right| \leq \begin{Vmatrix}{\ell }_{2}\end{Vmatrix}{\begin{Vmatrix}T\left( {f}_{1}\right) \end{Vmat... | Yes |
Theorem 5.6 There is an extended-valued non-negative function \( \widehat{m} \), defined on all subsets of \( \mathbb{R} \) with the following properties:\n\n(i) \( \widehat{m}\left( {{E}_{1} \cup {E}_{2}}\right) = \widehat{m}\left( {E}_{1}\right) + \widehat{m}\left( {E}_{2}\right) \) whenever \( {E}_{1} \) and \( {E}_... | From (i) we see that \( \widehat{m} \) is finitely additive; however it cannot be countably additive as the proof of the existence of non-measurable sets shows. (See Section 3, Chapter 1 in Book III.) | No |
Corollary 5.8 There is a non-negative function \( \widehat{m} \) defined on all subsets of \( \mathbb{R}/\mathbb{Z} \) so that:\n\n(i) \( \widehat{m}\left( {{E}_{1} \cup {E}_{2}}\right) = \widehat{m}\left( {E}_{1}\right) + \widehat{m}\left( {E}_{2}\right) \) for all disjoint subsets \( {E}_{1} \) and \( {E}_{2} \) .\n\... | We need only take \( \widehat{m}\left( E\right) = I\left( {\chi }_{E}\right) \), with \( I \) as in Theorem 5.7, where \( {\chi }_{E} \) denotes the characteristic function of \( E \) . | Yes |
Theorem 7.3 Let \( X \) be a compact metric space and \( C\left( X\right) \) the Banach space of continuous real-valued functions on \( X \) . Then, given any bounded linear functional \( \ell \) on \( C\left( X\right) \), there exists a unique finite signed Borel measure \( \mu \) on \( X \) so that\n\n\[ \ell \left( ... | Proof. By the proposition, there exist two positive linear functionals \( {\ell }^{ + } \) and \( {\ell }^{ - } \) so that \( \ell = {\ell }^{ + } - {\ell }^{ - } \) . Applying Theorem 7.1 to each of these positive linear functionals yields two finite Borel measures \( {\mu }_{1} \) and \( {\mu }_{2} \) . If we define ... | Yes |
Theorem 7.4 Suppose \( X \) is a metric space and \( \ell \) a positive linear functional on \( {C}_{b}\left( X\right) \) . For simplicity assume that \( \ell \) is normalized so that \( \ell \left( 1\right) = 1 \) . Assume also that for each \( \epsilon > 0 \), there is a compact set \( {K}_{\epsilon } \subset X \) so... | The proof of this theorem proceeds as that of Theorem 7.1, save for one key aspect. First we define\n\n\[ \rho \left( \mathcal{O}\right) = \sup \left\{ {\ell \left( f\right) ,\text{ where }f \in {C}_{b}\left( X\right) ,\operatorname{supp}\left( f\right) \subset \mathcal{O}\text{, and }0 \leq f \leq 1}\right\} .\n\]\n\n... | Yes |
Theorem 2.1 Suppose \( T \) is a linear mapping from \( {L}^{{p}_{0}} + {L}^{{p}_{1}} \) to \( {L}^{{q}_{0}} + \) \( {L}^{{q}_{1}} \) . Assume that \( T \) is bounded from \( {L}^{{p}_{0}} \) to \( {L}^{{q}_{0}} \) and from \( {L}^{{p}_{1}} \) to \( {L}^{{q}_{1}} \n\n\[ \left\{ \begin{array}{l} \parallel T\left( f\righ... | Null | No |
Lemma 2.2 (Three-lines lemma) Suppose \( \Phi \left( z\right) \) is a holomorphic function in the strip \( S = \{ z \in \mathbb{C} : 0 < \operatorname{Re}\left( z\right) < 1\} \), that is also continuous and bounded on the closure of \( S \) . If\n\n\[ \n{M}_{0} = \mathop{\sup }\limits_{{y \in \mathbb{R}}}\left| {\Phi ... | Proof. We begin by proving the lemma under the assumption that \( {M}_{0} = {M}_{1} = 1 \) and \( \mathop{\sup }\limits_{{0 \leq x \leq 1}}\left| {\Phi \left( {x + {iy}}\right) }\right| \rightarrow 0 \) as \( \left| y\right| \rightarrow \infty \) . In this case, let \( M = \sup \left| {\Phi \left( z\right) }\right| \) ... | Yes |
Corollary 2.3 With \( T \) as before:\n\n(a) The Riesz diagram of \( T \) is a convex set.\n\n(b) \( \log {M}_{x, y} \) is a convex function on this set. | Conclusion (a) means that if \( \left( {{x}_{0},{y}_{0}}\right) = \left( {1/{p}_{0},1/{q}_{0}}\right) \) and \( \left( {{x}_{1},{y}_{1}}\right) = \) \( \left( {1/{p}_{1},1/{q}_{1}}\right) \) are points in the Riesz diagram of \( T \), then so is the line segment joining them. This is an immediate consequence of Theorem... | Yes |
Corollary 2.4 If \( 1 \leq p \leq 2 \) and \( 1/p + 1/q = 1 \), then\n\n\[ \parallel T\left( f\right) {\parallel }_{{L}^{q}\left( \mathbb{Z}\right) } \leq \parallel f{\parallel }_{{L}^{p}\left( \left\lbrack {0,{2\pi }}\right\rbrack \right) }.\] | Note that since \( {L}^{2}\left( \left\lbrack {0,{2\pi }}\right\rbrack \right) \subset {L}^{1}\left( \left\lbrack {0,{2\pi }}\right\rbrack \right) \) and \( {L}^{2}\left( \mathbb{Z}\right) \subset {L}^{\infty }\left( \mathbb{Z}\right) \) we have \( {L}^{2}\left( \left\lbrack {0,{2\pi }}\right\rbrack \right) + {L}^{1}\l... | Yes |
Corollary 2.5 If \( 1 \leq p \leq 2 \) and \( 1/p + 1/q = 1 \), then\n\n\[{\begin{Vmatrix}{T}^{\prime }\left( \left\{ {a}_{n}\right\} \right) \end{Vmatrix}}_{{L}^{q}\left( \left\lbrack {0,{2\pi }}\right\rbrack \right) } \leq {\begin{Vmatrix}\left\{ {a}_{n}\right\} \end{Vmatrix}}_{{L}^{p}\left( \mathbb{Z}\right) }.\] | The proof is parallel to that of the previous corollary. The case \( {p}_{0} = \) \( {q}_{0} = 2 \) is, as has already been mentioned, a consequence of Parseval’s identity, while the case \( {p}_{1} = 1 \) and \( {q}_{1} = \infty \) follows directly from the fact that\n\n\[ \left| {\mathop{\sum }\limits_{{n = - \infty ... | Yes |
Corollary 2.6 If \( 1 \leq p \leq 2 \) and \( 1/p + 1/q = 1 \), then the Fourier transform \( T \) has a unique extension to a bounded map from \( {L}^{p} \) to \( {L}^{q} \), with \( \parallel T\left( f\right) {\parallel }_{{L}^{q}} \leq \parallel f{\parallel }_{{L}^{p}} \) | Null | No |
Theorem 3.2 Suppose \( 1 < p < \infty \) . Then the Hilbert transform \( H \), initially defined on \( {L}^{2} \cap {L}^{p} \) by (13) or (14), satisfies the inequality\n\n\[ \parallel H\left( f\right) {\parallel }_{{L}^{p}} \leq {A}_{p}\parallel f{\parallel }_{{L}^{p}},\;\text{ whenever }f \in {L}^{2} \cap {L}^{p}, \]... | ## 3.3 Proof of Theorem 3.2\n\nThe main idea of the proof was already outlined at the end of Section 1 in the context of Fourier series and the corresponding theorem for the conjugate function. While this proof, which depends on complex analysis, is elegant, its approach is essentially limited to this operator and cann... | No |
Theorem 4.1 Suppose \( f \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \) with \( 1 < p \leq \infty \) . Then \( {f}^{ * } \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \) , and (26) holds, namely\n\n\[ \n{\begin{Vmatrix}{f}^{ * }\end{Vmatrix}}_{{L}^{p}} \leq {A}_{p}\parallel f{\parallel }_{{L}^{p}}\n\]\n\nThe bound \( {A}_{p}... | Let us first see why \( {f}^{ * }\left( x\right) < \infty \), for a.e. \( x \), whenever \( f \in {L}^{p} \) . Observe that we can decompose \( f = {f}_{1} + {f}_{\infty } \), where \( {f}_{1}\left( x\right) = f\left( x\right) \) if \( \left| {f\left( x\right) }\right| > 1 \) , and \( {f}_{1}\left( x\right) = 0 \) else... | Yes |
