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Lemma 3.11. Let \( {\left( {\omega }_{\lambda }\right) }_{\lambda \in \Lambda } \) be a family of open subsets of \( \Omega \) . If \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) is 0 on \( {\omega }_{\lambda } \) for each \( \lambda \in \Lambda \), then \( u \) is 0 on the union \( \mathop{\bigcup }\limits_{... | Proof. Let \( \varphi \in {C}_{0}^{\infty }\left( \Omega \right) \) with support \( K \subset \mathop{\bigcup }\limits_{{\lambda \in \Lambda }}{\omega }_{\lambda } \) ; we must show that \( \langle u,\varphi \rangle = 0 \) . The compact set \( K \) is covered by a finite system of the \( {\omega }_{\lambda } \) ’s, say... | Yes |
Theorem 3.16. When \( u \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) and \( \varphi \in \mathcal{D}\left( {\mathbb{R}}^{n}\right) \), then \( \varphi * u \) equals the function of \( x \in {\mathbb{R}}^{n} \) defined by \( \langle u,\varphi \left( {x - \cdot }\right) \rangle \), it is in \( {C}^{\infty ... | Proof. Note first that \( x \mapsto \varphi \left( {x - \cdot }\right) \) is continuous from \( {\mathbb{R}}^{n} \) to \( \mathcal{D}\left( {\mathbb{R}}^{n}\right) \) . Then the map \( x \mapsto \langle u,\varphi \left( {x - \cdot }\right) \rangle \) is continuous from \( {\mathbb{R}}^{n} \) to \( \mathbb{C} \) (you ar... | Yes |
Lemma 3.17. Let \( {\left( {h}_{j}\right) }_{j \in \mathbb{N}} \) be a sequence as in (2.32). Then (for \( j \rightarrow \infty \) ) \( {h}_{j} * \varphi \rightarrow \varphi \) in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) when \( \varphi \) in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \), and \( {h... | Proof. For any \( \alpha ,{\partial }^{\alpha }\left( {{h}_{j} * \varphi }\right) = {h}_{j} * {\partial }^{\alpha }\varphi \rightarrow {\partial }^{\alpha }\varphi \) uniformly (cf. (2.40)). Then \( {h}_{j} * \varphi \rightarrow \varphi \) in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Moreover, \( {\left( ... | Yes |
Theorem 3.18. Let \( \Omega \) be open \( \subset {\mathbb{R}}^{n} \) . For any \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) there exists a sequence of functions \( {u}_{j} \in {C}_{0}^{\infty }\left( \Omega \right) \) so that \( {u}_{j} \rightarrow u \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \)... | Proof. Choose \( {K}_{j} \) and \( {\eta }_{j} \) as in Corollary \( {2.142}^{ \circ } \) ; then \( {\eta }_{j}u \rightarrow u \) for \( j \rightarrow \infty \) , and each \( {\eta }_{j}u \) identifies with a distribution in \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . For each \( j \), choose\n\n\( {k... | Yes |
Theorem 3.20. Let \( \Omega \) be a smooth open bounded subset of \( {\mathbb{R}}^{n} \) . Let \( k \in \mathbb{N} \) . If \( u \in {C}^{k - 1}\left( {\mathbb{R}}^{n}\right) \) is such that its \( k \) -th derivatives in \( \Omega \) and in \( {\mathbb{R}}^{n} \smallsetminus \bar{\Omega } \) exist and can be extended t... | Proof. For each boundary point \( x \) we have an open set \( {U}_{x} \) and a diffeomorphism \( {\kappa }_{x} : {U}_{x} \rightarrow B\left( {0,1}\right) \) according to Definition C.1; let \( {U}_{x}^{\prime } = {\kappa }_{x}^{-1}\left( {B\left( {0,\frac{1}{2}}\right) }\right) \) . Since \( \bar{\Omega } \) is compact... | Yes |
Some simple coordinate changes in \( {\mathbb{R}}^{n} \) that are often used are translation and dilation. | They lead to the coordinate change maps \( T\left( {\tau }_{a}\right) \) and \( T\left( {\mu }_{\lambda }\right) \), which look as follows for functions on \( {\mathbb{R}}^{n} \):\n\n\[ \left( {T\left( {\tau }_{a}\right) u}\right) \left( y\right) = u\left( {{\tau }_{a}^{-1}y}\right) = u\left( {y + a}\right) = u\left( x... | Yes |
