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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 |
Theorem 4.2 Let \( \left( {X,\mathcal{F},\mu }\right) \) be a finite measure space, that is, \( \mu \left( X\right) < \) \( \infty \) . Suppose that:\n\n(i) \( E \) is a closed subspace of \( {L}^{p}\left( {X,\mu }\right) \), for some \( 1 \leq p < \infty \), and\n\n(ii) \( E \) is contained in \( {L}^{\infty }\left( {... | Since \( E \subset {L}^{\infty } \), and \( X \) has finite measure, we find that \( E \subset {L}^{2} \) with\n\n\[ \parallel f{\parallel }_{{L}^{2}} \leq C\parallel f{\parallel }_{{L}^{\infty }}\;\text{ whenever }f \in E. \]\n\nThe essential idea in the proof of the theorem is to reverse this inequality, and then use... | 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 mea... | Yes |
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. H... | 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 |
Theorem 1.7\n\n(a) If \( \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\left| {c}_{n}\right| }^{2} < \infty \), then for almost every \( t \in \left\lbrack {0,1}\right\rbrack \) the function\n\n(10)\n\n\[ \n{f}_{t}\left( \theta \right) \sim \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\rho }_{n}\left( t\right)... | The proof is based on Khinchin's inequality, which like Lemma 1.6 is a further exploitation of the independence of the Rademacher functions.\n\nSuppose \( \left\{ {a}_{n}\right\} \) are complex numbers with \( \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\left| {a}_{n}\right| }^{2} < \infty \) . Let \( F\left( t\r... | 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 \( \parallel \mathbb{E}\left( f\right) {\parallel }_{{L}^{2}} \leq \parallel f{\parallel }_{{L}^{2}} \) . | 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}} = \mathop{\sup ... | Yes |
Proposition 2.6 Suppose \( \\left\\{ {f}_{k}\\right\\} \) is a sequence of integrable functions that are mutually independent and each have mean zero. Then there is an increasing family \( {\\mathcal{A}}_{n} \) of sub-algebras so that with respect to these \( {s}_{n} = \) \( \\mathop{\\sum }\\limits_{{k = 0}}^{n}{f}_{k... | To see this, we require further terminology. Let \( \\left\\{ {\\mathcal{B}}_{n}\\right\\} \) be a sequence of sub-algebras of \( \\mathcal{M} \) that are not assumed to be increasing. Then these are said to be mutually independent if for every \( N \) ,\n\n\[ \nm\\left( {\\mathop{\\bigcap }\\limits_{{j = 0}}^{N}{B}_{j... | 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. | See Exercise 7. | No |
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 \( \mathcal{B} \)... | No |
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 |
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 |
Theorem 3.1 The measures \( {\mu }_{N} \) on \( \mathcal{P} \) converge weakly to a measure as \( N \rightarrow \infty \) . This limit is the Wiener measure \( W \) . | There are two steps in the proof. The first, that the sequence \( {\mu }_{N} \) satisfies the tightness condition, is a little intricate. The second, that then \( {\mu }_{N} \) converges to the Wiener measure, is more direct. The second step is based on the central limit theorem. | No |
Lemma 3.2 We have as \( \lambda \rightarrow \infty \), \[ \mathop{\sup }\limits_{{n \geq 1}}m\left( \left\{ {x : \mathop{\sup }\limits_{{k \leq n}}\left| {{s}_{k}\left( x\right) }\right| > \lambda {n}^{1/2}}\right\} \right) = O\left( {\lambda }^{-p}\right) \] for 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... | \nThe 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... | Yes |
Proposition 5.1 Both \( {\tau }^{x} \) and \( {\tau }_{ * }^{x} \) are stopping times. | Proof. For simplicity of notation we take \( x = 0 \) ; we can then recover the general case by reducing to the situation where \( \mathcal{R} \) is replaced by \( \mathcal{R} - x \) . Now for any open set \( \mathcal{O} \) in \( {\mathbb{R}}^{d} \) define \( {\tau }_{\mathcal{O}}\left( \omega \right) = \inf \left\{ {t... | 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. \( {}^{9} \) | No |
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 \(... | No |
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 \( \left\{ {z \in {\mathbb{C}}^{n} : \left| z\right| < 1}\right\} \) . | Here we give a simple and elementary proof of this. Using more sophisticated arguments we shall see below that this property of \ | No |
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 |
