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Let \( b \) be a bounded function on \( {\mathbf{R}}^{n} \). Consider the symbol \[ {\sigma }_{b}\left( {x,\xi }\right) = \mathop{\sum }\limits_{{j \in \mathbf{Z}}}\left( {b * {\Psi }_{{2}^{-j}}}\right) \left( x\right) \widehat{\Psi }\left( {{2}^{-j}\xi }\right) ,\] where \( \widehat{\Psi } \) is a smooth function supp... | Indeed, every differentiation in \( x \) produces a factor of \( {2}^{j} \), while every differentiation in \( \xi \) produces a factor of \( {2}^{-j} \). But since \( \widehat{\Psi } \) is supported in \( \frac{1}{2} \cdot {2}^{j} \leq \left| \xi \right| \leq 2 \cdot {2}^{j} \), it follows that \( \left| \xi \right| \... | Yes |
Proposition 4.6.1. Let \( \Gamma \) be as in (4.6.1). Let \( f\left( \zeta \right) \) be a \( {\mathcal{C}}^{1} \) function on \( \Gamma \) that decays faster than \( C{\left| \zeta \right| }^{-1} \) as \( \left| \zeta \right| \rightarrow \infty \) . Then the limit in (4.6.3) exists as \( \varepsilon \rightarrow 0 \) f... | Proof. We show first that the limit in (4.6.3) exists as \( \varepsilon \rightarrow 0 \) . For \( z \in \Gamma \) and \( 0 < \varepsilon < 1 \) we write\n\n\[ \frac{1}{\pi i}\mathop{\int }\limits_{\substack{{\zeta \in \Gamma } \\ {\left| {\operatorname{Re}\zeta - \operatorname{Re}z}\right| > \varepsilon } }}\frac{f\lef... | Yes |
For \( s > 0 \), let \( {\theta }_{s} \) be a family of kernels satisfying (4.6.28) and (4.6.29) and let \( {\Theta }_{s} \) be the linear operator whose kernel is \( {\theta }_{s} \) . Suppose that for all \( s > 0 \) we have \[ {\Theta }_{s}\left( 1\right) = 0 \] (4.6.31) Then there is a constant \( {C}_{n,\delta } \... | Proof. We introduce Littlewood-Paley operators \( {Q}_{s} \) given by convolution with a smooth function \( {\Psi }_{s} = \frac{1}{{s}^{n}}\Psi \left( \frac{ \cdot }{s}\right) \) whose Fourier transform is supported in the annulus \( s/2 \leq \left| \xi \right| \leq {2s} \) that satisfies \[ {\int }_{0}^{\infty }{Q}_{s... | Yes |
Corollary 4.6.7. The Cauchy integral operator \( {\mathcal{C}}_{\Gamma } \) maps \( {L}^{2}\left( \mathbf{R}\right) \) to itself. | The corollary is a consequence of Theorem 4.6.6. Indeed, the crucial and important cancellation property\n\n\[ \n{\Theta }_{s}\left( {1 + i{A}^{\prime }}\right) = 0 \n\]\n\n(4.6.43)\n\nis valid for the accretive function \( 1 + i{A}^{\prime } \), when \( {\Theta }_{s} \) and \( {\theta }_{s} \) are as in (4.6.23) and (... | Yes |
Theorem 4.7.1. Let \( L \) be as in (4.7.2). Then there is a constant \( {C}_{n,\lambda ,\Lambda } \) such that for all smooth functions \( f \) with compact support, estimate (4.7.4) is valid. | The proof of this theorem requires certain estimates concerning elliptic operators. These are presented in the next subsection, while the proof of the theorem follows in the remaining four subsections. | No |
Lemma 4.7.2. Let \( E \) and \( F \) be two closed sets of \( {\mathbf{R}}^{n} \). Assume that the distance \( d = \operatorname{dist}\left( {E, F}\right) \) between them is positive. Then for all complex-valued functions \( f \) supported in \( E \) and all vector-valued functions \( \overrightarrow{f} \) supported in... | Proof. It suffices to obtain these inequalities whenever \( d \geq t > 0 \). Let us set \( {u}_{t} = \) \( {\left( I + {t}^{2}L\right) }^{-1}\left( f\right) \). For all \( v \in {L}_{1}^{2}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \n{\int }_{{\mathbf{R}}^{n}}{u}_{t}{vdx} + {t}^{2}{\int }_{{\mathbf{R}}^{n}}A\nabla... | Yes |
Lemma 4.7.3. Let \( {M}_{f} \) be the operator given by multiplication by a Lipschitz function \( f \) . Then there is a constant \( C \) that depends only on \( n,\lambda \), and \( \Lambda \) such that\n\n\[{\begin{Vmatrix}\left\lbrack {\left( I + {t}^{2}L\right) }^{-1},{M}_{f}\right\rbrack \end{Vmatrix}}_{{L}^{2} \r... | Proof. Set \( \overrightarrow{b} = A\nabla f,\overrightarrow{d} = {A}^{t}\nabla f \) and note that the operators given by pointwise multiplication by these vectors are \( {L}^{2} \) bounded with norms at most a multiple of \( C\parallel \nabla f{\parallel }_{{L}^{\infty }} \) . Write\n\n\[ \left\lbrack {{\left( I + {t}... | No |
Lemma 4.7.4. There exists a constant \( C \) depending only on \( n,\lambda \), and \( \Lambda \) such that for all \( Q \) cubes in \( {\mathbf{R}}^{n} \) with sides parallel to the axes, for all \( t \leq \ell \left( Q\right) \), and all Lipschitz functions \( f \) on \( {\mathbf{R}}^{n} \) we have\n\n\[ \frac{1}{\le... | Proof. We begin by proving (4.7.12), while we omit the proof of (4.7.13), since it is similar. By a simple rescaling, we may assume that \( \ell \left( Q\right) = 1 \) and that \( \parallel \nabla f{\parallel }_{{L}^{\infty }} = 1 \) . Set \( {Q}_{0} = {2Q} \) (i.e., the cube with the same center as \( Q \) with twice ... | Yes |
Lemma 4.7.5. For \( t > 0 \), let \( {U}_{t} \) be integral operators defined on \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) with measurable kernels \( {L}_{t}\left( {x, y}\right) \). Suppose that for some \( m > n \) and for all \( y \in {\mathbf{R}}^{n} \) and \( t > 0 \)\n\nwe have\n\[{\int }_{{\mathbf{R}}^{n}}{\lef... | Proof. We begin by observing that \( {U}_{t}^{ * }{U}_{t} \) has a kernel \( {K}_{t}\left( {x, y}\right) \) given by\n\n\[{K}_{t}\left( {x, y}\right) = {\int }_{{\mathbf{R}}^{n}}\overline{{L}_{t}\left( {z, x}\right) }{L}_{t}\left( {z, y}\right) {dz}.\n\]\n\nThe simple inequality \( \left( {1 + a + b}\right) \leq \left(... | Yes |
Lemma 4.7.6. Let \( {P}_{t} \) be as in Lemma 4.7.5. Then the operator \( {U}_{t} \) defined by \( {U}_{t}\left( \overrightarrow{f}\right) \left( x\right) = {\gamma }_{t}\left( x\right) \cdot {P}_{t}\left( \overrightarrow{f}\right) \left( x\right) - {Z}_{t}{P}_{t}\left( \overrightarrow{f}\right) \left( x\right) \) sati... | Proof. By the off-diagonal estimates of Lemma 4.7.2 for \( {Z}_{t} \) and the fact that \( p \) has support in the unit ball, it is simple to show that there is a constant \( C \) depending on \( n,\lambda \), and \( \Lambda \) such that for all \( y \in {\mathbf{R}}^{n} \) ,\n\n\[ \n\frac{1}{{t}^{n}}{\int }_{B\left( {... | Yes |
Lemma 4.7.7. The required estimate (4.7.4) follows from the Carleson measure estimate \[ \mathop{\sup }\limits_{Q}\frac{1}{\left| Q\right| }{\int }_{Q}{\int }_{0}^{\ell \left( Q\right) }{\left| {\gamma }_{t}\left( x\right) \right| }^{2}\frac{dxdt}{t} < \infty, \] where the supremum is taken over all cubes in \( {\mathb... | Proof. Indeed, (4.7.31) and Theorem 3.3.7 imply \[ {\int }_{{\mathbf{R}}^{n}}{\int }_{0}^{\infty }{\left| {P}_{t}^{2}\left( \nabla g\right) \left( x\right) \cdot {\gamma }_{t}\left( x\right) \right| }^{2}\frac{dxdt}{t} \leq C{\int }_{{\mathbf{R}}^{n}}{\left| \nabla g\right| }^{2}{dx} \] and together with (4.7.21) we de... | Yes |
Proposition 4.7.11. There exists an \( \varepsilon > 0 \) depending on \( n,\lambda \), and \( \Lambda \), and \( \eta = \) \( \eta \left( \varepsilon \right) > 0 \) such that for each unit vector \( w \) in \( {\mathbf{C}}^{n} \) and each cube \( Q \) with sides parallel to the axes, there exists a collection \( {\mat... | Proof. We begin by proving the following crucial estimate:\n\n\[ \left| {{\int }_{Q}\left( {1 - \nabla {f}_{Q, w}^{\varepsilon }\left( x\right) \cdot w}\right) {dx}}\right| \leq C{\varepsilon }^{\frac{1}{2}}\left| Q\right| \]\n\n(4.7.40)\n\nwhere \( C \) depends on \( n,\lambda \), and \( \Lambda \), but not on \( \var... | Yes |