Lemma 3.22. Let \( M \) be a subspace of \( {E}^{\prime } \) . If\n\n\[
\{ e \in E \mid \forall \eta \in M : \eta \left( e\right) = 0\} = \{ 0\} ,
\]\n\nthen \( M \) is weak* dense in \( {E}^{\prime } \) . | Proof. Assume that there is an \( {\eta }_{0} \) in \( {E}^{\prime } \) which does not lie in the weak* closure of \( M \) . Let \( U \) be an open convex neighborhood of \( {\eta }_{0} \) disjoint from \( M \) . According to a Hahn-Banach theorem there exists \( {e}_{0} \) in \( E \) and \( t \) in \( \mathbb{R} \) su... | Yes |
Theorem 3.23. The subspace \( \left\{ {{\Lambda }_{\varphi } \mid \varphi \in {C}_{0}^{\infty }\left( \Omega \right) }\right\} \) is weak* dense in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) . | Proof. It suffices to show that 0 is the only function \( \psi \) in \( {C}_{0}^{\infty }\left( \Omega \right) \) for which \( 0 = {\Lambda }_{\varphi }\left( \psi \right) = {\int }_{\Omega }{\varphi \psi dx} \) for every function \( \varphi \) in \( {C}_{0}^{\infty }\left( \Omega \right) \) ; this follows from the Du ... | Yes |
Theorem 3.24. Let there be given open sets \( \Omega \) in \( {\mathbb{R}}^{a} \) and \( \Xi \) in \( {\mathbb{R}}^{b}, a, b \in \mathbb{N} \) together with a weak-weak continuous linear map \( A \) of \( {C}_{0}^{\infty }\left( \Omega \right) \) into \( {C}_{0}^{\infty }\left( \Xi \right) \) . There is at most one wea... | Proof. The uniqueness is an immediate consequence of Theorem 3.23.\n\nIn the rest of the proof, we set \( E = {C}_{0}^{\infty }\left( \Omega \right), F = {C}_{0}^{\infty }\left( \Xi \right) \).\n\nAssume that \( \widetilde{A} \) exists as desired, then \( {\widetilde{A}}^{ \times } \) is a weak-weak continuous map of \... | Yes |
Let \( f \) be a function in \( {C}^{\infty }\left( \Omega \right) \) . The multiplication by \( f \) defines a continuous operator \( {M}_{f} : \varphi \mapsto {f\varphi } \) on \( {C}_{0}^{\infty }\left( \Omega \right) \) . | Since\n\n\[ \n{\int }_{\Omega }\left( {f\varphi }\right) {\psi dx} = {\int }_{\Omega }\varphi \left( {f\psi }\right) {dx},\varphi ,\psi \in {C}_{0}^{\infty }\left( \Omega \right) ,\n\]\n\nwe define \( {M}_{f}u = {fu} \) by\n\n\[ \n\left( {fu}\right) \left( \varphi \right) = u\left( {f\varphi }\right), u \in {\mathcal{D... | Yes |
For \( \alpha \in {\mathbb{N}}_{0}^{a},{\partial }^{\alpha } \) is a continuous operator on \( {C}_{0}^{\infty }\left( \Omega \right) \) . For \( \varphi \) and \( \psi \) in \( {C}_{0}^{\infty }\left( \Omega \right) \) ,\n\n\[ \n{\int }_{\Omega }\left( {{\partial }^{\alpha }\varphi }\right) {\psi dx} = {\left( -1\righ... | We therefore define a continuous operator \( {\partial }^{\alpha } \) on \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) by\n\n\[ \n\left( {{\partial }^{\alpha }u}\right) \left( \varphi \right) = {\left( -1\right) }^{\left| \alpha \right| }u\left( {{\partial }^{\alpha }\varphi }\right), u \in {\mathcal{D}}^{\prime }... | Yes |
When \( \varphi \) and \( \psi \) are in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \), then, as noted earlier, \( \varphi * \psi \) is in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and satisfies \( {\partial }^{\alpha }\left( {\varphi * \psi }\right) = \varphi * {\partial }^{\alpha }\psi \) for each... | For \( \varphi ,\psi \) and \( \chi \) in \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) we have, denoting \( \varphi \left( {-x}\right) \) by \( \check{\varphi }\left( x\right) \), that\n\n\[