Corollary 7.3 If \( f \) is a continuous function of compact support, then\n\n\[ \frac{\det \left( {I + A}\right) }{{\epsilon }^{n/2}}{\int }_{{\mathbb{R}}^{n}}{e}^{-\frac{\pi }{\epsilon }{\left( \left( I + A\right) v\right) }^{2}}f\left( {\xi + v}\right) {dv} \rightarrow f\left( \xi \right) \]\n\nuniformly in \( \xi \... | To prove the lemma note that \( \operatorname{Re}\left( {\left( \left( I + A\right) v\right) }^{2}\right) \geq {\left| v\right| }^{2} - \parallel A\parallel {\left| v\right| }^{2} \geq c{\left| v\right| }^{2} \) , with \( c > 0 \), so that the integral in (27) converges. A change of scale reduces the identity to the ca... | Yes |
Theorem 7.5 Suppose that the Levi form (18) has at least one strictly positive eigenvalue for each \( z \in M \) . Then for each \( {z}^{0} \in M \), there is a ball \( {B}^{\prime } \) centered at \( {z}^{0} \) so that whenever \( {F}_{0} \) is a continuous function on \( M \) that satisfies the tangential Cauchy-Riem... | To prove the theorem we first use Theorem 7.1 to find a ball \( {B}_{1} \) centered at \( {z}_{0} \) so that \( {F}_{0} \) can be uniformly approximated (on \( M \cap {B}_{1} \) ) by polynomials \( \left\{ {{p}_{n}\left( z\right) }\right\} \) . Then we invoke the corollary to Theorem 6.1 to find a ball \( {B}^{\prime }... | Yes |
Lemma 8.2 Suppose \( {B}_{1} \) and \( {B}_{2} \) are two open balls in \( {\mathbb{C}}^{n - 1} \), with \( {\bar{B}}_{1} \subset {B}_{2} \). Then, whenever \( f \) is holomorphic in \( {\mathbb{C}}^{n - 1} \n\n\[ \n\mathop{\sup }\limits_{{{z}^{\prime } \in {B}_{1}}}{\left| f\left( {z}^{\prime }\right) \right| }^{2} \l... | Indeed for sufficiently small \( \delta \), whenever \( {z}^{\prime } \in {B}_{1} \) then \( {B}_{\delta }\left( {z}^{\prime }\right) \subset {B}_{2} \), so since \( f \) is harmonic in \( {\mathbb{R}}^{{2n} - 2} \), the mean-value property and the Cauchy-Schwarz inequality gives\n\n\[ \n{\left| f\left( {z}^{\prime }\r... | No |
Theorem 8.5 Suppose \( F \in {H}^{2}\left( \mathcal{U}\right) \), and let \( {F}_{0} = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{F}_{\epsilon } \) as in Theorem 8.1. Then\n\n(38)\n\n\[ C\left( {F}_{0}\right) \left( z\right) = F\left( z\right) \] | The key lemma used is an observation giving a reproducing identity for a related space of entire functions on \( {\mathbb{C}}^{n - 1} \) . We consider the holomorphic functions \( f \) on \( {\mathbb{C}}^{n - 1} \) for which\n\n\[ {\int }_{{\mathbb{C}}^{n - 1}}{\left| f\left( {z}^{\prime }\right) \right| }^{2}{e}^{-{4\... | No |
Lemma 8.6 For \( f \) as above, we have\n\n(39)\n\n\[ f\left( {z}^{\prime }\right) = {\int }_{{\mathbb{C}}^{n - 1}}{K}_{\lambda }\left( {{z}^{\prime },{w}^{\prime }}\right) f\left( {w}^{\prime }\right) {e}^{-{4\pi \lambda }{\left| {w}^{\prime }\right| }^{2}}{dm}\left( {w}^{\prime }\right) \]\n\nwith \( {K}_{\lambda }\l... | Proof. In fact, consider first the case when \( {4\lambda } = 1 \), and \( {z}^{\prime } = 0 \) . Then (39), which states \( f\left( 0\right) = {\int }_{{\mathbb{C}}^{n - 1}}f\left( {w}^{\prime }\right) {e}^{-\pi {\left| {w}^{\prime }\right| }^{2}}{dm}\left( {w}^{\prime }\right) \), is a simple consequence of the mean-... | Yes |
Theorem 8.7 Suppose \( U \) is a distribution defined on \( \mathbb{C} \times \mathbb{R} \), so that \( \bar{L}\left( U\right) = f \) in a neighborhood of the origin. Then (41) must hold. | Proof. Assume first that \( U \) has compact support, and \( \bar{L}\left( U\right) = f \) everywhere. Then\n\n\[ C\left( f\right) \left( z\right) = \left\langle {f, S\left( {z,{u}_{2} + i{\left| {w}_{1}\right| }^{2}}\right) }\right\rangle = \left\langle {\bar{L}\left( U\right), S\left( {z,{u}_{2} + i{\left| {w}_{1}\ri... | Yes |
Proposition 1.1 The mapping \( f \mapsto A\left( f\right) \) is bounded from \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) to \( {L}_{k}^{2}\left( {\mathbb{R}}^{d}\right) \), with \( k = \frac{d - 1}{2} \) . | Proof. The proposition is a consequence of the identity\n\n(3)\n\n\[ \widehat{d\sigma }\left( \xi \right) = {2\pi }{\left| \xi \right| }^{-d/2 + 1}{J}_{d/2 - 1}\left( {{2\pi }\left| \xi \right| }\right) \]\n\nwhere \( \widehat{d\sigma }\left( \xi \right) = {\int }_{{S}^{d - 1}}{e}^{-{2\pi ix} \cdot \xi }{d\sigma }\left... | Yes |
Proposition 2.1 Suppose \( \left| {\nabla \Phi \left( x\right) }\right| \geq c > 0 \) for all \( x \) in the support of \( \psi \) . Then for every \( N \geq 0 \n\n\[ \left| {I\left( \lambda \right) }\right| \leq {c}_{N}{\lambda }^{-N},\;\text{ whenever }\lambda > 0. \] | Proof. We consider the following vector field\n\n\[ L = \frac{1}{i\lambda }\mathop{\sum }\limits_{{k = 1}}^{d}{a}_{k}\frac{\partial }{\partial {x}_{k}} = \frac{1}{i\lambda }\left( {a \cdot \nabla }\right) ,\]\n\nwith \( a = \left( {{a}_{1},\ldots ,{a}_{d}}\right) = \frac{\nabla \Phi }{{\left| \nabla \Phi \right| }^{2}}... | Yes |