Lemma 4.7.13. Let \( w, u, v \) be in \( {\mathbf{C}}^{n} \) such that \( \left| w\right| = 1 \) and let \( 0 < \varepsilon \leq 1 \) be such that\n\n\[ \left| {u - \left( {u \cdot \bar{w}}\right) w}\right| \leq \varepsilon \left| {u \cdot \bar{w}}\right| \]\n\n(4.7.44)\n\n\[ \operatorname{Re}\left( {v \cdot w}\right) ... | Proof. It follows from (4.7.45) that\n\n\[ \frac{3}{4}\left| {u \cdot \bar{w}}\right| \leq \left| {\left( {u \cdot \bar{w}}\right) \left( {v \cdot w}\right) }\right| \]\n\n(4.7.47)\n\nMoreover, (4.7.44) and the triangle inequality imply that\n\n\[ \left| u\right| \leq \left( {1 + \varepsilon }\right) \left| {u \cdot \b... | Yes |
Lemma 5.1.1. Let \( \delta > 0 \) be a given number. Then there exist a measurable subset \( E \) of \( {\mathbf{R}}^{2} \) and a finite collection of rectangles \( {R}_{j} \) in \( {\mathbf{R}}^{2} \) such that\n\n(1) The \( {R}_{j} \) ’s are pairwise disjoint.\n\n(2) We have \( 1/2 \leq \left| E\right| \leq 3/2 \) .\... | Proof. We start with an isosceles triangle \( {ABC} \) in the plane with height 1 and base \( {AB} \), where \( A = \left( {0,0}\right) \) and \( B = \left( {1,0}\right) \) . Given \( \delta > 0 \), we find a positive integer \( k \) such that \( k + 2 > {e}^{1/\delta } \) . For this \( k \) we set \( E = E\left( {1, k... | Yes |
Proposition 5.1.2. Let \( R \) be a rectangle whose center is the origin in \( {\mathbf{R}}^{2} \) and let \( v \) be a unit vector parallel to its longest side. Consider the half-plane\n\n\[ \mathcal{H} = \left\{ {x \in {\mathbf{R}}^{2} : x \cdot v \geq 0}\right\} \]\n\nand the multiplier operator\n\n\[ {S}_{\mathcal{... | Proof. Applying a rotation, we reduce the problem to the case \( R = \left\lbrack {-a, a}\right\rbrack \times \left\lbrack {-b, b}\right\rbrack \) , where \( 0 < a \leq b < \infty \), and \( v = {e}_{2} = \left( {0,1}\right) \) . Since the Fourier transform acts in each variable independently, we have the identity\n\n\... | Yes |
Theorem 5.1.5. Let \( n \geq 2 \) . The characteristic function of the unit ball in \( {\mathbf{R}}^{n} \) is not an \( {L}^{p} \) multiplier when \( 1 < p \neq 2 < \infty \) . | Proof. In view of Theorem 2.5.16 in [156], it suffices to prove the result in dimension \( n = 2 \) . By duality, matters reduce to the case \( p > 2 \) . To reach a contradiction, suppose that \( {\chi }_{B\left( {0,1}\right) } \in {\mathcal{M}}_{p}\left( {\mathbf{R}}^{2}\right) \) for some \( p > 2 \), say with norm ... | Yes |
Proposition 5.2.2. For all \( 1 \leq p \leq \infty \) and \( \operatorname{Re}\lambda > \frac{n - 1}{2},{B}^{\lambda } \) is a bounded operator on \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) with norm at most \( {C}_{1}{e}^{6{\left| \operatorname{Im}\lambda \right| }^{2}} \), where \( {C}_{1} \) depends smoothly on \( ... | Proof. The ingredients of the proof have already been discussed. | No |
Proposition 5.2.3. When \( \lambda > 0 \) and \( p \leq \frac{2n}{n + 1 + {2\lambda }} \) or \( p \geq \frac{2n}{n - 1 - {2\lambda }} \), the operators \( {B}^{\lambda } \) are not bounded on \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) . | Proof. Let \( h \) be a Schwartz function whose Fourier transform is equal to 1 on the ball \( B\left( {0,2}\right) \) and vanishes off the ball \( B\left( {0,3}\right) \) . Then\n\n\[ \n{B}^{\lambda }\left( h\right) \left( x\right) = {\int }_{\left| \xi \right| \leq 1}{\left( 1 - {\left| \xi \right| }^{2}\right) }^{\l... | Yes |
Theorem 5.2.6. There exists a constant \( C \) such that for all \( N \geq {10} \) and all \( f \) in \( {L}^{2}\left( {\mathbf{R}}^{2}\right) \) the following norm inequality is valid:\n\n\[ \mathop{\sup }\limits_{{a > 0}}{\begin{Vmatrix}{\mathcal{K}}_{N}^{a}\left( f\right) \end{Vmatrix}}_{{L}^{2}\left( {\mathbf{R}}^{... | Theorem 5.2.6 is a consequence of Theorem 5.3.5, in which the preceding estimate is proved for a more general maximal operator \( {\mathfrak{M}}_{{\sum }_{N}} \), which in particular controls \( {\mathcal{K}}_{N} \) and hence \( {\mathcal{K}}_{N}^{a} \) for all \( a > 0 \) . This maximal operator is introduced in the n... | No |
For \( N \in {\mathbf{Z}}^{ + } \), let\n\n\[ \sum = {\sum }_{N} = \left\{ {\left( {\cos \left( \frac{2\pi j}{N}\right) ,\sin \left( \frac{2\pi j}{N}\right) }\right) : j = 0,1,2,\ldots, N - 1}\right\} \]\n\nbe the set of \( N \) uniformly spread directions on the circle. Then we expect \( {\mathfrak{M}}_{{\sum }_{N}} \... | \[ {\mathcal{K}}_{N}\left( f\right) \leq {20}{\mathfrak{M}}_{{\sum }_{N}}\left( f\right) \] | Yes |
Proposition 5.3.4. There is a constant \( c \) such that for any \( N \geq {10} \) we have\n\n\[{\begin{Vmatrix}{\mathcal{K}}_{N}\end{Vmatrix}}_{{L}^{2}\left( {\mathbf{R}}^{2}\right) \rightarrow {L}^{2}\left( {\mathbf{R}}^{2}\right) } \geq c\log N\]\n\nand\n\n\[{\begin{Vmatrix}{\mathcal{K}}_{N}\end{Vmatrix}}_{{L}^{2}\l... | Proof. We consider the family of functions \( {f}_{N}\left( x\right) = \frac{1}{\left| x\right| }{\chi }_{3 \leq \left| x\right| \leq N} \) defined on \( {\mathbf{R}}^{2} \) for \( N \geq {10} \) . Then we have\n\n\[{\begin{Vmatrix}{f}_{N}\end{Vmatrix}}_{{L}^{2}\left( {\mathbf{R}}^{2}\right) } \leq {c}_{1}{\left( \log ... | Yes |
Lemma 5.3.6. Let \( N \in {\mathbf{Z}}^{ + } \) satisfy \( N \geq {1000} \) . Let \( {\omega }_{0} = 0 \) and define \( {\omega }_{k} = {2\pi }{2}^{k}/N \) for \( 1 \leq k < \left\lbrack {{\log }_{2}\left( {N/{64}}\right) }\right\rbrack \) and \( {\omega }_{\left\lbrack {\log }_{2}\left( N/{64}\right) \right\rbrack } =... | \[ {\omega }_{k} \leq \left| {{\theta }_{\alpha } - {\theta }_{\beta }}\right| < {\omega }_{k + 1} \] for some \( 0 \leq k < \left\lbrack {{\log }_{2}\left( {N/{64}}\right) }\right\rbrack \) . Let \[ {s}_{\alpha } = {16}\max \left( {{l}_{\alpha },{\omega }_{k}{L}_{\alpha }}\right) \] For an arbitrary \( x \in {R}_{\alp... | Yes |
For every \( 1 < p < \infty \) there exists a constant \( {c}_{p} \) such that\n\n\[{\begin{Vmatrix}{\mathcal{K}}_{N}\end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{2}\right) \rightarrow {L}^{p}\left( {\mathbf{R}}^{2}\right) } \leq {c}_{p}\left\{ \begin{array}{ll} {N}^{\frac{2}{p} - 1}{\left( \log N\right) }^{\frac{1}{{p}^... | Proof. We see that\n\n\[{\begin{Vmatrix}{\mathcal{K}}_{N}\end{Vmatrix}}_{{L}^{1}\left( {\mathbf{R}}^{2}\right) \rightarrow {L}^{1,\infty }\left( {\mathbf{R}}^{2}\right) } \leq {CN}\]\n\nwhich follows replacing a rectangle of dimensions \( a \times {aN} \) by a square of side length\n\n\( {aN} \) that contains it. Inter... | No |
Theorem 5.3.10. Let \( {p}_{n} = \frac{n + 1}{2} \) and \( N \geq {10} \) . Then there exists a constant \( {C}_{n} \) such that\n\n\[{\begin{Vmatrix}{\mathcal{K}}_{N}^{1}\end{Vmatrix}}_{{L}^{{p}_{n},1}\left( {\mathbf{R}}^{n}\right) \rightarrow {L}^{{p}_{n},\infty }\left( {\mathbf{R}}^{n}\right) } \leq {C}_{n}{N}^{\fra... | Proof. We begin by observing that (5.3.36) is a consequence of (5.3.29) and (5.3.34) using Theorem 1.3.2 in [156]. We also observe that (5.3.35) is a consequence of (5.3.34), while (5.3.34) is a consequence of (5.3.33) (see Exercise 5.3.8). We therefore concentrate on estimate (5.3.33).\n\nWe choose to work with the ce... | Yes |