{\int }_{{\mathbb{R}}^{n}}\varphi * \psi \left( y\right) \chi \left( y\right) {dy} = {\int }_{{\mathbb{R}}^{n}}{\int }_{... | Yes |
Example 3.30 (Change of Coordinates). Coordinate changes can also be handled in this way. Let \( \kappa \) be a \( {C}^{\infty } \) diffeomorphism of \( \Omega \) onto \( \Xi \) with the modulus of the functional determinant equal to \( J \) . Define \( T\left( \kappa \right) : {C}_{0}^{\infty }\left( \Omega \right) \r... | Then\n\n\[ \n\left( {T\left( \kappa \right) u}\right) \left( \psi \right) = u\left( {\psi \circ \kappa \cdot J}\right) ,\psi \in {C}_{0}^{\infty }\left( \Xi \right), u \in {\mathcal{D}}^{\prime }\left( \Omega \right) ,\n\]\n\ndefines a continuous linear map \( T\left( \kappa \right) \) of \( {\mathcal{D}}^{\prime }\lef... | Yes |
Lemma 4.3. The operators \( {A}_{\max } \) and \( {A}_{\min }^{\prime } \) are the adjoints of one another, as operators in \( {L}_{2}\left( \Omega \right) \) . | Proof. Since \( {A}_{\min }^{\prime } \) is the closure of \( {\left. {A}^{\prime }\right| }_{{C}_{0}^{\infty }\left( \Omega \right) } \), and \( {A}_{\max } \) is the Hilbert space adjoint of the latter operator, we see from the rule \( {T}^{ * } = {\left( \bar{T}\right) }^{ * } \) that \( {A}_{\max } = \) \( {\left( ... | Yes |
Lemma 4.6. \( {H}^{m}\left( \Omega \right) \) and \( {H}_{0}^{m}\left( \Omega \right) \) are Hilbert spaces. | Note that we have continuous injections\n\n\[ \n{C}_{0}^{\infty }\left( \Omega \right) \subset {H}_{0}^{m}\left( \Omega \right) \subset {H}^{m}\left( \Omega \right) \subset {L}_{2}\left( \Omega \right) \subset {\mathcal{D}}^{\prime }\left( \Omega \right) , \n\]\n\n(4.20)\n\nin particular, convergence in \( {H}^{m}\left... | No |
Lemma 4.8. Let \( u \in {H}^{m}\left( {\mathbb{R}}^{n}\right) \). Then \( {h}_{j} * u \in {C}^{\infty } \cap {H}^{m}\left( {\mathbb{R}}^{n}\right) \) with\n\n\[ \n{D}^{\alpha }\left( {{h}_{j} * u}\right) = {h}_{j} * {D}^{\alpha }u\;\text{ for }\left| \alpha \right| \leq m \n\]\n\n(4.22)\n\n\[ \n{h}_{j} * u \rightarrow ... | Proof. When \( u \in {H}^{m}\left( {\mathbb{R}}^{n}\right) \), then \( {h}_{j} * {D}^{\alpha }u \in {L}_{2}\left( {\mathbb{R}}^{n}\right) \cap {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) for each \( \left| \alpha \right| \leq m \), and \( {h}_{j} * {D}^{\alpha }u \rightarrow {D}^{\alpha }u \) in \( {L}_{2}\left( {\m... | Yes |
Theorem 4.9. Let \( m \) be integer \( \geq 0 \) and let \( \Omega \) be any open set in \( {\mathbb{R}}^{n} \) . Then \( {C}^{\infty }\left( \Omega \right) \cap {H}^{m}\left( \Omega \right) \) is dense in \( {H}^{m}\left( \Omega \right) \) . | Proof. We can assume that \( \Omega \) is covered by a locally finite sequence of open sets \( {V}_{j}, j \in {\mathbb{N}}_{0} \), with an associated partition of unity \( {\psi }_{j} \) as in Theorem 2.16 (cf. (2.4), (2.48)).\n\nLet \( u \in {H}^{m}\left( \Omega \right) \) ; we have to show that it can be approximated... | Yes |
Theorem 4.10. Let \( \Omega = {\mathbb{R}}^{n} \) or \( {\mathbb{R}}_{ + }^{n} \), or let \( \Omega \) be a smooth open bounded set. Then \( {C}_{\left( 0\right) }^{\infty }\left( \bar{\Omega }\right) \) is dense in \( {H}^{m}\left( \Omega \right) \) . | Proof. \( {1}^{ \circ } \) . The case \( \Omega = {\mathbb{R}}^{n} \) . We already know from Lemma 4.8 that \( {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \cap {H}^{m}\left( {\mathbb{R}}^{n}\right) \) is dense in \( {H}^{m}\left( {\mathbb{R}}^{n}\right) \) . Now let \( u \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) ... | No |