Proposition 2.2 In the above situation, \( \left| {{I}_{1}\left( \lambda \right) }\right| \leq c{\lambda }^{-1} \), all \( \lambda > 0 \), with \( c = 3 \) . | Proof. The proof uses the operator \( L \) that occurred in the previous proposition. We may assume \( {\Phi }^{\prime } > 0 \) on \( \left\lbrack {a, b}\right\rbrack \), because the case when \( {\Phi }^{\prime } < 0 \) follows by taking complex conjugates. So \( L = \frac{1}{{i\lambda }{\Phi }^{\prime }\left( x\right... | Yes |
Proposition 2.3 Under the above assumptions, and with \( {I}_{1}\left( \lambda \right) \) given by (7) we have\n\n\[ \left| {{I}_{1}\left( \lambda \right) }\right| \leq {c}^{\prime }{\lambda }^{-1/2}\;\text{ for all }\lambda > 0,\text{ with }{c}^{\prime } = 8. \] | Proof. We may assume that \( {\Phi }^{\prime \prime }\left( x\right) \geq 1 \) throughout the interval, because the case \( {\Phi }^{\prime \prime }\left( x\right) \leq - 1 \) follows from this by taking complex conjugates. Now \( {\Phi }^{\prime \prime }\left( x\right) \geq 1 \) implies that \( {\Phi }^{\prime }\left(... | Yes |
Corollary 2.4 Assume \( \Phi \) satisfies the hypotheses of Proposition 2.3. Then\n\n\[ \left| {{\int }_{a}^{b}{e}^{{i\lambda \Phi }\left( x\right) }\psi \left( x\right) {dx}}\right| \leq {c}_{\psi }{\lambda }^{-1/2} \]\n\nwhere \( {c}_{\psi } = 8\left( {{\int }_{a}^{b}\left| {{\psi }^{\prime }\left( x\right) }\right| ... | Proof. Let \( J\left( x\right) = {\int }_{a}^{x}{e}^{{i\lambda \Phi }\left( u\right) }{du} \) . We integrate by parts, using \( J\left( a\right) = \) 0 . Then\n\n\[ {\int }_{a}^{b}{e}^{{i\lambda \Phi }\left( x\right) }\psi \left( x\right) {dx} = - {\int }_{a}^{b}J\left( x\right) \frac{d\psi }{dx}{dx} + J\left( b\right)... | Yes |
Corollary 3.2 If \( M \) has at least \( m \) non-vanishing principal curvatures at each point of the support of \( {d\mu } \), then\n\n\[ \left| {\widehat{d\mu }\left( \xi \right) }\right| = O\left( {\left| \xi \right| }^{-m/2}\right) \;\text{ as }\left| \xi \right| \rightarrow \infty . \]\n | First some preliminary remarks. We can assume that the support of \( \psi \) is centered in a sufficiently small ball (so that in particular the representation (18) of \( M \) holds in it), because we can always write a given \( \psi \) as a finite sum of \( {\psi }_{j} \) of that type. Next, all our estimates can be m... | No |
Corollary 3.3 If \( M = \partial \Omega \) has non-vanishing Gauss curvature at each point, then\n\n\[ \n{\widehat{\chi }}_{\Omega }\left( \xi \right) = O\left( {\left| \xi \right| }^{-\frac{d + 1}{2}}\right) ,\;\text{ as }\left| \xi \right| \rightarrow \infty .\n\] | Proof. Using an appropriate partition of unity we can write\n\n\[ \n{\chi }_{\Omega } = \mathop{\sum }\limits_{{j = 0}}^{N}{\psi }_{j}{\chi }_{\Omega }\n\]\n\nwith each \( {\psi }_{j} \) a \( {C}^{\infty } \) function of compact support; \( {\psi }_{0} \) is supported in the interior of \( \Omega \), while each \( {\ps... | Yes |
Corollary 4.3 If we only assume that \( M \) has at least \( m \) non-vanishing principal curvatures, then the same conclusions hold with \( k = m/2 \), and \( p = \frac{m + 2}{m + 1}, q = m + 2 \) . | The proof of part (a) in the theorem is the same as that for the sphere once we invoke the decay (21), which implies that \( {\left( 1 + {\left| \xi \right| }^{2}\right) }^{k/2}\widehat{d\mu }\left( \xi \right) \) is bounded. Hence\n\n\[ \parallel A\left( f\right) {\parallel }_{{L}_{k}^{2}} = {\begin{Vmatrix}{\left( 1 ... | Yes |
Proposition 4.4 With the above assumptions,\n\n\[ \n{\begin{Vmatrix}{T}_{c}\end{Vmatrix}}_{{L}^{q}} \leq M\parallel f{\parallel }_{{L}^{p}} \n\]\n\nfor any \( c \) with \( a \leq c \leq b \), where \( c = \left( {1 - \theta }\right) a + {\theta b} \) and \( 0 \leq \theta \leq 1 \) ; and\n\n\[ \n\frac{1}{p} = \frac{1 - ... | Once we have formulated this result, we in fact observe that we can prove it by essentially the same argument as in Section 2 in Chapter 2.\n\nWe write \( s = a\left( {1 - z}\right) + {bz} \), so \( z = \frac{s - a}{b - a} \), and the strip \( S \) is thereby transformed into the strip \( 0 \leq \operatorname{Re}\left(... | Yes |