Proposition 5.1 Suppose \( f \in {L}^{p}\left( {\mathbb{R}}^{d}\right), p > 1 \), and \( f \) has bounded support. Then \( f \) belongs to \( {\mathbf{H}}_{r}^{1}\left( {\mathbb{R}}^{d}\right) \) if and only if \( {\int }_{{\mathbb{R}}^{d}}f\left( x\right) {dx} = 0 \) . | Note that \( f \) is automatically in \( {L}^{1} \), by Hölder’s inequality (see Proposition 1.4 in Chapter 1), and the cancelation condition is necessary as has been pointed out.\n\nTo prove the sufficiency we assume that \( f \) is supported in a ball \( {B}_{1} \) of unit radius, and that \( {\int }_{{B}_{1}}\left| ... | No |
Lemma 5.2 Suppose \( \Omega \subset {\mathbb{R}}^{d} \) is a non-trivial open set. Then there is a collection \( \left\{ {Q}_{j}\right\} \) of dyadic cubes with disjoint interiors so that \( \Omega = \) \( \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{Q}_{j} \), and\n\n\[ \operatorname{diam}\left( {Q}_{j}\right) \leq d\... | Proof. We claim first that every point \( \bar{x} \in \Omega \) belongs to some dyadic cube \( {Q}_{\bar{x}} \) for which (37) holds (with \( {Q}_{\bar{x}} \) in place of \( {Q}_{j} \) ).\n\nLet \( \delta = d\left( {\bar{x},{\Omega }^{c}}\right) > 0 \) . Now the dyadic cubes containing \( \bar{x} \) have diameters vary... | Yes |
Corollary 5.3 Fix \( p > 1 \) . Then any p-atom \( \mathfrak{a} \) is in \( {\mathbf{H}}_{r}^{1} \) . Moreover there is a bound \( {c}_{p} \), independent of the atom \( \mathfrak{a} \), so that\n\n\[ \parallel \mathfrak{a}{\parallel }_{{\mathbf{H}}_{r}^{1}} \leq {c}_{p} \] | Proof. One can rescale a \( p \) -atom \( \mathfrak{a} \), associated to a ball \( B \) of radius \( r \), by replacing \( \mathfrak{a} \) by \( {\mathfrak{a}}_{r} \), with \( {\mathfrak{a}}_{r}\left( x\right) = {r}^{d}\mathfrak{a}\left( {rx}\right) \) . Then clearly \( {\mathfrak{a}}_{r}\left( x\right) \) is supported... | Yes |
Theorem 5.4 If \( f \) belongs to the Hardy space \( {\mathbf{H}}_{r}^{1}\left( \mathbb{R}\right) \), then \( {H}_{\epsilon }\left( f\right) \in \) \( {L}^{1}\left( \mathbb{R}\right) \), for every \( \epsilon > 0 \) . Moreover \( {H}_{\epsilon }\left( f\right) \) (see (14)) converges in the \( {L}^{1} \) norm, as \( \e... | Proof. The argument below illustrates a nice feature of \( {\mathbf{H}}_{r}^{1}\left( \mathbb{R}\right) \) : to show the boundedness of an operator on \( {\mathbf{H}}_{r}^{1} \) it often suffices merely to verify it for atoms, and this is usually a simple task.\n\nLet us first see that for all atoms \( \mathfrak{a} \),... | Yes |
Theorem 6.1 Suppose \( \Phi \) is a \( {C}^{1} \) function with compact support on \( {\mathbb{R}}^{d} \) . With \( M \) defined by (41) we have that \( M\left( f\right) \in {L}^{1}\left( {\mathbb{R}}^{d}\right) \), whenever \( f \in \) \( {\mathbf{H}}_{r}^{1}\left( {\mathbb{R}}^{d}\right) \) . Moreover\n\n\[ \parallel... | Proof. Suppose \( f \) is in \( {\mathbf{H}}_{r}^{1}\left( {\mathbb{R}}^{d}\right) \) and \( f = \sum {\lambda }_{k}{\mathfrak{a}}_{k} \) is an atomic decomposition. Then clearly \( M\left( f\right) \leq \sum \left| {\lambda }_{k}\right| M\left( {\mathfrak{a}}_{k}\right) \), and thus it suffices to prove (42) when \( f... | Yes |
Theorem 6.2 Suppose \( g \in \mathrm{{BMO}} \) . Then the linear functional \( \ell \) defined by (44), initially considered for \( f \in {H}_{0}^{1} \), has a unique extension to \( {\mathbf{H}}_{r}^{1} \) that satisfies\n\n\[ \parallel \ell \parallel \leq c\parallel g{\parallel }_{\mathrm{{BMO}}} \]\n\nConversely, ev... | Proof. Let us first assume that \( g \in \) BMO is bounded. Start with a general \( f \in {\mathbf{H}}_{r}^{1} \), and let \( f = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }_{k}{\mathfrak{a}}_{k} \) be an atomic decomposition. Then by the convergence of the sum in the \( {L}^{1} \) norm we get \( \ell \left( f\... | Yes |
Proposition 1.1 Suppose \( F \) is a distribution and \( \psi \in \mathcal{D} \). Then\n\n(a) The two definitions of \( F * \psi \) given above coincide.\n\n(b) The distribution \( F * \psi \) is a \( {C}^{\infty } \) function. | Proof. Let us observe first that \( F\left( {\psi }_{x}^{ \sim }\right) \) is continuous in \( x \) and in fact indefinitely differentiable. Note that if \( {x}_{n} \rightarrow {x}_{0} \) as \( n \rightarrow \infty \), then \( {\psi }_{{x}_{n}}^{ \sim }\left( y\right) = \psi \left( {{x}_{n} - y}\right) \rightarrow \psi... | Yes |
Corollary 1.2 Suppose \( F \) is a distribution on \( {\mathbb{R}}^{d} \) . Then there exists a sequence \( \left\{ {f}_{n}\right\} \), with \( {f}_{n} \in {C}^{\infty } \), and \( {f}_{n} \rightarrow F \) in the weak sense. | Proof. Let \( \left\{ {\psi }_{n}\right\} \) be an approximation to the identity constructed as follows. Fix a \( \psi \in \mathcal{D} \) with \( {\int }_{{\mathbb{R}}^{d}}\psi \left( x\right) {dx} = 1 \) and set \( {\psi }_{n}\left( x\right) = {n}^{d}\psi \left( {nx}\right) \) . Form \( {F}_{n} = F * {\psi }_{n} \) . ... | Yes |
Proposition 1.3 Suppose \( F \) is a distribution whose support is \( {C}_{1} \), and \( \psi \) is in \( \mathcal{D} \) and has support \( {C}_{2} \) . Then the support of \( F * \psi \) is contained in \( {C}_{1} + {C}_{2} \) | Indeed for each \( x \) for which \( F\left( {\psi }_{x}^{ \sim }\right) \neq 0 \), we must have that the support of \( F \) intersects the support of \( {\psi }_{x}^{ \sim } \) . Since the support of \( {\psi }_{x}^{ \sim } \) is the set \( x - \) \( {C}_{2} \) this means that the set \( {C}_{1} \) and \( x - {C}_{2} ... | Yes |
Proposition 1.4 Suppose \( F \) is a tempered distribution. Then there is a positive integer \( N \) and a constant \( c > 0 \), so that\n\n\[ \left| {F\left( \varphi \right) }\right| \leq c\parallel \varphi {\parallel }_{N},\;\text{ for all }\varphi \in \mathcal{S}. \]\n | Proof. Assume otherwise. Then the conclusion fails and for each positive integer \( n \) there is a \( {\psi }_{n} \in \mathcal{S} \) with \( {\begin{Vmatrix}{\psi }_{n}\end{Vmatrix}}_{n} = 1 \), while \( \left| {F\left( {\psi }_{n}\right) }\right| \geq n \) . Take \( {\varphi }_{n} = {\psi }_{n}/{n}^{1/2} \) . Then \(... | Yes |