Corollary 4.11. \( {1}^{ \circ }{H}^{m}\left( {\mathbb{R}}^{n}\right) = {H}_{0}^{m}\left( {\mathbb{R}}^{n}\right) \) for all \( m \in {\mathbb{N}}_{0} \) ; i.e., \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is dense in \( {H}^{m}\left( {\mathbb{R}}^{n}\right) \) for all \( m \in {\mathbb{N}}_{0} \) . | Proof. \( {1}^{ \circ } \) follows from Theorem 4.10 and the fact that \( {C}_{\left( 0\right) }^{\infty }\left( {\mathbb{R}}^{n}\right) = {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) , cf. Definition \( {4.52}^{ \circ }. | Yes |
Theorem 4.14. When \( u \) and \( v \in {H}^{1}\left( I\right) \), then also \( {uv} \in {H}^{1}\left( I\right) \), and\n\n\[ \partial \left( {uv}\right) = \left( {\partial u}\right) v + u\left( {\partial v}\right) ;\;\text{ with } \] | Proof. Let \( u, v \in {H}^{1}\left( I\right) \) and let \( {u}_{n},{v}_{n} \in {C}^{1}\left( \bar{I}\right) \) with \( {u}_{n} \rightarrow u \) and \( {v}_{n} \rightarrow v \) in \( {H}^{1}\left( I\right) \) for \( n \rightarrow \infty \) . By Theorem 4.13, the convergences hold in \( {C}^{0}\left( \bar{I}\right) \), ... | Yes |
Theorem 4.15. Let \( I = \rbrack \alpha ,\beta \lbrack \) . The subspace \( {H}_{0}^{1}\left( I\right) \) of \( {H}^{1}\left( I\right) \) (the closure of \( {C}_{0}^{\infty }\left( I\right) \) in \( \left. {{H}^{1}\left( I\right) }\right) \) satisfies\n\n\[ \n{H}_{0}^{1}\left( I\right) = \left\{ {u \in {H}^{1}\left( I\... | Proof. When \( u = {\int }_{\alpha }^{x}g\left( s\right) {ds} + k \), then \( u\left( \alpha \right) = 0 \) if and only if \( k = 0 \), and when this holds, \( u\left( \beta \right) = 0 \) if and only if \( {\int }_{\alpha }^{\beta }g\left( s\right) {ds} = 0 \) . In view of (4.37), this proves the second equality in (4... | Yes |
Theorem 4.17. Let \( I = \rbrack \alpha ,\beta \lbrack \) . \( {1}^{ \circ }{H}^{m}\left( I\right) \) consists of the functions \( u \in {C}^{m - 1}\left( \bar{I}\right) \) such that \( {u}^{\left( m - 1\right) } \in \) \( {H}^{1}\left( I\right) \) . The inequality \[ \mathop{\sum }\limits_{{j \leq m - 1}}{\begin{Vmatr... | Proof. We can assume \( m > 1 \) . \( {1}^{ \circ } \) . It is clear from the definition that a function \( u \) is in \( {H}^{m}\left( I\right) \) if and only if \( u,{u}^{\prime },\ldots ,{u}^{\left( m - 1\right) } \) (defined in the distribution sense) belong to \( {H}^{1}\left( I\right) \) . This holds in particula... | Yes |
Theorem 4.18. \( {1}^{ \circ } \) An inequality (4.47) holds also when \( I \) is an unbounded interval of \( \mathbb{R} \). | Proof. \( {1}^{ \circ } \) . We already have (4.47) for bounded intervals \( {I}^{\prime } \), with constants \( C\left( {I}^{\prime }\right) \) . Since derivatives commute with translation,\n\n\[ \mathop{\max }\limits_{{x \in \overline{{I}^{\prime } + a}}}\begin{Vmatrix}{{\partial }^{j}u\left( {x - a}\right) }\end{Vma... | Yes |
Theorem 4.19. Let \( I \) be an open interval of \( \mathbb{R} \), let \( k \geq 1 \) and let \( u \in {\mathcal{D}}^{\prime }\left( I\right) \) . If \( {Du} = 0 \), then \( u \) equals a constant. If \( {D}^{k}u = 0 \), then \( u \) is a polynomial of degree \( \leq k - 1 \) . | Proof. The theorem is shown by induction in \( k \) . For the case \( k = 1 \), one has when \( {Du} = 0 \) that\n\n\[ 0 = \langle {Du},\varphi \rangle = - \langle u,{D\varphi }\rangle \;\text{ for all }\varphi \in {C}_{0}^{\infty }\left( I\right) . \]\n\nChoose a function \( h \in {C}_{0}^{\infty }\left( I\right) \) w... | Yes |