Proposition 4.5 The Fourier transform \( {\widehat{K}}_{s}\left( \xi \right) \) is analytically continuable into the half-plane \( - \frac{d - 1}{2} \leq \operatorname{Re}\left( s\right) \) and satisfies\n\n\[\n\mathop{\sup }\limits_{{\xi \in {\mathbb{R}}^{d}}}\left| {{\widehat{K}}_{s}\left( \xi \right) }\right| \leq M... | This is based on the following one-dimensional Fourier transform calculation. We suppose that \( F \) is a \( {C}^{\infty } \) function on \( \mathbb{R} \) with compact support, and let\n\n\[{I}_{s}\left( \rho \right) = s\left( {s + 1}\right) \cdots \left( {s + N}\right) {\int }_{0}^{\infty }{u}^{s - 1}F\left( u\right)... | No |
Lemma 4.6 \( {I}_{s}\left( \rho \right) \) initially given above for \( \operatorname{Re}\left( s\right) > 0 \), has an analytic continuation into the half-space \( \operatorname{Re}\left( s\right) > - N - 1 \) . | Proof. Write \( s\left( {s + 1}\right) \cdots \left( {s + N}\right) {u}^{s - 1} = {\left( \frac{d}{du}\right) }^{N + 1}{u}^{s + N} \) . Then an \( \left( {N + 1}\right) \) -fold integration by parts yields\n\n\[ \n{I}_{s}\left( \rho \right) = {\left( -1\right) }^{N + 1}{\int }_{0}^{\infty }{u}^{s + N}{\left( \frac{d}{d... | Yes |
Proposition 5.1 Suppose \( f \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \) is a radial function. Then \( \widehat{f} \) is continuous for \( \xi \neq 0 \) whenever \( 1 \leq p < {2d}/\left( {d + 1}\right) \) . Note the sequence of exponents \( \frac{2d}{\left( d + 1\right) } : 1,\frac{4}{3},\frac{3}{2},\frac{8}{5},\ldot... | Proof. Suppose \( f\left( x\right) = {f}_{0}\left( \left| x\right| \right) \) . Then \( \widehat{f}\left( \xi \right) = F\left( \left| \xi \right| \right) \) with \( F \) defined by (4), namely,\n\n(29)\n\n\[ F\left( \rho \right) = {2\pi }{\rho }^{-d/2 + 1}{\int }_{0}^{\infty }{J}_{d/2 - 1}\left( {2\pi \rho r}\right) {... | Yes |
Theorem 5.2 Suppose \( M \) has non-zero Gauss curvature at each point of the support of \( {d\mu } \) . Then the restriction inequality (31) holds for \( q = 2 \) and \( p = \frac{{2d} + 2}{d + 3} \) . | The proof starts with several quick observations. Let \( \mathcal{R} \) denote the restriction operator\n\n\[ \mathcal{R}\left( f\right) = {\left. \widehat{f}\left( \xi \right) \right| }_{M} = {\left. {\int }_{{\mathbb{R}}^{d}}{e}^{-{2\pi ix} \cdot \xi }f\left( x\right) dx\right| }_{M}, \]\n\nwhich is initially defined... | No |
Corollary 5.4 Under the assumptions of the theorem, the restriction inequality (31) holds for \( 1 \leq p \leq \frac{{2d} + 2}{d + 3} \) and \( q \leq \left( \frac{d - 1}{d + 1}\right) {p}^{\prime } \) . | This follows by combining the critical case \( p = \frac{{2d} + 2}{d + 3}, q \leq 2 \) (a consequence of the theorem and Hölder's inequality) with the trivial case \( p = 1, q = \infty \) via the Riesz interpolation theorem. | Yes |
Proposition 6.1 For each \( t \) :\n\n(i) \( {e}^{{it}\bigtriangleup } \) maps \( \mathcal{S} \) to \( \mathcal{S} \) .\n\n(ii) If we set \( u\left( {x, t}\right) = {e}^{{it}\bigtriangleup }\left( f\right) \left( x\right) \), with \( f \in \mathcal{S} \), then \( u \) is a \( {C}^{\infty } \) function of \( \left( {x, ... | Proof. That \( {e}^{{it}\bigtriangleup } \) maps \( \mathcal{S} \) to \( \mathcal{S} \) is clear because the multiplier \( {e}^{-{it4}{\pi }^{2}{\left| \xi \right| }^{2}} \) has the property that each derivative in \( \xi \) is of at most polynomial increase. Next, the Fourier inversion formula gives\n\n\[ u\left( {x, ... | Yes |
Proposition 6.2 For each \( t \) :\n\n(i) The operator \( {e}^{{it}\bigtriangleup } \) is unitary on \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \).\n\n(ii) For every \( f \), the mapping \( t \mapsto {e}^{{it}\bigtriangleup }\left( f\right) \) is continuous in the \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) norm.\n\n(iii... | Proof. Conclusion (i) is immediate from Plancherel's theorem, since the multiplier \( {e}^{-{it4}{\pi }^{2}{\left| \dot{\xi }\right| }^{2}} \) has absolute value one. Now if \( \widehat{f} \in {L}^{2}\left( {\mathbb{R}}^{d}\right) \) , then clearly \( {e}^{-{it4}{\pi }^{2}{\left| \xi \right| }^{2}}\widehat{f}\left( \xi... | Yes |
Theorem 6.4 The solution \( {e}^{t{\left( \frac{d}{dx}\right) }^{3}}\left( f\right) \) satisfies\n\n\[ \parallel u{\parallel }_{{L}^{q}\left( {\mathbb{R}}^{2}\right) } \leq c\parallel f{\parallel }_{{L}^{2}\left( \mathbb{R}\right) },\;\text{ with }q = 8. \] | The proof of this is result is parallel with that of the previous theorem and reduces to a restriction theorem on \( {\mathbb{R}}^{2} \) for the cubic curve\n\n\[ \Gamma = \left\{ {\left( {{\xi }_{1},{\xi }_{2}}\right) : {\xi }_{2} = - 4{\pi }^{2}{\xi }_{1}^{3}}\right\} \]\n\nAccording to Corollary 5.5, what is needed ... | No |