Proposition 1.5 Suppose \( F \) is a tempered distribution and \( \psi \in \mathcal{S} \) . Then \( F * \psi \) is a slowly increasing \( {C}^{\infty } \) function, which when considered as a tempered distribution satisfies \( {\left( F * \psi \right) }^{ \land } = {\psi }^{ \land }{F}^{ \land } \) . | Proof. The fact that \( F\left( {\psi }_{x}^{ \sim }\right) \) is slowly increasing follows from the proposition in Section 1.4 together with the observation that for any function \( \psi \in \mathcal{D} \) and \( N,{\begin{Vmatrix}{\psi }_{x}^{ \sim }\end{Vmatrix}}_{N} \leq c{\left( 1 + \left| x\right| \right) }^{N}\p... | Yes |
Proposition 1.6 If \( F \) is a distribution of compact support then its Fourier transform \( {F}^{ \land } \) is a slowly increasing \( {C}^{\infty } \) function. In fact, as a function of \( \xi \), one has \( {F}^{ \land }\left( \xi \right) = F\left( {e}_{\xi }\right) \) where \( {e}_{\xi } \) is the element of \( \... | Proof. If we invoke Proposition 1.4, we see immediately that \( \left| {F\left( {e}_{\xi }\right) }\right| \leq \) \( C{\begin{Vmatrix}{e}_{\xi }\end{Vmatrix}}_{N} \leq {c}^{\prime }{\left( 1 + \left| \xi \right| \right) }^{N} \) . By the same estimate, every difference quotient of \( F\left( {e}_{\xi }\right) \) conve... | Yes |
Theorem 1.7 Suppose \( F \) is a distribution supported at the origin. Then \( F \) is a finite sum\n\n\[ F = \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}{a}_{\alpha }{\partial }_{x}^{\alpha }\delta \]\n\nThat is,\n\n\[ F\left( \varphi \right) = \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}{\left( -1... | Proceeding with the proof of the theorem, we now apply the above lemma to \( {F}_{1} = F - \mathop{\sum }\limits_{{\left| \alpha \right| < N}}{a}_{\alpha }{\partial }_{x}^{\alpha }\delta \) where \( N \) is the index that guarantees the conclusion of Proposition 1.4, while the \( {a}_{\alpha } \) are chosen so that \( ... | Yes |
Lemma 1.8 Suppose \( {F}_{1} \) is a distribution supported at the origin that satisfies for some \( N \) the following two conditions:\n\n(a) \( \left| {{F}_{1}\left( \varphi \right) }\right| \leq c\parallel \varphi {\parallel }_{N} \), for all \( \varphi \in \mathcal{D} \).\n\n(b) \( {F}_{1}\left( {x}^{\alpha }\right... | In fact, let \( \eta \in \mathcal{D} \), with \( \eta \left( x\right) = 0 \) for \( \left| x\right| \geq 1 \), and \( \eta \left( x\right) = 1 \) when \( \left| x\right| \leq 1/2 \) , and write \( {\eta }_{\epsilon }\left( x\right) = \eta \left( {x/\epsilon }\right) \). Then since \( {F}_{1} \) is supported at the orig... | Yes |
Theorem 2.1 The distribution \( \operatorname{pv}\left( \frac{1}{x}\right) \) equals:\n\n(a) \( \frac{d}{dx}\left( {\log \left| x\right| }\right) \).\n\n(b) \( \frac{1}{2}\left( {\frac{1}{x - {i0}} + \frac{1}{x + {i0}}}\right) \).\n\nAlso, its Fourier transform equals \( \frac{\pi }{i}\operatorname{sign}\left( x\right)... | Regarding (a), note that \( \log \left| x\right| \) is a locally integrable function. Here \( \frac{d}{dx}\left( {\log \left| x\right| }\right) \) is its derivative taken as a distribution. Now in that sense\n\n\[ \left( {\frac{d}{dx}\log \left| x\right| }\right) \left( \varphi \right) = - {\int }_{-\infty }^{\infty }\... | Yes |
Proposition 2.2 Suppose \( F \) is a tempered distribution on \( {\mathbb{R}}^{d} \) that is homogeneous of degree \( \lambda \) . Then its Fourier transform \( {F}^{ \land } \) is homogeneous of degree \( - d - \lambda \) . | To deal with \( {\left( {F}^{ \land }\right) }_{a} \) we write successively,\n\n\[ \n{\left( {F}^{ \land }\right) }_{a}\left( \varphi \right) = {F}^{ \land }\left( {\varphi }^{a}\right) = F\left( {\left( {\varphi }^{a}\right) }^{ \land }\right) = F\left( {\left( {\varphi }^{ \land }\right) }_{a}\right) \n\] \n\n\[ \n= ... | Yes |
Theorem 2.3 If \( - d < \lambda < 0 \), then\n\n\[{\left( {H}_{\lambda }\right) }^{ \land } = {c}_{\lambda }{H}_{-d - \lambda },\;\text{with}\;{c}_{\lambda } = \frac{\Gamma \left( \frac{d + \lambda }{2}\right) }{\Gamma \left( \frac{-\lambda }{2}\right) }{\pi }^{-d/2 - \lambda }.\] | To prove the theorem we start with the fact that \( \psi \left( x\right) = {e}^{-\pi {\left| x\right| }^{2}} \) is its own Fourier transform. Then since \( {\left( {\psi }_{a}\right) }^{ \land } = {\left( {\psi }^{ \land }\right) }^{a} \) we get (with \( a = {t}^{1/2} \) )\n\n\[{\int }_{{\mathbb{R}}^{d}}{e}^{-{\pi t}{\... | Yes |
Theorem 2.4 The Fourier transform of a regular homogeneous distribution \( K \) of degree \( \lambda \) is a regular homogeneous distribution of degree \( - d - \lambda \), and conversely. | Proof. We already know from Proposition 2.2 that \( {K}^{ \land } \) is homogeneous of degree \( - d - \lambda \) . To prove that \( {K}^{ \land } \) agrees with a \( {C}^{\infty } \) function away from the origin, we decompose \( K = {K}_{0} + {K}_{1} \), with \( {K}_{0} \) supported near the origin and \( {K}_{1} \) ... | Yes |
Lemma 2.6 Suppose \( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n} \), are distinct real numbers and that for constants \( {a}_{j} \) and \( {b}_{j},1 \leq j \leq n \), we have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{a}_{j}{x}^{{\lambda }_{j}} + {b}_{j}{x}^{{\lambda }_{j}}\log x}\right) = 0\;\text{ for... | To prove the lemma we assume, as one may, that \( {\lambda }_{n} \) is the largest of the \( {\lambda }_{j} \) ’s. Then multiplying the identity by \( {x}^{-{\lambda }_{n}} \) and letting \( x \) tend to infinity we see that \( {b}_{n} \) as well as \( {a}_{n} \) must vanish. Thus we are reduced to the case when \( n \... | Yes |
Corollary 2.7 There is no distribution \( {K}_{0} \) that is homogeneous of degree \( - d \) and that agrees with the function \( 1/{\left| x\right| }^{d} \) away from the origin. | If such a \( {K}_{0} \) existed, then \( {K}_{0} - \left\lbrack \frac{1}{{\left| x\right| }^{d}}\right\rbrack \) would be supported at the origin, and hence equal to \( \mathop{\sum }\limits_{{\left| \alpha \right| \leq M}}{c}_{\alpha }{\partial }_{x}^{\alpha }\delta \) . Applying this difference to \( {\varphi }^{a} \... | Yes |
Theorem 2.8 For \( d \geq 3 \), the locally integrable function \( F \) defined by \( F\left( x\right) = {C}_{d}{\left| x\right| }^{-d + 2} \) is a fundamental solution for the operator \( \bigtriangleup \), with \( {C}_{d} = - \frac{\Gamma \left( {\frac{d}{2} - 1}\right) }{4{\pi }^{\frac{d}{2}}}. \) | This follows by taking \( \lambda = - d + 2 \) (in Theorem 2.3), then \( \Gamma \left( \frac{d + \lambda }{2}\right) = \) \( \Gamma \left( 1\right) = 1 \), while \( \Gamma \left( {d/2}\right) = \left( {d/2 - 1}\right) \Gamma \left( {d/2 - 1}\right) \) . Therefore \( \widehat{F}\left( \xi \right) \) equals \( 1/\left( {... | Yes |