Example 5.4.2. Property \( {R}_{1 \rightarrow \infty }\left( S\right) \) holds for any compact hypersurface \( S \) . | We denote by \( \mathcal{R}\left( f\right) = {\left. \widehat{f}\right| }_{S} \) the restriction of the Fourier transform on a hypersurface \( S \) . Let \( {d\sigma } \) be the canonically induced surface measure on \( S \) . Then for a function \( \varphi \) defined on \( S \) we have\n\n\[ \n{\int }_{S}\widehat{f}{\... | Yes |
Let \( {d\sigma } \) be surface measure on the unit sphere \( {\mathbf{S}}^{n - 1} \). Using the identity in Appendix B. 4 in [156], we have \[ \widehat{d\sigma }\left( \xi \right) = \frac{2\pi }{{\left| \xi \right| }^{\frac{n - 2}{2}}}{J}_{\frac{n - 2}{2}}\left( {{2\pi }\left| \xi \right| }\right) . \] | In view of the asymptotics in Appendix B. 8 in [156], the last expression is equal to \[ \frac{2}{{\left| \xi \right| }^{\frac{n - 1}{2}}}\cos \left( {{2\pi }\left| \xi \right| - \frac{\pi \left( {n - 1}\right) }{4}}\right) + O\left( {\left| \xi \right| }^{-\frac{n + 1}{2}}\right) \] as \( \left| \xi \right| \rightarro... | Yes |
Let \( \varphi \) be a Schwartz function on \( {\mathbf{R}}^{n} \) such that \( \widehat{\varphi } \geq 0 \) and \( \widehat{\varphi }\left( \xi \right) \geq 1 \) for all \( \xi \) in the closed ball \( \left| \xi \right| \leq 2 \) . For \( N \geq 1 \) define functions\n\n\[ \n{f}_{N}\left( {{x}_{1},{x}_{2},\ldots ,{x}... | We have seen that the restriction property \( {R}_{p \rightarrow q}\left( {\mathbf{S}}^{n - 1}\right) \) fails in the shaded region of Figure 5.10 but obviously holds on the closed line segment \( {CD} \) . It remains to investigate the validity of property \( {R}_{p \rightarrow q}\left( {\mathbf{S}}^{n - 1}\right) \) ... | Yes |
The presence of the logarithmic factor in estimate (5.4.28) is necessary. In fact, this estimate is sharp. | We prove this by showing that the corresponding estimate for the \ | No |
Theorem 6.1.1. There is a constant \( C > 0 \) such that for all square-integrable functions \( f \) on the line the following estimate is valid:\n\n\[ \parallel \mathcal{C}\left( f\right) {\parallel }_{{L}^{2,\infty }} \leq C\parallel f{\parallel }_{{L}^{2}} \] | Observe that it suffices to prove (6.1.2) for Schwartz functions \( f \) on the line. Indeed, suppose we know (6.1.2) for Schwartz functions \( h \) and we would like to prove it for all square-integrable functions \( f \) . Given \( f \) in \( {L}^{2}\left( \mathbf{R}\right) \) we pick a sequence of Schwartz functions... | Yes |
Lemma 6.1.11. Let \( {\mathbf{T}}_{j},{\mathbf{T}}_{j}^{\prime } \) be as previously. Let \( s \in {\mathbf{T}}_{j}^{\prime } \) and \( u \in {\mathbf{T}}_{k}^{\prime } \) . Then if \( {\omega }_{s} \subseteq \) \( {\omega }_{u\left( 1\right) } \), we have \( {I}_{u} \cap {I}_{\operatorname{top}\left( {\mathbf{T}}_{j}\... | Proof. We observe that if \( s \in {\mathbf{T}}_{j}^{\prime }, u \in {\mathbf{T}}_{k}^{\prime } \), and \( {\omega }_{s} \subseteqq {\omega }_{u\left( 1\right) } \), then the 2-trees \( {\mathbf{T}}_{j}^{\prime } \) and \( {\mathbf{T}}_{k}^{\prime } \) have different tops and therefore they cannot be the same tree; thu... | Yes |
Theorem 6.2.1. (a) There exist finite constants \( C,\kappa > 0 \) such that for any measurable subset \( F \) of the reals with finite measure we have\n\n\[ \left| \left\{ {x \in \mathbf{R} : \mathcal{C}\left( {\chi }_{F}\right) \left( x\right) > \alpha }\right\} \right| \leq C\left| F\right| \left\{ \begin{array}{ll}... | Proof. Assuming statement (a), we obtain\n\n\[ {\begin{Vmatrix}\mathcal{C}\left( {\chi }_{F}\right) \end{Vmatrix}}_{{L}^{p}}^{p} = p{\int }_{0}^{\infty }\left| \left\{ {\mathcal{C}\left( {\chi }_{F}\right) > \alpha }\right\} \right| {\alpha }^{p - 1}{d\alpha } \leq p{C}^{p}\left| F\right| {\int }_{0}^{\infty }\varphi \... | Yes |
Lemma 6.3.2. There is a positive constant \( c > 0 \) such that for all functions \( f \) in \( \mathop{\bigcup }\limits_{{1 \leq p < \infty }}{L}^{p}\left( \mathbf{R}\right) \) we have\n\n\[{\mathcal{C}}_{ * }\left( f\right) \leq {cM}\left( f\right) + M\left( {\mathcal{C}\left( f\right) }\right) ,\]\n\nwhere \( M \) i... | Proof. The proof of (6.3.3) is based on the classical inequality\n\n\[{H}^{\left( *\right) }\left( g\right) \leq {cM}\left( g\right) + M\left( {H\left( g\right) }\right)\]\n\ngiven in Theorem 5.3.4 in [156]. Applying this to the functions \( {M}^{\xi }\left( f\right) \) and taking the supremum over \( \xi \in \mathbf{R... | Yes |
We show that the operator given in (7.1.1) is well defined on \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \times {L}^{1}\left( {\mathbf{R}}^{n}\right) \) and is bounded from \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \times {L}^{1}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{1/2}\left( {\mathbf{R}}^{n}\right) \) . | Indeed, for \( f, g \geq 0 \) in \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \n{\int }_{{\mathbf{R}}^{n}}{\int }_{{\mathbf{R}}^{n}}f\left( {x + t}\right) g\left( {x - t}\right) {dtdx} = {\int }_{{\mathbf{R}}^{n}}{\int }_{{\mathbf{R}}^{n}}f\left( {t}^{\prime }\right) g\left( {{2x} - {t}^{\prime }}\right) d... | Yes |
Theorem 7.1.4. Let \( 0 < \alpha < n \) . The bilinear fractional integral\n\n\[ \n{I}_{\alpha }\left( {f, g}\right) \left( x\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x + t}\right) g\left( {x - t}\right) {\left| t\right| }^{\alpha - n}{dt} \n\]\n\nis of restricted weak types \( \left( {\frac{n}{\alpha },\infty ,\inf... | Proof. We first consider the point \( \left( {\frac{n}{\alpha },\infty ,\infty }\right) \) . Let \( A \) and \( B \) be measurable subsets of \( {\mathbf{R}}^{n} \) of finite measure. Let \( {v}_{n} \) be the volume of the unit ball in \( {\mathbf{R}}^{n} \) . We have\n\n\[ \n{\begin{Vmatrix}{I}_{\alpha }\left( {\chi }... | Yes |
Proposition 7.1.5. Let \( \mu \) be a nonnegative regular Borel measure on \( {\mathbf{R}}^{n} \times \cdots \times {\mathbf{R}}^{n} \) , which is not identically equal to zero. Suppose that the m-linear operator \( {T}^{\mu } \) maps \( {L}^{{p}_{1}}\left( {\mathbf{R}}^{n}\right) \times \cdots \times {L}^{{p}_{m}}\lef... | Proof. Fix \( 0 < {p}_{1},\ldots ,{p}_{m}, r \leq \infty \) . By translating \( \mu \) if necessary, we may assume that there exists a compact set \( E \subset {\left\lbrack 1, M\right\rbrack }^{n} \times \cdots \times {\left\lbrack 1, M\right\rbrack }^{n} \) for some \( M > 1 \) such that \( 0 < \mu \left( E\right) < ... | Yes |
Proposition 7.1.6. Let \( \mu \) be a nonnegative regular Borel measure on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \), where \( m \geq 1 \) . Suppose that the operator \( {T}^{\mu } \) maps \( {L}^{{p}_{1}}\left( {\mathbf{R}}^{n}\right) \times \cdots \times {L}^{{p}_{m}}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{r,\... | Proof. Let \( C \) be the norm of \( {T}^{\mu } \) as a bounded operator from \( {L}^{{p}_{1}}\left( {\mathbf{R}}^{n}\right) \times \cdots \times {L}^{{p}_{m}}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{r,\infty }\left( {\mathbf{R}}^{n}\right) \) . First we consider the case \( 0 < r < \infty \) . For a given \( R > 0... | Yes |
Proposition 7.1.8. There exists a nonnegative regular finite Borel measure \( \mu \) on \( \mathbf{R} \times \) \( \mathbf{R} \) with the property that \( {T}^{\mu } \) maps \( {L}^{1} \times {L}^{1} \) to \( {L}^{1/2,\infty } \) but not \( {L}^{1} \times {L}^{1} \) to \( {L}^{1/2} \) . | Proof. If an operator \( {T}^{\mu } \) maps \( {L}^{1}\left( \mathbf{R}\right) \times {L}^{1}\left( \mathbf{R}\right) \rightarrow {L}^{1/2,\infty }\left( \mathbf{R}\right) \), then necessarily \( \mu \) must be a finite measure in view of Proposition 7.1.6.\n\nWe define a measure\n\n\[ \mu = \mathop{\sum }\limits_{{j =... | Yes |