Lemma 6.5 Let \( I\left( \xi \right) = {\int }_{\mathbb{R}}{e}^{{2\pi i}\left( {{\xi }_{1}t + {\xi }_{2}{t}^{3}}\right) }\psi \left( t\right) {dt} \), where \( \psi \) is a \( {C}^{\infty } \) function of compact support. Then\n\n\[ I\left( \xi \right) = O\left( {\left| \xi \right| }^{-1/3}\right) ,\;\text{ as }\left| ... | Proof. First note that \( I\left( \xi \right) = O\left( {\left| {\xi }_{2}\right| }^{-1/3}\right) \) . In fact\n\n\[ I\left( \xi \right) = {\int }_{\left| t\right| \leq {\left| {\xi }_{2}\right| }^{-1/3}} + {\int }_{\left| t\right| > {\left| {\xi }_{2}\right| }^{-1/3}}. \]\n\nThe first integral is obviously \( O\left( ... | Yes |
Proposition 6.6 Suppose \( F \) is a \( {C}^{\infty } \) function on \( {\mathbb{R}}^{d} \times \mathbb{R} \) of compact support. Then \( S\left( F\right) \) is a \( {C}^{\infty } \) function that satisfies (43) and (44). | Proof. Write \( F = {e}^{{it}\bigtriangleup }G\left( {\cdot, t}\right) \) with \( G\left( {x, t}\right) = i{\int }_{0}^{t}{e}^{-{is}\bigtriangleup }F\left( {\cdot, s}\right) {ds} \) . Now \( F\left( {\cdot, s}\right) \) is in the Schwartz space \( \mathcal{S}\left( {\mathbb{R}}^{d}\right) \) for each \( s \) and depend... | Yes |
Proposition 6.8 If \( F \in {L}^{p}\left( {{\mathbb{R}}^{d} \times \mathbb{R}}\right) \) then \( S\left( F\right) \) can be corrected (that is, redefined on a set of measure zero) so that for each \( t, S\left( F\right) \left( {\cdot, t}\right) \) belongs to \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) and, moreover, th... | This is based on the inequality\n\n(52)\n\n\[ \n{\begin{Vmatrix}{\int }_{\alpha }^{\beta }{e}^{-{is}\bigtriangleup }F\left( \cdot, s\right) ds\end{Vmatrix}}_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } \leq c\parallel F{\parallel }_{{L}^{p}\left( {{\mathbb{R}}^{d} \times \mathbb{R}}\right) }, \]\n\nwith \( c \) independent... | Yes |
Proposition 7.4 Assume that\n\n\[ \n\begin{Vmatrix}{{T}_{k}{T}_{j}^{ * }}\end{Vmatrix} \leq {a}^{2}\left( {k - j}\right) \;\text{ and }\;\begin{Vmatrix}{{T}_{k}^{ * }{T}_{j}}\end{Vmatrix} \leq {a}^{2}\left( {k - j}\right) .\n\]\n\nThen for every \( r \) ,\n\n(72)\n\n\[ \n\begin{Vmatrix}{\mathop{\sum }\limits_{{k = 0}}^... | Proof. We write \( T = \mathop{\sum }\limits_{{k = 0}}^{r}{T}_{k} \) and recall that \( \parallel T{\parallel }^{2} = \begin{Vmatrix}{T{T}^{ * }}\end{Vmatrix} \) . Since \( T{T}^{ * } \) is self-adjoint we may use this identity repeatedly to obtain \( \parallel T{\parallel }^{2n} = \) \( \begin{Vmatrix}{\left( T{T}^{ *... | Yes |
Proposition 8.1 \( \mathop{\sum }\limits_{{k = 1}}^{\mu }{r}_{2}\left( k\right) = {\pi \mu } + O\left( {\mu }^{1/2}\right) \), as \( \mu \rightarrow \infty \) . | The proof depends on the realization that \( \mathop{\sum }\limits_{{k = 0}}^{\mu }{r}_{2}\left( k\right) \) represents the number of lattice points in the disc of radius \( R \) with \( {R}^{2} = \mu \) . In fact, with \( {\mathbb{Z}}^{2} \) denoting the lattice points in \( {\mathbb{R}}^{2} \), that is, the points in... | Yes |
Proposition 8.2 Suppose \( f \) belongs to the Schwartz space \( \mathcal{S}\left( {\mathbb{R}}^{d}\right) \) . Then\n\n\[ \mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}f\left( n\right) = \mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}\widehat{f}\left( n\right) \]\n\nHere \( {\mathbb{Z}}^{d} \) denotes the collectio... | For the proof consider two sums\n\n\[ \mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}f\left( {x + n}\right) \;\text{ and }\;\mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}\widehat{f}\left( n\right) {e}^{{2\pi in} \cdot x}. \]\n\nBoth are rapidly converging series (since \( f \) and \( \widehat{f} \) are in \( \mathca... | Yes |
Theorem 8.3 \( N\left( R\right) = \pi {R}^{2} + O\left( {R}^{2/3}\right) \), as \( R \rightarrow \infty \). | Proof. We replace the characteristic function \( {\chi }_{R} \) by a regularized version as follows. We fix a non-negative \ | No |
Theorem 8.5\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\mu }d\left( k\right) = \mu \log \mu + \left( {{2\gamma } - 1}\right) \mu + O\left( {{\mu }^{1/3}\log \mu }\right) \;\text{ as }\mu \rightarrow \infty .{}^{16} \] | Now as much as we might wish to follow the lines of the proof of Theorem 8.3, there are serious obstacles that seem to stand in the way. In fact, if \( {\chi }_{\mu } \) is the characteristic function of the region\n\n\[ \left\{ {\left( {{x}_{1},{x}_{2}}\right) \in {\mathbb{R}}^{2} : {x}_{1}{x}_{2} \leq \mu ,{x}_{1} > ... | Yes |