Theorem 2.10 \( F \) is a fundamental solution of \( L = \frac{\partial }{\partial t} - {\bigtriangleup }_{x} \) . | Proof. Since \( {LF}\left( \varphi \right) = F\left( {{L}^{\prime }\varphi }\right) \) with \( {L}^{\prime } = - \frac{\partial }{\partial t} - {\bigtriangleup }_{x} \), it suffices to see that \( F\left( {{L}^{\prime }\varphi }\right) \), which equals\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{\int }_{t \g... | Yes |
Theorem 2.11 Every constant coefficient (linear) partial differential equation \( L \) on \( {\mathbb{R}}^{d} \) has a fundamental solution. | Proof. After a possible change of coordinates consisting of a rotation and multiplication by a constant, we may assume that the characteristic polynomial of \( L \) will be of the form\n\n\[ P\left( \xi \right) = P\left( {{\xi }_{1},{\xi }^{\prime }}\right) = {\xi }_{1}^{m} + \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}{\... | No |
Theorem 2.12 Every elliptic operator has a regular parametrix. | Proof. Observe first by a straightforward inductive argument in \( k \) , that whenever \( \left| \alpha \right| = k \) and \( P \) is any polynomial\n\n\[{\left( \frac{\partial }{\partial \xi }\right) }^{\alpha }\left( \frac{1}{P\left( \xi \right) }\right) = \mathop{\sum }\limits_{{0 \leq \ell \leq k}}\frac{{q}_{\ell ... | Yes |
Corollary 2.13 Given any \( \epsilon > 0 \), the elliptic operator \( L \) has a regular parametrix \( {Q}_{\epsilon } \) that is supported in the ball \( \{ x : \left| x\right| \leq \epsilon \} \) . | In fact, let \( {\eta }_{\epsilon } \) be a cut-off function in \( \mathcal{D} \), that is 1 when \( \left| x\right| \leq \epsilon /2 \) , and that is supported where \( \left| x\right| \leq \epsilon \) . Set \( {Q}_{\epsilon } = {\eta }_{\epsilon }Q \), and observe that \( L\left( {{\eta }_{\epsilon }Q}\right) - {\eta... | Yes |
Theorem 2.14 Suppose the partial differential operator \( L \) has a regular parametrix. Assume \( U \) is a distribution given in an open set \( \Omega \subset {\mathbb{R}}^{d} \) and \( L\left( U\right) = f \), with \( f \) a \( {C}^{\infty } \) function in \( \Omega \) . Then \( U \) is also a \( {C}^{\infty } \) fu... | Proof of the theorem. It suffices to show that \( U \) agrees with a \( {C}^{\infty } \) function on any ball \( B \) with \( \bar{B} \subset \Omega \) . Fix such a ball (say of radius \( \rho \) ), and let \( {B}_{1} \) be the concentric ball having radius \( \rho + \epsilon \), with \( \epsilon > 0 \) so small that \... | Yes |
Theorem 3.2 Let \( T \) be the operator \( T\left( f\right) = f * K \), with \( K \) as in Proposition 3.1. Then \( T \) initially defined for \( f \) in \( \mathcal{S} \) extends to a bounded operator on \( {L}^{p}\left( {\mathbb{R}}^{d}\right) \), for \( 1 < p < \infty \) . | This means that for each \( p,1 < p < \infty \), there is a bound \( {A}_{p} \) so that\n\n(23)\n\n\[ \parallel {Tf}{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{d}\right) } \leq {A}_{p}\parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{d}\right) } \]\n\nfor \( f \in \mathcal{S} \) . Thus by Proposition 5.4 in Chapter 1 ... | Yes |
Lemma 3.3 For each \( f \) in \( {L}^{1}\left( {\mathbb{R}}^{d}\right) \) and \( \alpha > 0 \), we can find an open set \( {E}_{\alpha } \) and a decomposition \( f = g + b \) so that:\n\n(a) \( m\left( {E}_{\alpha }\right) \leq \frac{c}{\alpha }\parallel f{\parallel }_{{L}^{1}\left( {\mathbb{R}}^{d}\right) } \) .\n\n(... | The proof of the lemma is a simplified version of the argument used to prove Proposition 5.1 in the previous chapter; in particular, here we use the full maximal function \( {f}^{ * } \) instead of the truncated version \( {f}^{ \dagger } \) . The guiding idea is to try to cut the domain of \( f \) into the set when \(... | Yes |
Theorem 1.1 Every complete metric space \( X \) is of the second category in itself, that is, \( X \) cannot be written as the countable union of nowhere dense sets. | Proof of the theorem. We argue by contradiction, and assume that \( X \) is a countable union of nowhere dense sets \( {F}_{n} \) ,\n\n\[ X = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{F}_{n} \]\n\nBy replacing each \( {F}_{n} \) by its closure, we may assume that each \( {F}_{n} \) is closed. It now suffices to find... | Yes |
Corollary 1.2 In a complete metric space, a generic set is dense. | To prove the corollary, we argue by contradiction and assume that \( E \subset X \) is generic but not dense. Then there exists a closed ball \( \bar{B} \) entirely contained in \( {E}^{c} \) . Since \( E \) is generic we can write \( {E}^{c} = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{F}_{n} \) where each \( {F}_{n... | Yes |
Lemma 1.4 Suppose \( \left\{ {f}_{n}\right\} \) is a sequence of continuous functions on a complete metric space \( X \), and \( {f}_{n}\left( x\right) \rightarrow f\left( x\right) \) for each \( x \) as \( n \rightarrow \infty \) . Then, given an open ball \( B \subset X \) and \( \epsilon > 0 \), there exists an open... | Proof. Let \( Y \) denote a closed ball contained in \( B \) . Note that \( Y \) is itself a complete metric space. Define\n\n\[ \n{E}_{\ell } = \left\{ {x \in Y : \mathop{\sup }\limits_{{j, k \geq \ell }}\left| {{f}_{j}\left( x\right) - {f}_{k}\left( x\right) }\right| \leq \epsilon }\right\} \n\]\n\nThen, since \( {f}... | Yes |
Lemma 1.6 For every \( M > 0 \), the set \( {\mathcal{P}}_{M} \) of zig-zag functions is dense in \( C\left( \left\lbrack {0,1}\right\rbrack \right) \) . | Proof. It is plain that given \( \epsilon > 0 \) and a continuous function \( f \) , there exists a function \( g \in \mathcal{P} \) so that \( \parallel f - g\parallel \leq \epsilon \) . Indeed, since \( f \) is continuous on the compact set \( \left\lbrack {0,1}\right\rbrack \) it must be uniformly continuous, and th... | Yes |
Theorem 2.1 Suppose that \( \mathcal{B} \) is a Banach space, and \( \mathcal{L} \) is a collection of continuous linear functionals on \( \mathcal{B} \) .\n\n(i) If \( \mathop{\sup }\limits_{{\ell \in \mathcal{L}}}\left| {\ell \left( f\right) }\right| < \infty \) for each \( f \in \mathcal{B} \), then\n\n\[ \mathop{\s... | Proof. It suffices to show (ii) since by Baire’s theorem, \( \mathcal{B} \) is of the second category. So suppose that \( \mathop{\sup }\limits_{{\ell \in \mathcal{L}}}\left| {\ell \left( f\right) }\right| < \infty \) for all \( f \in E \), where \( E \) is of the second category.\n\nFor each positive integer \( M \), ... | Yes |