Proposition 7.2.1. Let \( \left( {{X}_{j},{\mu }_{j}}\right), j = 1,\ldots, m,\left( {Y, v}\right) \) be \( \sigma \) -finite measure spaces. Let \( T \) be an operator defined on \( S\left( {X}_{1}\right) \times \cdots \times S\left( {X}_{m}\right) \) and taking values into the set of measurable functions on \( Y \) t... | Proof. The proof of (7.2.3) is based on a straightforward multilinear extension of Lemma 1.4.20 in [156]. Exercise 7.2.4 outlines the steps of the solution. | No |
Theorem 7.2.2. Let \( m \) be a positive integer, and let \( {X}_{1},\ldots ,{X}_{m} \) be \( \sigma \) -finite measure spaces. Suppose that \( T \) is a multisublinear operator defined on \( S\left( {X}_{1}\right) \times \cdots \times S\left( {X}_{m}\right) \) that takes values in the set of measurable functions of a ... | This theorem is proved in Subsection 7.2.2. | Yes |
Proposition 7.2.3. Let \( \\left( {{X}_{j},{\\mu }_{j}}\\right) ,\\left( {Y, v}\\right) \) be \( \\sigma \) -finite measure spaces, and let \( T \) be a multilinear operator defined on \( S\\left( {X}_{1}\\right) \\times \\cdots \\times S\\left( {X}_{m}\\right) \) and taking values in the set of measurable functions on... | Proof. We show that (7.2.11) is valid for general functions in \( {L}^{{p}_{1},{t}_{1}} \\times \\cdots \\times {L}^{{p}_{m},{t}_{m}} \) . For any \( j = 1,2,\\ldots, m \) and \( {f}_{j} \\in {L}^{{p}_{j},{t}_{j}} \), in view of the density of \( {S}_{0}\\left( {X}_{j}\\right) \) in \( {L}^{{p}_{j},{t}_{j}} \) , which ... | Yes |
Corollary 7.2.4. Under the hypotheses of Theorem 7.2.2, assume additionally that \( {\gamma }_{j} \neq 0 \) for all \( j = 1,\ldots, m \) and that, instead of (7.2.8), the inequality holds:\n\n\[ \n\frac{1}{q} \leq \frac{1}{{p}_{1}} + \cdots + \frac{1}{{p}_{m}} \n\]\n\n(7.2.13)\n\nThen, for any \( 0 < \delta < \min \le... | Proof. Using (7.2.6), we see that if \( {p}_{i} = \infty \) for some \( i \), then \( {\gamma }_{0} = 0 \) . Thus \( {p}_{j} < \infty \) for all \( j \), and we pick \( {s}_{j} = {p}_{j} < \infty \) in (7.2.10) and define \( s \) by \( \frac{1}{s} = \frac{1}{{p}_{1}} + \cdots + \frac{1}{{p}_{m}} \), so that (7.2.8) is ... | Yes |
Proposition 1.4.5 (6) in [156] and Exercise 1.1.5(c) in [156], together with the sublinearity of \( T \) and the quasilinearity of Lorentz norms, imply\n\n\[ \n{\begin{Vmatrix}T\left( {f}_{1},\ldots ,{f}_{m}\right) \end{Vmatrix}}_{{L}^{q, s}} \n\]\n\n\[ \n= {\begin{Vmatrix}{t}^{\frac{1}{q}}T{\left( {f}_{1},\ldots ,{f}_... | \[ \n\leq {\begin{Vmatrix}{t}^{\frac{1}{q}}{\left( \mathop{\sum }\limits_{{{i}_{1},\ldots ,{i}_{m} \in \{ 1, - 1\} }}\left| T\left( {f}_{1,{i}_{1}, t},\ldots ,{f}_{m,{i}_{m}, t}\right) \right| \right) }^{ * }\left( t\right) \end{Vmatrix}}_{{L}^{s}\left( {{dt}/t}\right) } \n\]\n\n\[ \n\leq {\begin{Vmatrix}{t}^{\frac{1}{... | Yes |
Lemma 7.2.7. For all \( j \in {\Lambda }^{\prime } \) and all \( \ell \in \left\{ {1,2,\ldots ,{2}^{m}}\right\} \), when \( {p}_{j} > {r}_{\ell, j} \), we have\n\n\[ \n{\begin{Vmatrix}{f}_{j, - 1,1}\end{Vmatrix}}_{{L}^{{r}_{\ell, j},\delta }} \leq {C}_{1}\left( {{r}_{\ell, j},{p}_{j},\delta }\right) {\lambda }_{j}^{\fr... | Now we bound the \( {L}^{s}\left( {{dt}/t}\right) \) quasi-norm of (7.2.31). First apply Lemma 7.2.7 when \( j \in {\Lambda }^{\prime } \), then apply Hölder’s inequality with exponents \( {s}_{j}, j \in \Lambda \), noting that \( \frac{1}{s} = \mathop{\sum }\limits_{{j \in \Lambda }}\frac{1}{{s}_{j}} \), and finally a... | Yes |
Example 7.2.8. We recall the operator \( {I}_{\alpha } \) of Theorem 7.1.4. Let \( 0 < \alpha < n \) . It was shown that the bilinear fractional integral | \[ {I}_{\alpha }\left( {f, g}\right) \left( x\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x + t}\right) g\left( {x - t}\right) {\left| t\right| }^{\alpha - n}{dt} \] is of restricted weak types \( \left( {\frac{n}{\alpha },\infty ,\infty }\right) ,\left( {\infty ,\frac{n}{\alpha },\infty }\right) ,\left( {1,\infty ,\fr... | Yes |
Corollary 7.2.11. Let \( T \) be an \( m \) -linear operator defined on the \( m \) -fold product of spaces of finitely simple functions of \( \sigma \) -finite measure spaces \( \left( {{X}_{i},{\mu }_{i}}\right) \) and taking values in the set of measurable functions of another \( \sigma \) -finite measure space \( \... | Proof. Take \( {T}_{z} = T \) in Theorem 7.2.9, and use Exercise 1.3.8 in [156]. | No |
Theorem 7.2.12. Let \( 0 < p < \infty, A, B > 0 \), and let \( f \) be a measurable function on a \( \sigma \) -finite measure space \( \left( {X,\mu }\right) \). (i) Suppose that \( \parallel f{\parallel }_{{L}^{p,\infty }} \leq A \). Then for every measurable set \( E \) of finite measure there exists a measurable su... | Proof. Define \( {E}^{\prime } = E \smallsetminus \left\{ {\left| f\right| > A{2}^{\frac{1}{p}}\mu {\left( E\right) }^{-\frac{1}{p}}}\right\} \). Since the set \( \left\{ {\left| f\right| > A{2}^{\frac{1}{p}}\mu {\left( E\right) }^{-\frac{1}{p}}}\right\} \) has measure at most \( \mu \left( E\right) /2 \), it follows t... | Yes |
Theorem 7.2.13. Let \( \\left( {X,\\mu }\\right) ,\\left( {{X}_{1},{\\mu }_{1}}\\right) ,\\ldots ,\\left( {{X}_{m},{\\mu }_{m}}\\right) \) be \( \\sigma \) -finite measure spaces. Suppose that an m-linear operator \( T \) is defined on the space of simple functions on \( {X}_{1} \\times \\cdots \\times {X}_{m} \) and t... | Proof. First we prove the claim for \( \\left( {1/{q}_{1},\\ldots ,1/{q}_{m},1/q}\\right) = \\left( {1/{p}_{1},\\ldots ,1/{p}_{m},1/p}\\right) \) . Let \( M \) be the supremum in (7.2.66). For notational uniformity we set \( {\\mu }_{0} = \\mu \) .\n\nCase 1: Suppose \( \\frac{\\mu \\left( {A}_{0}\\right) }{\\sqrt[m]{{... | Yes |
Theorem 7.3.1. Let \( \left( {{X}_{j},{\mu }_{j}}\right) ,\left( {Y, v}\right) \) be \( \sigma \) -finite measure spaces.\n\n(a) Let \( T \) be an \( m \) -linear operator that maps\n\n\[ \n{L}^{{p}_{1}}\left( {{X}_{1},{\mu }_{1}}\right) \times \cdots \times {L}^{{p}_{m}}\left( {{X}_{m},{\mu }_{m}}\right) \rightarrow {... | Proof. To prove the inequality in part (a), we recall the Rademacher functions \( {r}_{j} \) that satisfy the following inequality:\n\n\[ \n{B}_{q}^{m}{\left( \mathop{\sum }\limits_{{k}_{1}}\cdots \mathop{\sum }\limits_{{k}_{m}}{\left| {c}_{{k}_{1},\ldots ,{k}_{m}}\right| }^{2}\right) }^{\frac{1}{2}} \leq {\begin{Vmatr... | Yes |
Proposition 7.3.4. Let \( 0 < {p}_{1},\ldots ,{p}_{m} \leq \infty \) . Then the following statements are valid:\n\n(i) If \( \lambda \in \mathbf{C},\sigma ,{\sigma }_{1} \) and \( {\sigma }_{2} \) are in \( {\mathcal{M}}_{{p}_{1},\ldots ,{p}_{m}} \), then so are \( {\lambda \sigma } \) and \( {\sigma }_{1} + {\sigma }_... | Proof. Item (i) is trivial while (ii)-(iv) are proved by a straightforward change of variables: translation, dilation, and rotation. Item \( \left( v\right) \) easily follows by applying Fa-tou’s lemma on \( {L}^{p} \) (where \( p \) is related to \( {p}_{1},\ldots ,{p}_{m} \) via (7.3.13)) since\n\n\[ {T}_{\sigma }\le... | Yes |