Corollary 8.7 The conclusions for \( {\mathfrak{J}}_{a, b}^{ - } \) are the same as those for \( {\mathfrak{J}}_{a, b}^{ + } \) stated in Proposition 8.6, except that (i) should be modified to read that uniformly in \( a, b \) ,\n\n\[ \left( {\mathrm{i}}^{\prime }\right) {\mathfrak{J}}_{a, b}^{ - } = O\left( {\left| \l... | The only change occurs in the treatment of \( {II} \), namely \( \int {e}^{{i\lambda \Phi }\left( u\right) }\alpha \left( u\right) \frac{du}{u} \) , where now \( \Phi \left( u\right) = u - 1/u \) . In this case \( {\Phi }^{\prime }\left( u\right) = 1 + 1/{u}^{2} > 1 \), and there is no critical point. So Proposition 2.... | Yes |
Theorem 8.9 Let \( \widehat{f} \) be the Fourier transform of \( f\left( {x, y}\right) = {f}_{0}\left( {xy}\right) \) . Then \( \widehat{f} \) is a continuous function where \( {\xi \eta } \neq 0 \) . It is given by\n\n\[ \widehat{f}\left( {\xi ,\eta }\right) = 2{\int }_{0}^{\infty }{\mathfrak{J}}^{ + }\left( {-{2\pi }... | Proof. We approximate \( f \) by \( {f}_{\epsilon } \), with \( {f}_{\epsilon }\left( {x, y}\right) = {f}_{0}\left( {xy}\right) {\eta }_{\epsilon }\left( x\right) {\eta }_{\epsilon }\left( y\right) \) . Then each \( {f}_{\epsilon } \) is a \( {C}^{\infty } \) function of compact support, and clearly \( {f}_{\epsilon } ... | Yes |
Corollary 8.10 The Fourier transforms \( {\widehat{f}}_{\epsilon } \) and \( \widehat{f} \) satisfy the following estimate, uniformly in \( \epsilon \) :\n\n(94)\n\n\[ \left| {{\widehat{f}}_{\epsilon }\left( {\xi ,\eta }\right) }\right| \leq {A}_{N}{\left| \xi \eta \right| }^{-N}\;\text{ when }\left| {\xi \eta }\right|... | This is a consequence of the asymptotic behavior of \( {\mathfrak{J}}^{ \pm }\left( \lambda \right) \) for \( \lambda \) as given in Proposition 8.6 and its corollary together with the fact that \( {\int }_{0}^{\infty }{e}^{-{4\pi i\rho }{\left| \xi \eta \right| }^{1/2}}{f}_{0}\left( {\rho }^{2}\right) {\rho d\rho } \)... | Yes |
Theorem 4.1 Let (C3) and (C4) be satisfied. If for some \( \delta > 0,\mathrm{E}{\left| {\xi }_{1}\right| }^{2 + \delta } < \infty \), and \( \alpha \left( n\right) = O\left( {n}^{-\theta }\right) \) with \( \theta > \left( {2 + \delta }\right) /\delta \), then\n\n\[ \n{\sigma }_{n}{\left( x\right) }^{-1}\left\{ {{g}_{... | Proof of Theorem 4.1 For convenient writing, omitting everywhere the argument \( x \) and setting \( {S}_{n}^{* * } = {\sigma }_{n}^{-1}\left( {{g}_{n} - \mathrm{E}{g}_{n}}\right) \), then we have\n\n\[ \n{S}_{n}^{* * } = \mathop{\sum }\limits_{{i = 1}}^{n}{\sigma }_{n}^{-1}{w}_{ni}{\varepsilon }_{ni} = \mathop{\sum }\... | Yes |
Theorem 3.1 Let \( f : G \rightarrow {\mathbb{R}}^{n} \) is a continuous function whose derivative \( {f}_{x} \) exist and is continuous on \( G \), and \( g : G \rightarrow {\mathbb{R}}^{n} \) is continuously differentiable functions. Let \( f \) satisfies (i), (ii). Further assume that \( u : \left\lbrack {a, b}\righ... | Proof We borrow some ideas from [2].\n\nAccording to the assumptions, there exist positive constants \( A, B, K, H \) such that\n\n\[ \begin{Vmatrix}{{f}_{x}\left( {x, t}\right) }\end{Vmatrix} \leq A\;\begin{Vmatrix}{{f}_{x}\left( {x, t}\right) - {f}_{x}\left( {y, t}\right) }\end{Vmatrix} \leq B\parallel x - y\parallel... | Yes |
Example 2.1 Let \( X = Y = {\mathbb{R}}^{2}, D = {\mathbb{R}}_{ + }^{2}, e = \left( {1,1}\right) \in \operatorname{int}\left( D\right), S = \left\{ {x = \left( {{x}_{1},{x}_{2}}\right) \in {\mathbb{R}}^{2}}\right. \) : \( \left. {2{x}_{1} + {x}_{2} > 2,{x}_{1} \geq 0,{x}_{2} \geq 0}\right\} \), and for any map \( \eta ... | In fact, let \( \bar{x} = \left( {4,0}\right) ,\bar{z} = \left( {0,4}\right) ,\bar{t} = \frac{1}{2} \), then It yields that \( \bar{x},\bar{z} \in S \) , \( \left( {1,0}\right) \in F\left( \bar{x}\right) ,\left( {0,2}\right) \in F\left( \bar{z}\right) \) . For any mapping \( \eta \left( {x, z}\right) \) , \[ F\left( {z... | Yes |
Theorem 3.1 Let \( S \subset X \) be an invex set with respect to \( \eta, e \in \operatorname{int}\left( D\right) \) and the set-valued mapping \( F : S \rightarrow {2}^{Y} \) be \( D \) -subpreinvex on \( S \) with respect to \( e \) and \( \eta \) . Suppose that \( \bar{x} \in S,\bar{y} \in F\left( \bar{x}\right) \)... | Proof For any \( x \in S \) and \( y \in F\left( x\right) \), defined a sequence \( {\left( {x}_{n},{y}_{n}\right) }_{n \in \mathbb{N}} \) as follows:\n\n\[ {x}_{n} \mathrel{\text{:=}} \bar{x} + \frac{1}{n} \cdot \eta \left( {x,\bar{x}}\right) ,\;{y}_{n} \mathrel{\text{:=}} \left( {1 - \frac{1}{n}}\right) \frac{1}{n} \... | Yes |