Lemma 2.3 \( \begin{Vmatrix}{\ell }_{N}\end{Vmatrix} = {L}_{N} \) for all \( N \geq 0 \) . | Proof. We already know from the above that \( \begin{Vmatrix}{\ell }_{N}\end{Vmatrix} \leq {L}_{N} \) . To prove the reverse inequality, it suffices to find a sequence of continuous functions \( \left\{ {f}_{k}\right\} \) so that \( \begin{Vmatrix}{f}_{k}\end{Vmatrix} \leq 1 \), and \( {\ell }_{N}\left( {f}_{k}\right) ... | Yes |
Lemma 2.4 There is a constant \( c > 0 \) so that \( {L}_{N} \geq c\log N \) . | Proof. Since \( \left| {\sin y}\right| /\left| y\right| \leq 1 \) for all \( y \), and \( \sin y \) is an odd function, we see that \( {}^{1} \)\n\n\[ \n{L}_{N} \geq c{\int }_{0}^{\pi }\frac{\left| \sin \left( N + 1/2\right) y\right| }{\left| y\right| }{dy} \]\n\n\[ \n\geq c{\int }_{0}^{\left( {N + 1/2}\right) \pi }\fr... | Yes |
Corollary 3.2 If \( X \) and \( Y \) are Banach spaces, and \( T : X \rightarrow Y \) is a continuous bijective linear transformation, then the inverse \( {T}^{-1} : Y \rightarrow X \) of \( T \) is also continuous. Hence there are constants \( c, C > 0 \) with\n\n\[ c\parallel f{\parallel }_{X} \leq \parallel T\left( ... | This follows immediately from the discussion preceding Theorem 3.1. | No |
Corollary 3.3 Suppose the vector space \( V \) is equipped with two norms \( \parallel \cdot {\parallel }_{1} \) and \( \parallel \cdot {\parallel }_{2} \) . If\n\n\[ \parallel v{\parallel }_{1} \leq C\parallel v{\parallel }_{2}\;\text{ for all }v \in V, \]\n\nand \( V \) is complete with respect to both norms, then \(... | Indeed, the hypothesis implies that the identity mapping \( I : \left( {V,\parallel \cdot {\parallel }_{2}}\right) \rightarrow \) \( \left( {V,\parallel \cdot {\parallel }_{1}}\right) \) is continuous, and since it is clearly bijective, its inverse \( I \) : \( \left( {V,\parallel \cdot {\parallel }_{1}}\right) \righta... | Yes |
Theorem 3.4 The mapping \( T : {\mathcal{B}}_{1} \rightarrow {\mathcal{B}}_{2} \) given by \( T\left( f\right) = \{ \widehat{f}\left( n\right) \} \) is linear, continuous and injective, but not surjective. | Proof. We first note that \( T \) is clearly linear, and also continuous with \( \parallel T\left( f\right) {\parallel }_{\infty } \leq \parallel f{\parallel }_{{L}^{1}} \) . Moreover, \( T \) is injective since \( T\left( f\right) = 0 \) implies that \( \widehat{f}\left( n\right) = 0 \) for all \( n \), which then imp... | Yes |
Theorem 4.1 Suppose \( X \) and \( Y \) are two Banach spaces. If \( T : X \rightarrow Y \) is a closed linear map, then \( T \) is continuous. | Proof. Since the graph of \( T \) is a closed subspace of the Banach space \( X \times Y \) with the norm \( \parallel \left( {x, y}\right) {\parallel }_{X \times Y} = \parallel x{\parallel }_{X} + \parallel y{\parallel }_{Y} \), the graph \( {G}_{T} \) is itself a Banach space. Consider the two projections \( {P}_{X} ... | Yes |
Lemma 4.3 Under the assumptions of the theorem, there exists \( A > 0 \) so that\n\n\[ \parallel f{\parallel }_{{L}^{\infty }} \leq A\parallel f{\parallel }_{{L}^{2}}\;\text{ for all }f \in E. \] | Proof. If \( 1 \leq p \leq 2 \), then Hölder’s inequality with the conjugate exponents \( r = 2/p \) and \( {r}^{ * } = 2/\left( {2 - p}\right) \) yields\n\n\[ \int {\left| f\right| }^{p} \leq {\left( \int {\left| f\right| }^{2}\right) }^{p/2}{\left( \int 1\right) }^{\frac{2 - p}{2}}.\]\n\nSince \( X \) has finite meas... | Yes |
Theorem 5.1 The collection of sets in \( \mathcal{K} \) of two-dimensional Lebesgue measure zero is generic. | In particular, this collection is non-empty, and in fact dense. Loosely stated, the key to the argument is to show that sets \( K \) in \( \mathcal{K} \) whose horizontal slices \( \{ x : \left( {x, y}\right) \in K\} \) have \ | No |
Lemma 5.2 For each fixed \( {y}_{0} \) and \( \epsilon \), the collection of sets \( \mathcal{K}\left( {{y}_{0},\epsilon }\right) \) is open and dense in \( \mathcal{K} \) . | To prove that \( \mathcal{K}\left( {{y}_{0},\epsilon }\right) \) is open, suppose \( K \in \mathcal{K}\left( {{y}_{0},\epsilon }\right) \) and pick \( \eta \) so that \( {K}^{\eta } \) satisfies the condition above. Suppose \( {K}^{\prime } \in \mathcal{K} \) with \( \operatorname{dist}\left( {K,{K}^{\prime }}\right) <... | Yes |
Proposition 1.1 For each integer \( N \geq 1 \) , \[ {\begin{Vmatrix}{S}_{N}\end{Vmatrix}}_{{L}^{2}} = {N}^{1/2}. \] | This proposition follows from the fact that \( \left\{ {{r}_{n}\left( t\right) }\right\} \) is an orthonormal system on \( {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) . Indeed, we have that \( {\int }_{0}^{1}{r}_{n}\left( t\right) {dt} = 0 \) because each \( {r}_{n} \) is equal to 1 on a set of measure \( ... | Yes |
Corollary 1.2 \( {S}_{N}/N \) converges to 0 in probability. | In fact,\n\n\[ m\left( \left\{ {\left| {{S}_{N}\left( x\right) /N}\right| > \epsilon }\right\} \right) = m\left( \left\{ {\left| {{S}_{N}\left( x\right) }\right| > {\epsilon N}}\right\} \right) \leq \frac{1}{{\epsilon }^{2}{N}^{2}}\int {\left| {S}_{N}\left( x\right) \right| }^{2}{dm}, \]\n\nby Tchebychev’s inequality. ... | Yes |
Corollary 1.5 Let \( {S}_{N}\left( t\right) = \mathop{\sum }\limits_{{n = 1}}^{N}{r}_{n}\left( t\right) \) . Then \( {S}_{N}\left( t\right) /N \rightarrow 0 \), as \( N \rightarrow \) \( \infty \) for almost every \( t \) . In fact, if \( \alpha > 1/2 \), then \( {S}_{N}\left( t\right) /{N}^{\alpha } \rightarrow 0 \) f... | Proof. Fix \( 1/2 < \beta < \alpha \), and let \( {a}_{n} = {n}^{-\beta } \) and \( {b}_{n} = {n}^{\beta } \) . Clearly \( \sum {a}_{n}^{2} < \infty \) . Set \( {\widetilde{S}}_{N}\left( t\right) = \mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n}{r}_{n}\left( t\right) \) . Then, by summation by parts, setting \( {\widetilde{... | Yes |
Lemma 1.8 For each \( p < \infty \) there is a bound \( {A}_{p} \) so that\n\n\[ \parallel F{\parallel }_{{L}^{p}} \leq {A}_{p}\parallel F{\parallel }_{{L}^{2}} \]\n\nfor all \( F \in {L}^{p}\left( \left\lbrack {0,1}\right\rbrack \right) \) of the form \( F\left( t\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\inf... | It clearly suffices to prove the corresponding statement when the \( {a}_{n} \) are assumed real and have been normalized so that \( \parallel F{\parallel }_{{L}^{2}}^{2} = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}^{2} = 1 \) .\n\nNow observe that the defining property (3) shows that whenever \( \left\{ {f}_{... | Yes |