Theorem 7.3.5. Let \( 0 < {p}_{1},\ldots ,{p}_{m} < \infty \) and \( 0 < p < \infty \), and let \( \sigma \) be a locally integrable function defined on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \) that satisfies (7.3.11) for some \( N \geq 0 \) . If \( N = 0 \) , suppose that \( \varphi \) lies in \( {L}^{1}\left( {\ma... | Proof. Let us denote by \( {M}^{b}\left( f\right) \left( x\right) = {e}^{{2\pi ib} \cdot x}f\left( x\right) \) the modulation operator acting on a function \( f \) . For functions \( {f}_{1},\ldots ,{f}_{m} \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \), an easy calculation based on a change of variables gives that f... | Yes |
Proposition 7.3.7. Let \( \sigma \) be a locally integrable function defined on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \) that satisfies (7.3.11). Suppose that the multilinear convolution operator \( {T}_{\sigma } \) is nonzero and maps the \( m \) -fold product \( {L}^{{p}_{1}} \times \cdots \times {L}^{{p}_{m}} \) ... | Proof. Assume first that the kernel \( {K}_{0} \) of \( {T}_{\sigma } \) is supported in \( {\overline{B\left( {0, M}\right) }}^{m} \) . Fix \( {f}_{1},\ldots ,{f}_{m} \) in \( {\mathcal{C}}_{0}^{\infty }\left( {\mathbf{R}}^{n}\right) \), and suppose that all \( {f}_{j} \) are supported in \( \overline{B\left( {0,{M}^{... | Yes |
Let \( {K}_{0}\left( \overrightarrow{u}\right) \) be a function on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \smallsetminus \{ \overrightarrow{0}\} \) that satisfies the size condition \( \left| {{K}_{0}\left( \overrightarrow{u}\right) }\right| \leq {A}^{\prime }{\left| \overrightarrow{u}\right| }^{-{mn}} \) and the reg... | We verify the second assertion. For each \( j \) in \( \{ 1,\ldots, m\} \) , fix \( {y}_{j}^{\prime } \) and take points \( {y}_{k}, k = 1,\ldots, m \) and \( x \) not all equal to each other, satisfying (7.4.3). Then by the mean value theorem, we bound the left-hand side of (7.4.2) by \( \frac{C{A}^{\prime }\left| {{y... | Yes |
Theorem 7.4.6. Let \( T \) be an \( m \) -linear operator associated with a kernel \( K \) in \( m \) - \( \operatorname{CZK}\left( {A,\varepsilon }\right) \), where \( m \geq 2 \) . Assume that for some \( 1 \leq {q}_{1},{q}_{2},\ldots ,{q}_{m} \leq \infty \) and some \( 0 < q < \infty \) satisfying\n\n\[ \n\frac{1}{{... | Proof. Set \( B = \parallel T{\parallel }_{{L}^{{q}_{1}} \times \cdots \times {L}^{{q}_{m}} \rightarrow {L}^{q,\infty }} \) . For \( 1 \leq j \leq m \) fix step functions \( {f}_{j} \) . Assume that each \( {f}_{j} \) is a step function given by a finite linear combination of characteristic functions of disjoint dyadic... | Yes |
Proposition 7.4.7. Given \( K \) in \( m - {CZK}\left( {A,\varepsilon }\right) \) and \( 0 < \delta < 1/4 \), define\n\n\[ \n{K}_{\delta }\left( {{y}_{0},{y}_{1},\ldots ,{y}_{m}}\right) = K\left( {{y}_{0},{y}_{1},\ldots ,{y}_{m}}\right) \left\lbrack {{\chi }_{\left\{ \mathop{\sum }\limits_{{j, k = 0}}^{m}\left| {y}_{j}... | Proof. We pick a smooth function \( \Phi \left( t\right) \) on the real line with values in \( \left\lbrack {0,1}\right\rbrack \), which is equal to 1 when \( t \geq 2 \) and which vanishes when \( t \leq 1 \) . We define the function\n\n\[ \ns\left( {{y}_{0},{y}_{1},\ldots ,{y}_{m}}\right) = s\left( {{y}_{0},\overrigh... | Yes |
Theorem 7.4.9. Fix a \( {\mathcal{C}}^{\infty } \) function \( \eta \) on \( {\mathbf{R}}^{n} \) supported in \( B\left( {0,2}\right) \) that satisfies \( 0 \leq \) \( \eta \left( x\right) \leq 1 \) and \( \eta \left( x\right) = 1 \) when \( \left| x\right| \leq 1 \) . Let \( {\eta }_{k}\left( x\right) = \eta \left( {x... | Proof. We begin by observing that the necessity of the conditions in (7.4.22) is a consequence of Proposition 7.4.8. Thus, the main implication in the proof is contained in their sufficiency, i.e., the fact that if (7.4.22) holds, then \( T \) extends to a bounded operator from \( {L}^{{q}_{1}} \times \cdots \times {L}... | Yes |
Corollary 7.4.10. Let \( {K}_{0}\left( {{u}_{1},\ldots ,{u}_{m}}\right) \) be a function on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \smallsetminus \{ 0\} \) that satisfies the size estimate\n\n\[ \left| {{K}_{0}\left( {{u}_{1},\ldots ,{u}_{m}}\right) }\right| \leq A{\left| \left( {u}_{1},\ldots ,{u}_{m}\right) \right|... | Proof. Let \( W \) be a tempered distribution that coincides with \( {K}_{0} \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) . In view of Theorem 5.4.1 in [156], we have that the Fourier transform of \( W \) is a bounded function whose \( {L}^{\infty } \) norm is controlled by a multiple of \( A \) . We note that co... | Yes |
(a) For any \( r \in \{ 0,1,\ldots, q - 1\} \) there is a constant \( c = c\left( {n, p,{b}_{1},{b}_{2},\Psi }\right) \) such that for all \( {L}^{2} \) functions \( F \) we have\n\n\[ \parallel F{\parallel }_{{L}^{p}} \leq c{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{k = r{\;\operatorname{mod}\;q}}}{\left| {\Delta... | Proof. (a) Assume that the expression on the right-hand side in (7.5.9) is finite; otherwise there is nothing to prove. It follows from Corollary 2.2.10 that there is a polynomial \( Q \) such that \( F - Q \) lies in \( {H}^{p}\left( {\mathbf{R}}^{n}\right) \) and that\n\n\[ \parallel F - Q{\parallel }_{{H}^{p}} \leq ... | Yes |
Theorem 7.5.3. Suppose that a bounded function \( \sigma \) on \( {\left( {\mathbf{R}}^{n}\right) }^{2} \smallsetminus \{ \left( {0,0}\right) \} \) satisfies\n\n\[ \left| {{\partial }^{{\alpha }_{1}}{\partial }^{{\alpha }_{2}}\sigma \left( {{\xi }_{1},{\xi }_{2}}\right) }\right| \leq {C}_{{\alpha }_{1},{\alpha }_{2}}{\... | Proof. We first assume that \( {p}_{1},{p}_{2} < \infty \) . We fix a Schwartz function \( \Psi \) whose Fourier transform is nonnegative, supported in the set \( \left\{ {\xi \in {\mathbf{R}}^{n} : \frac{6}{7} \leq \left| \xi \right| \leq 2}\right\} \), is equal to 1 on the set \( \left\{ {\xi \in {\mathbf{R}}^{n} : 1... | Yes |
Proposition 2.1. Suppose that the group \( G \) acts transitively on the set \( \Omega \) , and let \( H \) be the stabiliser of \( a \in \Omega \) . Then \( G \) acts primitively on \( \Omega \) if and only if \( H \) is a maximal subgroup of \( G \) . | Proof. We prove both directions of this in the contrapositive form. First assume that \( H \) is not maximal, and choose a subgroup \( K \) with \( H < K < G \) . Then the points of \( \Omega \) are in bijection with the (right) cosets of \( H \) in \( G \) . Now the cosets of \( K \) in \( G \) are unions of \( H \) -... | Yes |
Lemma 2.2. If \( n \geq 7 \) and \( {A}_{n - 1} \cong H \leq {A}_{n} \), then \( H \) is the stabiliser of one of the \( n \) points on which \( {A}_{n} \) acts. | Proof. By the above remark, \( H \) cannot act on a non-trivial orbit of length less than \( n - 1 \), so if it is not a point stabiliser then it must act transitively on the \( n \) points. For \( n = 7 \) this is impossible, as 7 does not divide the order of \( {A}_{6} \) . For \( n > 8 \), each element of \( H \) wh... | Yes |
Theorem 2.3. If \( n \geq 7 \) then \( \operatorname{Aut}\left( {A}_{n}\right) \cong {S}_{n} \) . | Proof. Any automorphism of \( {A}_{n} \) permutes the subgroups, and in particular permutes the \( n \) subgroups isomorphic to \( {A}_{n - 1} \) . But these subgroups are in natural one-to-one correspondence with the \( n \) points of \( \Omega \), and therefore any automorphism acts as a permutation of \( \Omega \), ... | No |