Theorem 4.20. Let \( I \) be an open interval of \( \mathbb{R} \). Let \( m \geq 1 \). If \( u \in {\mathcal{D}}^{\prime }\left( I\right) \) and \( {D}^{m}u \in {L}_{2}\left( I\right) \), then \( u \in {H}^{m}\left( {I}^{\prime }\right) \) for each bounded subinterval \( {I}^{\prime } \) of \( I \). | Proof. It suffices to show the result for a bounded interval \( I = \rbrack \alpha ,\beta \lbrack \). By successive integration of \( {D}^{m}u \) we obtain a function\n\n\[ v\left( t\right) = {i}^{m}{\int }_{\alpha }^{t}d{s}_{1}{\int }_{\alpha }^{{s}_{1}}d{s}_{2}{\int }_{\alpha }^{{s}_{2}}\cdots {\int }_{\alpha }^{{s}_... | Yes |
Theorem 4.23. Let \( I = \rbrack \alpha ,\beta \lbrack \) and let \( m > 0 \) . Let \( A = {D}^{m} \) . Then\n\n\[ D\left( {A}_{\max }\right) = {H}^{m}\left( I\right) \]\n\n(4.56)\n\n\[ D\left( {A}_{\min }\right) = {H}_{0}^{m}\left( I\right) \]\n\nThere exists a constant \( {C}_{1} > 0 \) so that\n\n\[ {\left( \paralle... | Proof. Clearly, \( {H}^{m}\left( I\right) \subset D\left( {A}_{\max }\right) \) . The opposite inclusion follows from Theorem 4.20. The first inequality in (4.57) follows from the definition of the \( m \) -norm; thus the injection of \( {H}^{m}\left( I\right) \) into \( D\left( {A}_{\max }\right) \) (which is a Hilber... | Yes |
Theorem 4.24. The trace map \( {\gamma }_{0} : u\left( {{x}^{\prime },{x}_{n}}\right) \mapsto u\left( {{x}^{\prime },0}}\right) \) that sends \( {C}_{\left( 0\right) }^{\infty }\left( {\mathbb{R}}_{ + }^{n}\right) \) into \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n - 1}\right) \) extends by continuity to a continuous ma... | Proof. As earlier, we denote \( \left( {{x}_{1},\ldots ,{x}_{n - 1}}\right) = {x}^{\prime } \) . For \( u \in {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), we have the inequality (using that \( \left| {2\operatorname{Re}{ab}}\right| \leq {\left| a\right| }^{2} + {\left| b\right| }^... | Yes |
Theorem 4.27. The Friedrichs extension \( T \) of \( S = {\left. -\Delta \right| }_{{C}_{0}^{\infty }\left( \Omega \right) } \) is a selfadjoint realization of \( - \Delta \) . Its lower bound \( m\left( T\right) \) equals \( m\left( S\right) \) . \( T \) is the variational operator determined from the triple \( \left(... | Proof. It remains to account for the second paragraph. The uniqueness follows from Corollary 12.25. Note that since \( T \) is a selfadjoint extension of \( {A}_{\min }, T \subset {A}_{\max } \), so it acts like \( - \Delta \) . In formula (4.66), the inclusion ’ \( \subset \) ’ is clear since \( T \subset {A}_{\max } ... | Yes |
Theorem 4.28. The variational operator \( {T}_{1} \) defined from the triple (4.69) is a selfadjoint realization of \( - \Delta \), with \( m\left( {T}_{1}\right) \geq 0 \) ( \( = 0 \) if \( \Omega \) is bounded), and | \[ D\left( {T}_{1}\right) = \left\{ {u \in {H}^{1}\left( \Omega \right) \cap D\left( {A}_{\max }\right) \mid \left( {-{\Delta u}, v}\right) = {s}_{1}\left( {u, v}\right) \text{ for all }v \in {H}^{1}\left( \Omega \right) }\right\} . \] The domain \( D\left( {T}_{1}\right) \) is dense in \( {H}^{1}\left( \Omega \right) ... | No |
For \( 1 \leq p \leq \infty ,\mathcal{S}\left( {\mathbb{R}}^{n}\right) \) is continuously injected in \( {L}_{p}\left( {\mathbb{R}}^{n}\right) . | We have for \( \varphi \) in \( \mathcal{S}\left( {\mathbb{R}}^{n}\right), M \) in \( {\mathbb{N}}_{0} \) and \( \alpha \) in \( {\mathbb{N}}_{0}^{n} \) with \( \left| \alpha \right| \leq M \) that\n\n\[ \left| {{D}^{\alpha }\varphi \left( x\right) }\right| \leq {p}_{M}\left( \varphi \right) \langle x{\rangle }^{-M} \]... | Yes |