Theorem 2.1 Suppose \( \\left\\{ {f}_{n}\\right\\} \) is a sequence of functions that are mutually independent, are identically distributed, and have mean \( {m}_{0} \) . Then\n\n\[ \n\\frac{1}{N}\\mathop{\\sum }\\limits_{{n = 0}}^{{N - 1}}{f}_{n}\\left( x\\right) \\rightarrow {m}_{0}\\;\\text{ for almost every }x \\in... | The possibility of reducing this theorem to the ergodic theorem depends on the device of replacing the sequence \( \\left\\{ {f}_{n}\\right\\} \) by another sequence that is \ | No |
Lemma 2.2 If \( \left\{ {f}_{N}\right\} \) and \( \left\{ {g}_{N}\right\} \) have the same joint distribution, then so do the sequences \( \left\{ {{\Phi }_{N}\left( f\right) }\right\} \) and \( \left\{ {{\Phi }_{N}\left( g\right) }\right\} \) . Here \( {\Phi }_{N}\left( f\right) = {\Phi }_{N}\left( {{f}_{1},\ldots ,{f... | To see this, note that if \( B \subset {\mathbb{R}}^{N} \) is a Borel set, and \( \Phi = \left( {{\Phi }_{1},\ldots ,{\Phi }_{N}}\right) \) , then \( {B}^{\prime } = {\Phi }^{-1}\left( B\right) \) is also a Borel set in \( {\mathbb{R}}^{N} \), so if \( f = \left( {{f}_{1},\ldots ,{f}_{N}}\right) \) and \( g = \left( {{... | Yes |
Lemma 2.3 If \( \left\{ {F}_{N}\right\} \) and \( \left\{ {G}_{N}\right\} \) have the same joint distribution, then \( {F}_{N}\left( x\right) \rightarrow {m}_{0} \) almost everywhere as \( N \rightarrow \infty \) if and only if \( {G}_{N}\left( y\right) \rightarrow {m}_{0} \) almost everywhere as \( N \rightarrow \inft... | To prove this lemma, note that if we define \( {E}_{N, k} = \left\{ {x : \mathop{\sup }\limits_{{r \geq N}} \mid {F}_{r}\left( x\right) - }\right. \) \( \left. {{m}_{0} \mid \leq 1/k}\right\} \), then \( {F}_{N} \rightarrow {m}_{0} \) almost everywhere if and only if \( m\left( {E}_{N, k}\right) \rightarrow \) 1, as \(... | Yes |
Proposition 2.4 Given an integrable function \( f \) and a sub-algebra \( \mathcal{A} \) of \( \mathcal{M} \), there is a unique \( {}^{9} \) function \( F \) so that:\n\n(i) \( F \) is \( \mathcal{A} \) -measurable.\n\n(ii) \( {\int }_{A}{Fdm} = {\int }_{A}{fdm} \) for any set \( A \in \mathcal{A} \) . | Proof. We denote by \( {m}^{\prime } \) the restriction of the measure \( m \) to \( \mathcal{A} \) . Define a ( \( \sigma \) -finite) signed measure \( \nu \) on \( \mathcal{A} \) by \( \nu \left( A\right) = {\int }_{A}f\;{dm} \), for \( A \in \mathcal{A} \) . Then since \( \nu \) is clearly absolutely continuous with... | Yes |
(a) If \( f \in {L}^{2} \), then \( \mathbb{E}\left( f\right) \in {L}^{2} \) and \( \parallel \mathbb{E}\left( f\right) {\parallel }_{{L}^{2}} \leq \parallel f{\parallel }_{{L}^{2}} \) . | Proof. To establish (a) observe that if \( g \) is bounded and \( \mathcal{A} \) -measurable, then by the proposition above, \( {\int }_{X}{gfdm} = {\int }_{X}\mathbb{E}\left( {gf}\right) {dm} = {\int }_{X}g\mathbb{E}\left( f\right) {dm} \) . But\n\n\[ \parallel \mathbb{E}\left( f\right) {\parallel }_{{L}^{2}} = \matho... | Yes |
Lemma 2.7 Suppose \( {\mathcal{B}}_{0},\ldots ,{\mathcal{B}}_{n} \) are mutually independent algebras. Then for each \( k < n \), the algebras \( \mathop{\bigvee }\limits_{{j = 0}}^{k}{\mathcal{B}}_{j} \) and \( {\mathcal{B}}_{n} \) are mutually independent. | Null | No |
Theorem 2.8 Suppose \( {f}_{0},\ldots ,{f}_{n},\ldots \) are independent functions that are square integrable, and that each has mean zero, and variance \( {\sigma }_{n}^{2} = {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{{L}^{2}}^{2} \) . Assume that\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{\sigma }_{n}^{2} < \infty \... | We begin the proof of the theorem by noting that under its assumptions the sequence \( {s}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{f}_{k} \) converges in the \( {L}^{2} \) norm, as \( n \rightarrow \infty \) . Indeed, since the \( {f}_{n} \) are mutually independent and \( {\int }_{X}{f}_{n}{dm} = 0 \), then by (4) t... | Yes |
Corollary 2.9 If \( \mathop{\sup }\limits_{n}{\sigma }_{n} < \infty \), then for each \( \alpha > 1/2 \)\n\n\[ \frac{{s}_{n}}{{n}^{\alpha }} \rightarrow 0\;\text{ almost everywhere as }n \rightarrow \infty . \]\n\nNote that here, unlike in Theorem 2.1, we have not assumed that the \( {f}_{n} \) are identically distribu... | We begin the proof of the theorem by noting that under its assumptions the sequence \( {s}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{f}_{k} \) converges in the \( {L}^{2} \) norm, as \( n \rightarrow \infty \) . Indeed, since the \( {f}_{n} \) are mutually independent and \( {\int }_{X}{f}_{n}{dm} = 0 \), then by (4) t... | Yes |
Theorem 2.10 Suppose \( {s}_{\infty } \) is an integrable function, and \( {s}_{n} = {\mathbb{E}}_{n}\left( {s}_{\infty }\right) \) , where the \( {\mathbb{E}}_{n} \) are conditional expectations for an increasing family \( \left\{ {\mathcal{A}}_{n}\right\} \) of sub-algebras of \( \mathcal{M} \) . Then:\n\n(a) \( m\le... | For the proof of part (a) we may assume that \( {s}_{\infty } \) is non-negative, for otherwise we may proceed with \( \left| {s}_{\infty }\right| \) instead of \( {s}_{\infty } \) and then obtain the result once we observe that \( \left| {{\mathbb{E}}_{n}\left( {s}_{\infty }\right) }\right| \leq {\mathbb{E}}_{n}\left(... | Yes |
Theorem 2.11 If the algebras \( {\mathcal{A}}_{0},{\mathcal{A}}_{1},\ldots ,{\mathcal{A}}_{n},\ldots \) are mutually independent then every element of the tail algebra has either measure zero or one. | Proof. Let \( \mathcal{B} \) denote the tail algebra. Note that \( {\mathcal{A}}_{r} \) is automatically independent from \( \mathop{\bigvee }\limits_{{k = r + 1}}^{\infty }{\mathcal{A}}_{k} \), by Lemma 2.7. Hence each \( {\mathcal{A}}_{r} \) is independent of \( \mathcal{B} \), and thus the algebras \( \mathcal{B} \)... | Yes |
Corollary 2.12 Suppose \( {f}_{0},{f}_{1},\ldots ,{f}_{n},\ldots \) are mutually independent functions. The set where \( \mathop{\sum }\limits_{{k = 0}}^{\infty }{f}_{k} \) converges has measure zero or one. | Proof. Set \( {\mathcal{A}}_{n} = {\mathcal{A}}_{{f}_{n}} \) . Then these algebras are independent. Now with \( {s}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{f}_{k} \), and a fixed positive integer \( {n}_{0} \), we have by the Cauchy criterion that\n\n\[ \left\{ {x : \lim {s}_{n}\left( x\right) \text{ exists }}\right\... | Yes |