Theorem 3.2 Let \( S \) be an invex set with respect to \( \eta, e \in \operatorname{int}\left( D\right) \) and \( F \) be a \( D \) - subpreinvex set-valued map on \( S \) with respect to \( \eta \) and \( e \) . Then, for problem (SOP), every locally weak efficient element is a globally \( {\epsilon e} \) -efficient ... | Proof Suppose that \( \left( {\bar{x},\bar{y}}\right) \) is a locally weak efficient element of (SOP). Then there is a neighborhood \( N\left( \bar{x}\right) \) of \( \bar{x} \) such that\n\n\[ \left( {F\left( {S \cap N\left( \bar{x}\right) }\right) - \bar{y}}\right) \cap \left( {-\operatorname{int}\left( D\right) }\ri... | Yes |
Theorem 3.3 Let \( e \in \operatorname{int}\left( D\right) \) and \( \left( {\bar{x},\bar{y}}\right) \in \operatorname{graph}\left( F\right) \) . In problem (SOP), suppose that \( S \) is invex set with respect to \( \eta \), and the contingent epiderivative \( {DF}\left( {\bar{x},\bar{y}}\right) \) exists. If \( F \) ... | Proof Since the contingent epiderivative \( {DF}\left( {\bar{x},\bar{y}}\right) \) exists and (3.6) holds, we get that \[ \{ {DF}\left( {\bar{x},\bar{y}}\right) \left( {\eta \left( {x,\bar{x}}\right) }\right) \} \cap \left( {-\operatorname{int}\left( D\right) }\right) = \varnothing ,\forall x \in S, \] which implies \[... | Yes |
Theorem 3.4 Let \( e \in \operatorname{int}\left( D\right) \) and \( \left( {\bar{x},\bar{y}}\right) \in \operatorname{graph}\left( F\right) \) . In problem (SOP), suppose that \( S \) is invex set with respect to \( \eta \), and the contingent epiderivative \( {DF}\left( {\bar{x},\bar{y}}\right) \) exists. if \( F \) ... | Proof Since \( {FD} \) -subpreinvex set-valued map on \( S \) with respect to \( \eta \) and \( e \), it follows from Theorem 2.7 that \( F\left( x\right) - \bar{y} + e \subset {DF}\left( {\bar{x},\bar{y}}\right) \left( {\eta \left( {x,\bar{x}}\right) }\right) + D,\forall x \in S \) . Furthermore, by (3.8), we obtain t... | Yes |
Lemma 2.1 Assume \( 3 < p < \infty \) and \( \left( {{v}_{0},{w}_{0}}\right) \in {L}^{p}\left( {\mathbb{R}}^{3}\right) \) with \( \nabla \cdot {v}_{0} = 0 \) in the sense of distributions. Then there exist a constant \( T > 0 \) and a unique strong solution \( \left( {v, w}\right) \) of the 3D micropolar fluid equation... | \[ v \in {BC}\left( {\lbrack 0, T);{L}^{p}\left( {\mathbb{R}}^{3}\right) }\right) ,{t}^{\frac{1}{2}}\nabla u \in {BC}\left( {\lbrack 0, T);{L}^{p}\left( {\mathbb{R}}^{3}\right) }\right) . \] | Yes |
Lemma 1 Let \( H \) be a set of complex numbers satisfying \( \overline{\operatorname{dens}}\{ \left| z\right| : z \in H\} > 0 \), and let \( A\left( z\right) \) and \( B\left( z\right) \) be entire functions such that for some constants \( \alpha \geq 0,\mu > 0 \), we have\n\n\[ \left| {A\left( z\right) }\right| \leq ... | Proof Using the similar arguments as in Theorem D, we can easily prove the result. | No |
Lemma 3 Suppose \( f\left( z\right) \) and \( g\left( z\right) \) are two nonconstant meromorphic functions in the complex plane with \( {\sigma }_{2}\left( f\right) \) and \( {\sigma }_{2}\left( g\right) \) as the hyper-order of \( f\left( z\right) \) and \( g\left( z\right) \), respectively. Then\n\n\[ \n{\sigma }_{2... | Proof Using the similar arguments as in Lemma 2, we can easily prove the result. | No |
Lemma 5 Suppose \( f\left( z\right) \) and \( g\left( z\right) \) are two nonconstant meromorphic functions in the complex plane with \( \sigma \left( f\right) \) and \( \sigma \left( g\right) \) as the order of \( f\left( z\right) \) and \( g\left( z\right) \), respectively. If \( \sigma \left( f\right) = \) \( \infty... | Proof Suppose that \( \sigma \left( {fg}\right) < \infty \) . Then\n\n\[ \sigma \left( f\right) = \sigma \left( {{fg} \cdot \frac{1}{g}}\right) \leq \max \left\{ {\sigma \left( {fg}\right) ,\sigma \left( \frac{1}{g}\right) }\right\} = \max \{ \sigma \left( {fg}\right) ,\sigma \left( g\right) \} < \infty . \]\n\nThis is... | Yes |
Lemma 2.1 Let \( X \) be a Hilbert space and \( C \) be a nonempty closed and convex subset of \( X,\Gamma : C \rightrightarrows X \) be an any correspondence and \( {P}_{C} : X \rightarrow C \) be the metric projection. Then, \( {x}^{ * } \) is a solution to \( \operatorname{GVI}\left( {C,\Gamma }\right) \) if and onl... | \[ {x}^{ * } \in \operatorname{Fix}\left( {{P}_{C} \circ \left( {{\operatorname{id}}_{\mathrm{C}} - {\lambda \Gamma }}\right) }\right) \text{ for some function }\lambda : X \rightarrow {R}_{+ + }, \] where \( \operatorname{Fix}\left( {{P}_{C} \circ \left( {{\mathrm{{id}}}_{\mathrm{C}} - {\lambda \Gamma }}\right) }\righ... | Yes |