Lemma 2.5. Every non-trivial normal subgroup \( N \) of a primitive group \( H \) is transitive. | Proof. Otherwise the orbits of \( N \) form a system of imprimitivity for \( H \) . | No |
Lemma 2.6. Any two distinct minimal normal subgroups \( {N}_{1} \) and \( {N}_{2} \) of any group \( H \) commute. | Proof. By normality, \( \left\lbrack {{N}_{1},{N}_{2}}\right\rbrack \leq {N}_{1} \cap {N}_{2} \trianglelefteq H \), so by minimality\n\n\[ \left\lbrack {{N}_{1},{N}_{2}}\right\rbrack = {N}_{1} \cap {N}_{2} = 1 \] | Yes |
Lemma 2.8. If \( K \) is characteristically simple then it is a direct product of isomorphic simple groups. | Proof. If \( T \) is any minimal normal subgroup of \( K \), then so is \( {T}^{\alpha } \) for any \( \alpha \in \) Aut \( K \) . So by the proof of Lemma 2.6 either \( {T}^{\alpha } = T \) or \( T \cap {T}^{\alpha } = 1 \) . In the latter case \( T{T}^{\alpha } = T \times {T}^{\alpha } \) is a direct product. Since \... | Yes |
Corollary 2.9. Every minimal normal subgroup \( N \) of a finite group \( H \) is a direct product of isomorphic simple groups (not necessarily non-abelian). | Proof. By minimality, \( N \) is characteristically simple. | No |
Lemma 2.10. If \( H \) is primitive, and \( N \) is a non-trivial normal subgroup of \( H \), then either \( {C}_{H}\left( N\right) \) is trivial, or \( {C}_{H}\left( N\right) \) is regular and \( \left| {{C}_{H}\left( N\right) }\right| = \left| \Omega \right| \) . | Proof. Clearly \( {C}_{H}\left( N\right) \) is normal in \( H \), so by Lemma 2.5, if \( {C}_{H}\left( N\right) \neq 1 \) then both \( N \) and \( {C}_{H}\left( N\right) \) are transitive. Moreover, if \( 1 \neq x \in {C}_{H}\left( N\right) \) has any fixed points, then the set of fixed points of \( x \) is preserved b... | Yes |
Corollary 2.11. If \( H \) is primitive, and \( {N}_{1} \) and \( {N}_{2} \) are non-trivial normal subgroups of \( H \), and \( \left\lbrack {{N}_{1},{N}_{2}}\right\rbrack = 1 \), then \( {N}_{2} = {C}_{H}\left( {N}_{1}\right) \) and vice versa. In particular, \( H \) contains at most two minimal normal subgroups, and... | Proof. By Lemma 2.5, \( {N}_{1} \) is transitive, and by Lemma 2.10, \( {C}_{H}\left( {N}_{2}\right) \) is regular. But \( {N}_{1} \subseteq {C}_{H}\left( {N}_{2}\right) \), and therefore \( {N}_{1} \) and \( {C}_{H}\left( {N}_{2}\right) \) have the same order and are equal. | No |
Corollary 2.12. With the same notation, \( {N}_{1} \cong {N}_{2} \) . | Proof. The result is trivial if \( {N}_{1} = {N}_{2} \), so assume \( {N}_{1} \neq {N}_{2} \), and therefore \( {N}_{1} \cap {N}_{2} = 1 \) . Fix a point \( x \in \Omega \), and let \( K \) be the stabiliser of \( x \) in the group \( {N}_{1}{N}_{2} \) . Then \( K \cap {N}_{1} = K \cap {N}_{2} = 1 \) as \( {N}_{1} \) a... | Yes |
Lemma 2.13. Suppose that \( H \) is primitive and \( N \) is a non-trivial normal subgroup of \( H \) . Let \( K \) be the stabiliser in \( H \) of a point. Then \( {KN} = H \) . | Proof. By Lemma 2.5, \( N \) is transitive, so the result follows by the orbit-stabiliser theorem. | No |
Lemma 2.14. If \( K, X \) and \( N \) are subgroups of a group \( G \) and \( X \leq N \), then \( N \cap \left( {KX}\right) = \left( {N \cap K}\right) X. \) | Proof. It is obvious that \( \left( {N \cap K}\right) X \leq N \cap \left( {KX}\right) \) . Conversely, if \( k \in K \) and \( x \in X \) satisfy \( {kx} \in N \), then also \( k \in N \), so \( {kx} \in \left( {N \cap K}\right) X \) . | Yes |
Lemma 2.15. Suppose that \( H \) is primitive and \( N \) is a minimal normal subgroup of \( H \) . Let \( K \) be the stabiliser in \( H \) of a point. Then \( K \cap N \) is maximal among \( K \) -invariant proper subgroups of \( N \) . | Proof. If \( K \cap N < X < N \) and \( K \leq {N}_{H}\left( X\right) \) then \( {KX} \) is a subgroup of \( H \) . Moreover, \( X \) contains elements (of \( N \) ) not in \( K \), so \( K < {KX} \) ; and \( H \) contains elements (of \( N \) ) not in \( X \), so \( N \cap \left( {KX}\right) = \left( {N \cap K}\right)... | Yes |
Theorem 3.1. If \( G \) is a finite perfect group, acting faithfully and primitively on a set \( \Omega \), such that the point stabiliser \( H \) has a normal abelian subgroup \( A \) . whose conjugates generate \( G \), then \( G \) is simple. | Proof. For otherwise, there is a normal subgroup \( K \) with \( 1 < K < G \), which does not fix all the points of \( \Omega \), so we may choose a point stabiliser \( H \) with \( K \nleq H \), and therefore \( G = {HK} \) since \( H \) is a maximal subgroup of \( G \) . So any \( g \in G \) can be written \( g = {hk... | Yes |
Theorem 3.3. If \( \left( {V, f}\right) \) and \( \left( {W, g}\right) \) are isometric spaces, with \( f \) and \( g \) nonsingular, and either alternating bilinear, or conjugate-symmetric sesquilinear, or symmetric bilinear in odd characteristic, then any isometry \( \alpha \) between subspaces \( X \) of \( V \) and... | Proof. Suppose for a contradiction that Witt's Lemma is false, and pick a counterexample such that \( \dim V \) is minimal, and \( X \) is as large as possible in \( V \) . We divide into two cases, according as \( X \) contains a non-trivial nonsingular subspace \( U \), or \( X \) is totally isotropic. In the first c... | Yes |
Theorem 3.4. If the finite group \( G \) acts faithfully and primitively on a set \( \Omega \), and \( A \) is a normal subgroup of the point stabiliser \( H \), such that the \( G \) - conjugates of \( A \) generate \( G \), then any proper quotient of \( G \) is isomorphic to a quotient of \( A \) . | Proof. Suppose that \( K \) is a non-trivial normal subgroup of \( G \) . Then \( K \) is not contained in \( H \), since the action of \( G \) is faithful, so \( G = {HK} \) by maximality of \( H \), since the action is primitive. Therefore, by the same argument as in Theorem 3.1, \( G = {AK} \), whence \( G/K = {AK}/... | Yes |
Theorem 3.5. Any subgroup of \( {\mathrm{{GL}}}_{n}\left( q\right) \) not containing \( {\mathrm{{SL}}}_{n}\left( q\right) \) is contained in one of the following subgroups:\n\n(i) a reducible group \( {q}^{km} : \left( {{\mathrm{{GL}}}_{k}\left( q\right) \times {\mathrm{{GL}}}_{m}\left( q\right) }\right) \), the stabi... | Proof. Given any subgroup \( H \) of \( G = {\operatorname{PGL}}_{n}\left( q\right) \) not containing \( {\operatorname{PSL}}_{n}\left( q\right) \), let \( \widetilde{H} \) denote its preimage in \( \widetilde{G} = {\mathrm{{GL}}}_{n}\left( q\right) \) . The socle of \( H \), written \( \operatorname{soc}H \), is the p... | Yes |
Theorem 3.6. If \( {G}_{0} \) is a finite simple classical group, \( {G}_{0} \leq G \leq \operatorname{Aut}\left( {G}_{0}\right) \) , and \( G \) does not involve the triality automorphism of \( {\mathrm{{P\Omega }}}_{8}^{ + }\left( q\right) \) or the graph automorphism of \( {\operatorname{PSp}}_{4}\left( {2}^{a}\righ... | This form of the theorem is stated and proved in [171]. Its proof does not require any more work than we have done already. However, if we want to provide more detail of the structures of the corresponding subgroups, and to decide which ones are in fact maximal, then we need to do a lot more work. The book of Kleidman ... | No |