Lemma 2.14 \( \widehat{\mu }\left( {\xi /{N}^{1/2}}\right) = 1 - 2{\sigma }^{2}{\pi }^{2}{\xi }^{2}/N + o\left( {1/N}\right) \), as \( N \rightarrow \infty \) . | Proof. Indeed, when \( \xi \) is fixed\n\n\[ \n{e}^{-{2\pi i\xi t}/{N}^{1/2}} = 1 - {2\pi i\xi t}/{N}^{1/2} - 2{\pi }^{2}{\xi }^{2}{t}^{2}/N + {E}_{N}\left( t\right) \n\] \n\nwith \( {E}_{N}\left( t\right) = O\left( {{t}^{2}/N}\right) \), but also \( {E}_{N}\left( t\right) = O\left( {{t}^{3}/{N}^{3/2}}\right) \). Integ... | Yes |
Lemma 2.15 Suppose \( \left\{ {\mu }_{N}\right\}, N = 1,2,\ldots \), and \( \nu \) are non-negative \( {fi} \) - nite Borel measures on \( \mathbb{R} \), and that \( \nu \) is continuous. Assume that \( {\widehat{\mu }}_{N}\left( \xi \right) \rightarrow \) \( \widehat{\nu }\left( \xi \right) \), as \( N \rightarrow \in... | Proof. We prove first that\n\n(16)\n\n\[ \n{\mu }_{N}\left( \varphi \right) \rightarrow \nu \left( \varphi \right) \;\text{ as }N \rightarrow \infty \n\] \n\nfor any \( \varphi \) that is \( {C}^{\infty } \) and has compact support, where we have used the notation \( {\mu }_{N}\left( \varphi \right) = {\int }_{-\infty ... | Yes |
Theorem 2.17 Under the above conditions on \( \left\{ {f}_{n}\right\} \), the measures \( {\mu }_{N} \) converge weakly to \( {\nu }_{{\sigma }^{2}} \) as \( N \rightarrow \infty \) . | The proof proceeds essentially as in the case of real-valued functions, showing first the analog of (16) for smooth functions with compact support, and then proceeding as in Corollary 2.16 for continuous functions. The calculation of the characteristic function of the Gaussian is given in Exercise 32. | No |
Lemma 2.1 The \( \sigma \) -algebra \( \mathcal{C} \) is the same as the \( \sigma \) -algebra \( \mathcal{B} \) of Borel sets. | Proof. If \( \mathcal{O} \) is an open set in \( {\mathbb{R}}^{dk} \), then clearly\n\n\[ \left\{ {\mathrm{p} \in \mathcal{P} : \left( {\mathrm{p}\left( {t}_{1}\right) ,\mathrm{p}\left( {t}_{2}\right) ,\ldots ,\mathrm{p}\left( {t}_{k}\right) }\right) \in \mathcal{O}}\right\} \]\n\nis open in \( \mathcal{P} \), and henc... | Yes |
Corollary 2.3 Suppose the sequence of probability measures \( \left\{ {\mu }_{N}\right\} \) is tight, and for each \( 0 \leq {t}_{1} \leq {t}_{2} \leq \cdots \leq {t}_{k} \) the measures \( {\mu }_{N}^{\left( {t}_{1},\ldots ,{t}_{k}\right) } \) converge weakly to a measure \( {\mu }_{{t}_{1},\ldots ,{t}_{k}} \), as \( ... | Proof. First, by Lemma 2.2, there is a subsequence \( \left\{ {\mu }_{{N}_{m}}\right\} \) that converges weakly to a measure \( \mu \) . Next, \( {\mu }_{{N}_{m}}^{\left( {t}_{1},\ldots ,{t}_{k}\right) } \rightarrow {\mu }^{\left( {t}_{1},\ldots ,{t}_{k}\right) } \) weakly. In fact, if \( {\pi }^{{t}_{1},{t}_{2},\ldots... | Yes |
Lemma 2.4 A closed set \( K \subset \mathcal{P} \) is compact if for each positive \( T \) there is a positive bounded function \( h \mapsto {w}_{T}\left( h\right) \), defined for \( h \in (0,1\rbrack \) with \( {w}_{T}\left( h\right) \rightarrow 0 \) as \( h \rightarrow 0 \), and so that\n\n(7)\n\n\[ \mathop{\sup }\li... | The condition (7) implies that the functions on \( K \) are equicontinuous on each interval \( \left\lbrack {0, T}\right\rbrack \) . The lemma then essentially follows from the Arzela-Ascoli criterion. (Recall, this criterion was used in a special setting in Section 3, Chapter 8 of Book II.) | Yes |
Lemma 3.2 We have as \( \lambda \rightarrow \infty \) ,\n\n\[ \mathop{\sup }\limits_{{n \geq 1}}m\left( \left\{ {x : \mathop{\sup }\limits_{{k \leq n}}\left| {{s}_{k}\left( x\right) }\right| > \lambda {n}^{1/2}}\right\} \right) = O\left( {\lambda }^{-p}\right) \]\n\nfor every \( p \geq 2 \) . | To prove the lemma we apply the martingale maximal theorem of the previous chapter (Theorem 2.10, in the form that it takes in Exercise 29, part (b)) to the stopped sequence \( \left\{ {s}_{k}^{\prime }\right\} \) defined as \( {s}_{k}^{\prime } = {s}_{k} \) if \( k \leq n,{s}_{k}^{\prime } = \) \( {s}_{n} \) if \( k \... | Yes |
Theorem 4.1 The following are also Brownian motion processes:\n\n(a) \( {\delta }^{-1/2}{B}_{t\delta } \) for every fixed \( \delta > 0 \) .\n\n(b) \( \mathfrak{o}\left( {B}_{t}\right) \) whenever \( \mathfrak{o} \) is an orthogonal linear transformation on \( {\mathbb{R}}^{d} \) .\n\n(c) \( {B}_{t + {\sigma }_{0}} - {... | We need only check that these new processes satisfy the conditions B-1, B-2, and B-3 defining Brownian motion. Thus the assertion (a) of the theorem is clear once we observe that for any function \( f \), the covariance matrix of \( {\delta }^{-1/2}f \) is \( {\delta }^{-1} \) times the covariance matrix of \( f \) . T... | Yes |
Theorem 4.2 With \( W \) the Wiener measure on \( \mathcal{P} \) we have:\n\n(a) If \( 0 < a < 1/2 \) and \( T > 0 \), then, with respect to \( W \) almost every path \( \mathrm{p} \) satisfies\n\n\[ \mathop{\sup }\limits_{{0 \leq t \leq T,0 < h \leq 1}}\frac{\left| \mathrm{p}\left( t + h\right) - \mathrm{p}\left( t\ri... | The first conclusion is implicit in our construction of Brownian motion. Indeed, suppose \( {K}^{\left( T\right) } \) is the set arising in the proof of Theorem 3.1. Then we have seen that \( {\mu }_{N}\left( {K}^{\left( T\right) }\right) \geq 1 - \epsilon \) for every \( N \) . Thus the same holds for the weak limit o... | Yes |
Lemma 5.2 \( {\mathcal{A}}_{0 + } = {\mathcal{A}}_{0} \) . | Proof of the lemma. Fix a bounded continuous function \( f \) on \( {\mathbb{R}}^{kd} \), and a sequence \( 0 \leq {t}_{1} < {t}_{2} < \cdots < {t}_{k} \) . For any \( \delta > 0 \), set\n\n\[ \n{f}_{\delta } = f\left( {{B}_{{t}_{1} + \delta } - {B}_{\delta },{B}_{{t}_{2} + \delta } - {B}_{{t}_{1} + \delta },\ldots ,{B... | Yes |
Theorem 5.3 Suppose \( {B}_{t} \) is a Brownian motion and \( \sigma \) is a stopping time. Then the process \( {B}_{t}^{ * } \), defined by\n\n\[ \n{B}_{t}^{ * }\left( \omega \right) = {B}_{t + \sigma \left( \omega \right) }\left( \omega \right) - {B}_{\sigma \left( \omega \right) }\left( \omega \right) \n\]\n\nis als... | Proof. We have already noted that if \( \sigma \left( \omega \right) \) is a constant, \( \sigma \left( \omega \right) = \) \( {\sigma }_{0} \), then \( {B}_{t + {\sigma }_{0}} - {B}_{{\sigma }_{0}} \) is a Brownian motion (see Theorem 4.1), so the assertion in the theorem holds in this case.\n\nNext assume that \( \si... | Yes |
Proposition 6.2 Suppose \( x \in \partial \mathcal{R} \) and \( x + \Gamma \) is disjoint from \( \mathcal{R} \), for some truncated cone \( \Gamma \) . Then \( x \) is a regular point. | Proof. We assume \( x = 0 \), and consider the set \( A \) of Brownian paths starting at the origin that enter \( \Gamma \) for an infinite sequence of times tending to zero. Let \( {A}_{n} = \mathop{\bigcup }\limits_{{{r}_{k} < 1/n}}\left\{ {\omega : {B}_{{r}_{k}}\left( \omega \right) \in \Gamma }\right\} \) where \( ... | Yes |