Theorem 4.1 Let \( \left( {X, \succcurlyeq }\right) \) be a separable Hilbert lattice and \( C \) be a weakly compact and convex \( \succcurlyeq \) -sublattice of \( X \) . If \( \Gamma : C \rightrightarrows X \) is a compact-valued mapping such that \( {\operatorname{id}}_{C} - {\lambda \Gamma } \) is lower \( \succcu... | Proof Define \( \Psi : C \rightrightarrows X \) by \( \Psi = {\operatorname{id}}_{C} - {\lambda \Gamma } \), and define \( f : C \rightrightarrows C \) by \( f = {P}_{C} \circ \Psi \) . Next, let us show that \( f \) satisfies the conditions of Theorem 3.1.\n\nFirstly, we claim that \( f \) is lower \( \succcurlyeq \) ... | Yes |
Theorem 5.1 Let \( \left( {X,{ \succcurlyeq }_{X}}\right) \) be a separable Hilbert lattice, \( \left( {\Omega ,{ \succcurlyeq }_{\Omega }}\right) \) and \( \left( {\Theta ,{ \succcurlyeq }_{\Theta }}\right) \) be posets, \( C : \Omega \rightrightarrows X \) and \( \Gamma : X \times \Theta \rightrightarrows X \) be set... | Proof Define \( f : X \times \Omega \times \Theta \rightrightarrows X \) by\n\n\[ f\left( {x,\omega ,\theta }\right) = {P}_{C\left( \omega \right) }\left( {x - \lambda \left( x\right) \Gamma \left( {x,\theta }\right) }\right) ,\]\n\n(5.4)\n\nwhere \( \lambda \) is the map given in condition (ii). Obviously, \( f \) is ... | Yes |
Corollary 5.1 Let \( \left( {X,{ \succcurlyeq }_{X}}\right) \) be a separable Hilbert lattice and \( C \) be a weakly compact and convex \( { \succcurlyeq }_{X} \) -sublattice of \( X,\left( {\Theta ,{ \succcurlyeq }_{\Theta }}\right) \) be a poset, and \( \Gamma : X \times \Theta \rightrightarrows X \) be a set-valued... | Proof Define \( {C}^{ * } : \Omega \rightrightarrows X \) by \( {C}^{ * }\left( \omega \right) = C \) for any \( \omega \in \Omega \), and choose \( Y = C \) . It is easy to check that \( {C}^{ * } \) and \( \Gamma \) satisfy all the conditions of Theorem 5.1. | No |
Theorem 4.2 Let \( {H}_{1} = {F}_{0} - {G}_{0} \) be a strong symmetric \( P \) -regular splitting. If \( \lambda \) is an eigenvalue of the preconditioned matrix \( P \) and \( \mu \) is an eigenvalue of \( Q \), then\n\n\[ \lambda = \frac{1 - \mu }{\alpha } \]\n\n(4.5)\n\nand\n\n\[ \kappa \left( P\right) = \kappa \le... | Proof From the assumptions in the theorem, we have\n\n\[ P = {\left( F + \alpha {H}_{2}\right) }^{-1}A = \frac{1}{\alpha }\left( {{\left( {F}_{0} + {H}_{2}\right) }^{-1}A}\right) = \frac{1}{\alpha }\left( {I - Q}\right) . \]\n\nThus, (4.5) and (4.6) hold true. | Yes |
Lemma 1.2 \( {}^{\left\lbrack 7\right\rbrack } \) Let \( {q}_{1},{q}_{2} \in \left( {1,\infty }\right) \) be given so that \( \frac{1}{p} = \frac{1}{{q}_{1}} + \frac{1}{{q}_{2}} \), and \( k \) be a natural number. Then, there exists a constant \( m = m\left( {k, p,{q}_{1},{q}_{2}}\right) > 0 \) such that for any \( \d... | \[ \leq m{\left( \frac{n}{\delta }\right) }^{k}{\left( 1 + \mathop{\max }\limits_{{1 \leq i \leq n}}{\begin{Vmatrix}{F}_{i}\end{Vmatrix}}_{k, k{q}_{1}}\right) }^{k}\mu {\left( \mathop{\bigcap }\limits_{{i = 1}}^{n}\left\{ {a}_{i} - \delta < {F}_{i}\left( z\right) < {b}_{i} + \delta \right\} \right) }^{1/{q}_{2}}, \] | Yes |
Lemma 1.3 For any \( f \in V \) and \( \varphi > 0,{t}_{i} \geq 0,{u}_{i} > 0 \), we set\n\n\[ \n{F}_{\varphi }^{\left( i\right) }\left( w\right) = {\begin{Vmatrix}\varphi \left( \frac{w\left( {{t}_{i} + {u}_{i} \cdot }\right) - w\left( {t}_{i}\right) }{\sqrt{{u}_{i}}}\right) - f\end{Vmatrix}}_{\alpha }, i = 1,2,\cdots... | Proof By applying Lemma 2.1 of [7], the proof is similar to the one done in Lemma 2.2. | No |
Lemma 1.4 Let \( k\left( \alpha \right) > 0 \) and \( \gamma \) be defined as in (1.2), then for any \( \tau > 0 \) and \( f \in V \) we have\n\n\[ \mathop{\lim }\limits_{{\varphi \rightarrow 0}}{\varphi }^{1/\gamma }\log {C}_{r, p}\left( {{\begin{Vmatrix}\frac{w\left( {t + u \cdot }\right) - w\left( t\right) }{\sqrt{u... | Proof By applying Thereom 4.4 in [2], the proof is similar to the one done in Lemma 2.4 of [7]. | No |
Theorem 2.1 We have\n\n\[ \n\mathop{\liminf }\limits_{{u \rightarrow \infty }}{z}_{u}{}^{\gamma }{a}_{u}{}^{-\frac{1}{2}}\mathop{\inf }\limits_{{t \in \left\lbrack {0,1 - {b}_{u}}\right\rbrack }}\parallel \Delta \left( {t, u}\right) {\parallel }_{\alpha } = {\left( k\left( \alpha \right) \right) }^{\gamma },\;{C}_{r, p... | Proof We divide the proof of Theorem 2.1 into the following lemmas. (2.1) follows from Lemma 2.1 and Lemma 2.2, while (2.2) follows from Lemma 2.1 and Lemma 2.3. | No |
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