Theorem 3.10. If \( q \) is odd, \( n \geq 2 \) and \( M \) is any subgroup of \( {\mathrm{{GO}}}_{{2n} + 1}\left( q\right) \) not containing \( {\Omega }_{{2n} + 1}\left( q\right) \) then either \( M \) is almost simple modulo \( \{ \pm 1\} \), so that \( M \) is the normaliser of a simple group \( S \), and the repre... | (i) \( {q}^{k\left( {k - 1}\right) /2} \cdot {q}^{k\left( {{2n} - {2k} + 1}\right) } \cdot \left( {{\mathrm{{GL}}}_{k}\left( q\right) \times {\mathrm{{GO}}}_{{2n} - {2k} + 1}\left( q\right) }\right) \), the stabiliser of a totally isotropic \( k \) -space, with \( 1 \leq k \leq n \) ;\n\n(ii) \( {\mathrm{{GO}}}_{2k}^{\... | Yes |
Theorem 3.11. If \( n \geq 3 \) and \( M \) is any subgroup of \( {\mathrm{{GO}}}_{2n}^{ - }\left( q\right) \) not containing \( {\Omega }_{2n}^{ - }\left( q\right) \) then either \( M \) is almost simple modulo \( \{ \pm 1\} \), so that \( M \) is the nor-maliser of a quasisimple group \( S \), and the representation ... | (i) \( {q}^{k\left( {k - 1}\right) /2} \cdot {q}^{{2k}\left( {n - k}\right) } \cdot \left( {{\mathrm{{GL}}}_{k}\left( q\right) \times {\mathrm{{GO}}}_{2\left( {n - k}\right) }^{ - }\left( q\right) }\right) \), the stabiliser of a totally isotropic subspace of dimension \( k,1 \leq k < n \) ;\n\n(ii) \( {\mathrm{{GO}}}_... | Yes |
Theorem 5.1. If \( \Lambda \) is a 24-dimensional even (integral) lattice containing no vectors of norm 2, 196560 vectors of norm 4, 16773120 vectors of norm 6 and 398034000 vectors of norm 8, then \( \Lambda \) is isomorphic to the Leech lattice. | Proof. The same counting argument as above (5.34) shows that in any such lattice the vectors of norm 8 form coordinate frames. Writing the lattice with respect to a basis such that one such coordinate frame consists of the vectors of shape \( \left( {\pm 8,{0}^{23}}\right) \), we know that \( \left( {\pm 8, \pm 8,{0}^{... | Yes |
Let \( \Omega = \mathbb{R} \) and consider the Heaviside function \( H\left( x\right) \) ; it is defined by\n\n\[ H\left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ for }x > 0 \\ 0 & \text{ for }x \leq 0 \end{array}\right. \]\n\nIt is locally integrable on \( \mathbb{R} \) . But there is no locally integrable func... | For, assume that \( v \) were such a function, and let \( \varphi \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) with \( \varphi \left( 0\right) = 1 \) and set \( {\varphi }_{N}\left( x\right) = \varphi \left( {Nx}\right) \) . Note that \( \max \left| {\varphi \left( x\right) }\right| = \max \left| {{\varphi }_{N}\le... | Yes |
Lemma 2.1. \( {1}^{ \circ } \) Let \( R > r > 0 \) . There is a function \( {\chi }_{r, R}\left( x\right) \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) with the properties: \( {\chi }_{r, R}\left( x\right) = 1 \) for \( \left| x\right| \leq r,{\chi }_{r, R}\left( x\right) \in \left\lbrack {0,1}\right\rbrack \)... | Proof. \( {1}^{ \circ } \) . The function\n\n\[ f\left( t\right) = \left\{ \begin{array}{ll} {e}^{-1/t} & \text{ for }t > 0 \\ 0 & \text{ for }t \leq 0 \end{array}\right. \]\n\nis a \( {C}^{\infty } \) -function on \( \mathbb{R} \) . For \( t \neq 0 \) this is obvious. At the point \( t = 0 \) we have that \( f\left( t... | Yes |
Lemma 2.2. Let \( \Omega \) be a nonempty open subset of \( {\mathbb{R}}^{n} \). There exists a sequence of compact subsets \( {\left( {K}_{j}\right) }_{j \in \mathbb{N}} \) such that\n\n\[ \n{K}_{1} \subset {K}_{2}^{ \circ } \subset {K}_{2} \subset \cdots \subset {K}_{j}^{ \circ } \subset {K}_{j} \subset \ldots ,\;\ma... | Proof. We can for example take\n\n\[ \n{K}_{j} = \left\{ {x \in \Omega \left| \right| x \mid \leq j\text{ and }\operatorname{dist}\left( {x,\complement \Omega }\right) \geq \frac{1}{j}}\right\} \n\]\n\nthe interior of this set is defined by the formula with \( \leq \) and \( \geq \) replaced by \( < \) and \( > \) . (I... | Yes |
Lemma 2.3. Let \( J = \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \times \cdots \times \left\lbrack {{a}_{n},{b}_{n}}\right\rbrack \) be a closed box in \( {\mathbb{R}}^{n} \) and let \( {f}_{l} \) be a sequence of functions in \( {C}^{1}\left( J\right) \) such that \( {f}_{l} \rightarrow f \) and \( {\partial }_{j}{f}... | Proof. For each \( j \) we use the above-mentioned theorem in situations where all but one coordinate are fixed. This shows that \( f \) has continuous partial derivatives \( {\partial }_{j}f = {g}_{j} \) at each point of \( J \) . | No |
Lemma 2.4. \( {1}^{ \circ } \) For each \( k \in {\mathbb{N}}_{0},{C}^{k}\left( \Omega \right) \) is a Fréchet space when provided with the family of seminorms \( {\left\{ {p}_{k, j}\right\} }_{j \in \mathbb{N}} \) . | Proof. \( {1}^{ \circ } \) . The family \( {\left\{ {p}_{k, j}\right\} }_{j \in \mathbb{N}} \) is separating, for when \( f \in {C}^{k}\left( \Omega \right) \) is \( \neq 0 \) , then there is a point \( {x}_{0} \) where \( f\left( {x}_{0}\right) \neq 0 \), and \( {x}_{0} \in {K}_{j} \) for \( j \) sufficiently large; f... | Yes |
Theorem 2.5. The topology on \( {C}_{0}^{\infty }\left( \Omega \right) \) has the following properties:\n\n(a) A sequence \( {\left( {\varphi }_{l}\right) }_{l \in \mathbb{N}} \) of test functions converges to \( {\varphi }_{0} \) in \( {C}_{0}^{\infty }\left( \Omega \right) \) if and only if there is a \( j \in \mathb... | \[ \mathop{\sup }\limits_{{x \in {K}_{j}}}\left| {{\partial }^{\alpha }{\varphi }_{l}\left( x\right) - {\partial }^{\alpha }{\varphi }_{0}\left( x\right) }\right| \rightarrow 0\;\text{ for }l \rightarrow \infty ,\] (2.14)\n\nfor all \( \alpha \in {\mathbb{N}}_{0}^{n} \) . | Yes |
Theorem 2.6. \( {1}^{ \circ } \) The mapping \( {\partial }^{\alpha } : \varphi \mapsto {\partial }^{\alpha }\varphi \) is a continuous linear operator in \( {C}_{0}^{\infty }\left( \Omega \right) \) . The same holds for \( {D}^{\alpha } \) . | Proof. Clearly, \( {\partial }^{\alpha } \) and \( {M}_{f} \) are linear operators from \( {C}_{0}^{\infty }\left( \Omega \right) \) to itself. As for the continuity it suffices, according to 2.5 (c), to show that \( {\partial }^{\alpha } \) resp. \( {M}_{f} \) is continuous from \( {C}_{{K}_{j}}^{\infty }\left( \Omega... | Yes |
Lemma 2.9. When \( u \in {L}_{1,\operatorname{loc}}\left( {\mathbb{R}}^{n}\right) \), then \( {h}_{j} * u \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \), and\n\n\[{\partial }^{\alpha }\left( {{h}_{j} * u}\right) = \left( {{\partial }^{\alpha }{h}_{j}}\right) * u\text{ for all }\alpha \in {\mathbb{N}}_{0}^{n}.\]\n\n... | Proof. Let \( {x}_{0} \) be an arbitrary point of \( {\mathbb{R}}^{n} \) ; we shall show that \( {h}_{j} * u \) is \( {C}^{\infty } \) on a neighborhood of the point and satisfies (2.35) there. When \( x \in B\left( {{x}_{0},1}\right) \) , then \( {h}_{j}\left( {x - y}\right) \) vanishes for \( y \notin B\left( {{x}_{0... | Yes |
Theorem 2.10. \( {1}^{ \circ } \) When \( v \) is continuous and has compact support in \( {\mathbb{R}}^{n} \) , i.e., \( v \in {C}_{0}^{0}\left( {\mathbb{R}}^{n}\right) \) (cf. (C.8)), then \( {h}_{j} * v \rightarrow v \) for \( j \rightarrow \infty \) uniformly, hence also in \( {L}_{p}\left( {\mathbb{R}}^{n}\right) ... | Proof. \( {1}^{ \circ } \) . When \( v \) is continuous with compact support, then \( v \) is uniformly continuous and one has for \( x \in {\mathbb{R}}^{n} \) :\n\n\[ \left| {\left( {{h}_{j} * v}\right) \left( x\right) - v\left( x\right) }\right| = \left| {{\int }_{B\left( {0,\frac{1}{j}}\right) }v\left( {x - y}\right... | Yes |