Corollary 6.3 Suppose the bounded open set \( \mathcal{R} \) satisfies the outside cone condition. Assume \( f \) is a given continuous function on \( \partial \mathcal{R} \). Then there is a unique function \( u \) that is continuous in \( \overline{\mathcal{R}} \), harmonic in \( \mathcal{R} \), and such that \( {\le... | Proof. Theorem 6.1 and Proposition 6.2 show that \( u \) is continuous in \( \overline{\mathcal{R}} \) and \( {\left. u\right| }_{\partial \mathcal{R}} = f \). The uniqueness is a consequence of the well-known maximum principle. \( {}^{9} \) | Yes |
Proposition 1.2 Suppose \( f \) and \( g \) are a pair of holomorphic functions in a region \( {}^{1}\Omega \), and \( f \) and \( g \) agree in a neighborhood of a point \( {z}^{0} \in \Omega \) . Then \( f \) and \( g \) agree throughout \( \Omega \) . | Proof. We may assume that \( g = 0 \) . If we fix any point \( {z}^{\prime } \in \Omega \) , it suffices to prove that \( f\left( {z}^{\prime }\right) = 0 \) . Using the pathwise connectedness of \( \Omega \) we can find a sequence of points \( {z}^{1},\ldots ,{z}^{N} = {z}^{\prime } \) in \( \Omega \) and polydiscs \(... | Yes |
Theorem 2.1 Suppose \( F \) is holomorphic in \( \Omega = \left\{ {z \in {\mathbb{C}}^{n},\rho < \left| z\right| < 1}\right\} \) , for some fixed \( \rho ,0 < \rho < 1 \) . Then \( F \) can be analytically continued into the ball \( \left\{ {z \in {\mathbb{C}}^{n} : \left| z\right| < 1}\right\} \) . | Proof. Consider the integral\n\n\[ I\left( {{z}_{1},{z}_{2}}\right) = \frac{1}{2\pi i}{\int }_{\left| {\zeta }_{1}\right| = a + \epsilon }\frac{F\left( {{\zeta }_{1},{z}_{2}}\right) }{{\zeta }_{1} - {z}_{1}}d{\zeta }_{1} \]\n\nwhich is well-defined for small positive \( \epsilon \), when \( \left( {{z}_{1},{z}_{2}}\rig... | Yes |
Lemma 2.2 If the function \( F \) is holomorphic in a region \( \mathcal{O} \) that contains the union \( {K}_{1} \cup {K}_{2} \) then \( F \) extends analytically to an open set \( \widetilde{\mathcal{O}} \) containing the product set\n\n\[ \left\{ {\left( {{z}_{1},{z}_{2}}\right) : \left| {z}_{1}\right| \leq a,{b}_{2... | Proof. Consider the integral\n\n\[ I\left( {{z}_{1},{z}_{2}}\right) = \frac{1}{2\pi i}{\int }_{\left| {\zeta }_{1}\right| = a + \epsilon }\frac{F\left( {{\zeta }_{1},{z}_{2}}\right) }{{\zeta }_{1} - {z}_{1}}d{\zeta }_{1} \]\n\nwhich is well-defined for small positive \( \epsilon \), when \( \left( {{z}_{1},{z}_{2}}\rig... | Yes |
Proposition 3.1 Suppose \( f \) is continuous and has compact support on \( \mathbb{C} \) . Then:\n\n(a) \( {ugivenby}\left( 6\right) \) is also continuous and satisfies (5) in the sense of distributions.\n\n(b) If \( f \) is in the class \( {C}^{k}, k \geq 1 \), then so is \( u \), and \( u \) satisfies (5) in the usu... | Proof. Note first that\n\n\[ u\left( {z + h}\right) - u\left( z\right) = \frac{1}{\pi }{\int }_{{\mathbb{C}}^{1}}f\left( {z + h - \zeta }\right) - f\left( {z - \zeta }\right) \frac{d\zeta }{\zeta }, \]\n\nand that this tends to zero as \( h \rightarrow 0 \), by the uniform continuity of \( f \) and the fact that the fu... | Yes |
Proposition 3.2 Suppose \( n \geq 2 \) . If \( {f}_{j},1 \leq j \leq n \), are functions of class \( {C}^{k} \) of compact support that satisfy (7), then there exists a function \( u \) of class \( {C}^{k} \) and of compact support that satisfies the inhomogeneous Cauchy-Riemann equations (4). \( {}^{2} \) | Proof. Write \( z = \left( {{z}^{\prime },{z}_{n}}\right) \), where \( {z}^{\prime } = \left( {{z}_{1},\ldots ,{z}_{n - 1}}\right) \in {\mathbb{C}}^{n - 1} \) and set\n\n(8)\n\n\[ u\left( z\right) = \frac{1}{\pi }{\int }_{{\mathbb{C}}^{1}}{f}_{n}\left( {{z}^{\prime },{z}_{n} - \zeta }\right) \frac{{dm}\left( \zeta \rig... | Yes |
Theorem 4.1 Assume \( \Omega \) is a bounded region in \( {\mathbb{C}}^{n} \), whose boundary is of class \( {C}^{3} \), and suppose the complement of \( \bar{\Omega } \) is connected. If \( {F}_{0} \) is a function of class \( {C}^{3} \) on \( \partial \Omega \) that satisfies the tangential Cauchy-Riemann equations, ... | The proof of this theorem is in the same spirit as the previous one, but the details are different. The function \( {F}_{0} \) of class \( {C}^{3}\left( {\partial \Omega }\right) \) can, by definition, be thought of as a function of class \( {C}^{3} \) on the whole space. Now \( {F}_{0} \) satisfies the tangential Cauc... | Yes |
Proposition 5.1 Near any point \( {z}^{0} \in \partial \Omega \) we can introduce holomorphic coordinates \( \left( {{z}_{1},\ldots ,{z}_{n}}\right) \) centered at \( {z}^{0} \) so that\n\n\[ \Omega = \left\{ {\operatorname{Im}\left( {z}_{n}\right) > \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{\lambda }_{j}{\left| {z}_{j... | Proof of the proposition. As in (10), we see that we can introduce complex coordinates (with an affine complex linear change of variables) so that near \( {z}^{0} \) the set \( \Omega \) is given by\n\n\[ \operatorname{Im}\left( {z}_{n}\right) > \varphi \left( {{z}^{\prime },{x}_{n}}\right) \]\n\nwith \( z = \left( {{z... | Yes |
Corollary 6.2 Suppose the Levi form, as given by (18), has at least one strictly positive eigenvalue for each \( z \in M \) . Under these circumstances, for every \( {z}^{0} \in M \) there is a ball \( {B}^{\prime } \) centered at \( {z}^{0} \) so that whenever \( F \) is holomorphic in \( {\Omega }_{ - } \) and contin... | The theorem we have just proved tells us that when an eigenvalue of the Levi form is positive, the control of the restriction of a holomorphic function to a small piece of the boundary gives us a corresponding control of the function in an interior region. This is a strong hint that for such boundaries a local version ... | No |
Theorem 7.1 Suppose \( M \subset {\mathbb{C}}^{n} \) is a hypersurface of class \( {C}^{2} \) as above. Given a point \( {z}^{0} \in M \), there are open balls \( {B}^{\prime } \) and \( B \), centered at \( {z}^{0} \), with \( {\bar{B}}^{\prime } \subset B \), so that: if \( F \) is a continuous function in \( M \cap ... | Proof. We shall first take \( B \) small enough so that in \( B \), the hypersurface \( M \) has been represented by \( M = \left\{ {{y}_{n} = \varphi \left( {{z}^{\prime },{x}_{n}}\right) }\right\} \) where \( {z}^{0} \) corresponds to the origin. Besides \( \varphi \left( {0,0}\right) = 0 \), we can also suppose that... | Yes |
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