For every \( p \in \left\lbrack {1,\infty \left\lbrack {,{C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) }\right. }\right. \) is dense in \( {L}_{p,\operatorname{loc}}\left( {\mathbb{R}}^{n}\right) \) . | Proof. Note first that \( \chi \left( {x/N}\right) u \rightarrow u \) in \( {L}_{p,\text{ loc }} \) for \( N \rightarrow \infty \) (cf. (2.3)). Namely, \( \chi \left( {x/N}\right) = 1 \) for \( \left| x\right| \leq N \), and hence for any \( j \) ,\n\n\[ \n{p}_{j}\left( {\chi \left( {x/N}\right) u - u}\right) \equiv {\... | Yes |
Lemma 2.12. Let \( u \in {L}_{p,\operatorname{loc}}\left( \Omega \right) \) for some \( p \in \lbrack 1,\infty \lbrack \) and let \( \varepsilon > 0 \) . When \( j > 1/\varepsilon \), then\n\n\[ \n{v}_{j}\left( x\right) = \left( {{h}_{j} * u}\right) \left( x\right) = {\int }_{B\left( {0,\frac{1}{j}}\right) }{h}_{j}\lef... | Proof. Let \( j > 1/\varepsilon \), then \( {v}_{j}\left( x\right) \) is defined for \( x \in {\Omega }_{\varepsilon } \) . In the calculation of the integral (2.45), when \( j > 2/\varepsilon \) one only uses the values of \( u \) on \( {K}_{\varepsilon, R} = \) \( {\bar{\Omega }}_{\varepsilon } \cap \underline{B}\lef... | Yes |
Theorem 2.13. Let \( M \) be a subset of \( {\mathbb{R}}^{n} \), let \( \varepsilon > 0 \), and set \( {M}_{k\varepsilon } = \bar{M} + \) \( \underline{B}\left( {0,{k\varepsilon }}\right) \) for \( k > 0 \) . There exists a function \( \eta \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) which is 1 on \( {M}_{\varep... | Proof. The function\n\n\[ \psi \left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ on }{M}_{2\varepsilon }, \\ 0 & \text{ on }{\mathbb{R}}^{n} \smallsetminus {M}_{2\varepsilon }, \end{array}\right. \]\n\n(2.47)\n\nis in \( {L}_{1,\operatorname{loc}}\left( {\mathbb{R}}^{n}\right) \), and for \( j \geq 1/\varepsilon \... | Yes |
Corollary 2.14. \( {1}^{ \circ } \) Let \( \Omega \) be open and let \( K \) be compact \( \subset \Omega \) . There is a function \( \eta \in {C}_{0}^{\infty }\left( \Omega \right) \) taking values in \( \left\lbrack {0,1}\right\rbrack \) such that \( \eta = 1 \) on a neighborhood of \( K \) . | Proof. We use Theorem 2.13, noting that \( \operatorname{dist}\left( {K,\complement \Omega }\right) > 0 \) and that for all \( j \) , \( \operatorname{dist}\left( {{K}_{j},\complement {K}_{j + 1}}\right) > 0. | No |
Theorem 2.15. Let \( \Omega \) be open \( \subset {\mathbb{R}}^{n} \). \( {1}^{ \circ }{C}_{0}^{\infty }\left( \Omega \right) \) is dense in \( {C}^{\infty }\left( \Omega \right) \). | Proof. \( {1}^{ \circ } \). Let \( u \in {C}^{\infty }\left( \Omega \right) \). Choosing \( {\eta }_{j} \) as in Corollary 2.14 one has that \( {\eta }_{j}u \in {C}_{0}^{\infty }\left( \Omega \right) \) and \( {\eta }_{j}u \rightarrow u \) in \( {C}^{\infty }\left( \Omega \right) \) for \( j \rightarrow \infty \) (sinc... | Yes |
Theorem 2.16. Let the open set \( \Omega \) be a union of bounded open sets \( {V}_{j} \) with \( {\bar{V}}_{j} \subset \Omega, j \in {\mathbb{N}}_{0} \), for which there are open subsets \( {V}_{j}^{\prime } \) such that \( \overline{{V}_{j}^{\prime }} \subset {V}_{j} \) and we still have \( \mathop{\bigcup }\limits_{... | Proof. Since \( \overline{{V}_{j}^{\prime }} \) is a compact subset of \( {V}_{j} \), we can for each \( j \) choose a function \( {\zeta }_{j} \in {C}_{0}^{\infty }\left( {V}_{j}\right) \) that is 1 on \( {V}_{j}^{\prime } \) and takes values in \( \left\lbrack {0,1}\right\rbrack \), by Corollary 2.14. Now\n\n\[ \Psi ... | Yes |
Theorem 2.17. Let \( K \) be a compact subset of \( {\mathbb{R}}^{n} \), and let \( {\left\{ {V}_{j}\right\} }_{j = 0}^{N} \) be a bounded open cover of \( K \) (i.e., the \( {V}_{j} \) are bounded and open in \( {\mathbb{R}}^{n} \), and \( \left. {K \subset \mathop{\bigcup }\limits_{{j = 0}}^{N}{V}_{j}}\right) \) . Th... | Proof. Let us first show that there exist open sets \( {V}_{j}^{\prime } \subset {V}_{j} \), still forming a cover \( {\left\{ {V}_{j}^{\prime }\right\} }_{j = 0}^{N} \) of \( K \), such that \( \overline{{V}_{j}^{\prime }} \) is a compact subset of \( {V}_{j} \) for each \( j \) . For this, let \( {V}_{jl} = \left\{ {... | Yes |
Lemma 3.2. When \( f \in {L}_{1,\text{ loc }}\left( \Omega \right) \) with \( \int f\left( x\right) \varphi \left( x\right) {dx} = 0 \) for all \( \varphi \in {C}_{0}^{\infty }\left( \Omega \right) \) , then \( f = 0 \) . | Proof. Let \( \varepsilon > 0 \) and consider \( {v}_{j}\left( x\right) = \left( {{h}_{j} * f}\right) \left( x\right) \) for \( j > 1/\varepsilon \) as in Lemma 2.12. When \( x \in {\Omega }_{\varepsilon } \), then \( {h}_{j}\left( {x - y}\right) \in {C}_{0}^{\infty }\left( \Omega \right) \), so that \( {v}_{j}\left( x... | Yes |
Lemma 3.6. Let \( R > 0 \), and let \( \Omega = B\left( {0, R}\right) \) in \( {\mathbb{R}}^{n} \) ; define also \( {\Omega }_{ \pm } = \Omega \cap {\mathbb{R}}_{ \pm }^{n} \) . Let \( k > 0 \), and let \( u \in {C}^{k - 1}\left( \bar{\Omega }\right) \) with \( k \) -th derivatives defined in \( {\Omega }_{ + } \) and ... | Proof. Let \( \left| \alpha \right| = k \), and write \( {\partial }^{\alpha } = {\partial }_{j}{\partial }^{\beta } \), where \( \left| \beta \right| = k - 1 \) . When \( \varphi \in \) \( {C}_{0}^{\infty }\left( \Omega \right) \), we have if \( j = n \) (using the notation \( {x}^{\prime } = \left( {{x}_{1},\ldots ,{... | Yes |
Lemma 3.7 (The Leibniz formula). When \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right), f \in {C}^{\infty }\left( \Omega \right) \) and \( \alpha \in {\mathbb{N}}_{0}^{n} \), then\n\n\[ \n{\partial }^{\alpha }\left( {fu}\right) = \mathop{\sum }\limits_{{\beta \leq \alpha }}\left( \begin{array}{l} \alpha \\ \beta \... | Proof. When \( f \) and \( u \) are \( {C}^{\infty } \) -functions, the first formula is obtained by induction from the simplest case\n\n\[ \n{\partial }_{j}\left( {fu}\right) = \left( {{\partial }_{j}f}\right) u + f{\partial }_{j}u. \n\]\n\n(3.28)\n\nThe same induction works in the distribution case, if we can only sh... | Yes |
Theorem 3.8. Let \( T \) be a continuous linear operator in \( \mathcal{D}\left( \Omega \right) \). Then the adjoint operator in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \), defined by\n\n\[ \left\langle {{T}^{ \times }u,\varphi }\right\rangle = \langle u,{T\varphi }\rangle \text{ for }u \in {\mathcal{D}}^{\prim... | Proof. Let \( W \) be a neighborhood of 0 in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \). Then \( W \) contains a neighborhood \( {W}_{0} \) of 0 of the form\n\n\[ {W}_{0} = W\left( {{\varphi }_{1},\ldots ,{\varphi }_{N},\varepsilon }\right) \]\n\n\[ \equiv \left\{ {v \in {\mathcal{D}}^{\prime }\left( \Omega \ri... | Yes |
Theorem 3.9 (THE LIMIT THEOREM). A sequence of distributions \( {u}_{k} \in \) \( {\mathcal{D}}^{\prime }\left( \Omega \right) \left( {k \in \mathbb{N}}\right) \) is convergent in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) for \( k \rightarrow \infty \) if and only if the sequence \( \left\langle {{u}_{k},\varp... | Proof. When the topology is defined by the seminorms (3.1) (cf. Theorem B.5), then \( {u}_{k} \rightarrow v \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) if and only if\n\n\[ \left\langle {{u}_{k} - v,\varphi }\right\rangle \rightarrow 0\text{ for }k \rightarrow \infty \]\n\nholds for all \( \varphi \in {C}_... | Yes |
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