Theorem 5.4. \( {1}^{ \circ } \) The Fourier transform \( \mathcal{F} \) is a continuous linear map of \( {L}_{1}\left( {\mathbb{R}}^{n}\right) \) into \( {C}_{{L}_{\infty }}\left( {\mathbb{R}}^{n}\right) \), such that when \( f \in {L}_{1}\left( {\mathbb{R}}^{n}\right) \), then\n\n\[ \parallel \widehat{f}{\parallel }_... | Proof. \( {1}^{ \circ } \) . The inequality\n\n\[ \left| {\widehat{f}\left( \xi \right) }\right| = \left| {\int {e}^{{ix} \cdot \xi }f\left( x\right) }\right| {dx} \leq \int \left| {f\left( x\right) }\right| {dx} \]\n\nshows the first statement in (5.8), so \( \mathcal{F} \) maps \( {L}_{1} \) into \( {L}_{\infty } \) ... | Yes |
Theorem 5.5 (PARSEVAL-PLANCHEREL THEOREM). \( {1}^{ \circ } \) The Fourier transform \( \mathcal{F} : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) extends in a unique way to an isometric isomorphism \( {\mathcal{F}}_{2} \) of \( {L}_{2}\left( {{\mathbb{R}}^{n},{dx}}... | Proof. \( {1}^{ \circ } \) . We first show (5.14) for \( f, g \in \mathcal{S} \) . By Theorem \( {5.43}^{ \circ } \) ,\n\n\[ g\left( x\right) = {\left( 2\pi \right) }^{-n}\int {e}^{{i\xi } \cdot x}\widehat{g}\left( \xi \right) {d\xi } \]\n\nso by the Fubini theorem,\n\n\[ \int f\left( x\right) \overline{g\left( x\right... | Yes |
Theorem 5.6. When \( f, g \in {L}_{1}\left( {\mathbb{R}}^{n}\right) \), then\n\n\[ \mathcal{F}\left( {f * g}\right) = \mathcal{F}f \cdot \mathcal{F}g. \] | Proof. We find by use of the Fubini theorem and a simple change of variables:\n\n\[ \mathcal{F}\left( {f * g}\right) \left( \xi \right) = {\int }_{{\mathbb{R}}^{n}}{e}^{-{i\xi } \cdot x}\left( {{\int }_{{\mathbb{R}}^{n}}f\left( {x - y}\right) g\left( y\right) {dy}}\right) {dx} \]\n\n\[ = {\int }_{{\mathbb{R}}^{n}}g\lef... | Yes |
Lemma 5.7. When \( \varphi \) and \( \psi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), then \( \varphi * \psi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), and \( \psi \mapsto \varphi * \psi \) is a continuous operator on \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . | Proof. Since \( \varphi \) and \( \psi \) belong to \( {L}_{1}\left( {\mathbb{R}}^{n}\right) \cap {C}_{{L}_{\infty }}\left( {\mathbb{R}}^{n}\right) \), the rules (5.18) and (5.19) show that \( \varphi * \psi \in {L}_{1}\left( {\mathbb{R}}^{n}\right) \cap {C}_{{L}_{\infty }}\left( {\mathbb{R}}^{n}\right) \) . Since \( \... | Yes |
Lemma 5.9. \( \mathcal{D}\left( {\mathbb{R}}^{n}\right) = {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is a dense subset of \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), with a stronger topology. | Proof. As already noted, \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \subset \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), and the neighborhood basis (5.5) for \( \mathcal{S} \) at zero intersects \( {C}_{0}^{\infty } \) with open neighborhoods of 0 there, so that the topology induced on \( {C}_{0}^{\infty } \) ... | Yes |
Theorem 5.10. The map \( J : \Lambda \mapsto {\Lambda }^{\prime } \) from \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) to \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) defined by restriction of \( \Lambda \) to \( \mathcal{D}\left( {\mathbb{R}}^{n}\right) \), \[ \left\langle {{\Lambda }^{\p... | Proof. The map \( J : \Lambda \mapsto {\Lambda }^{\prime } \) is injective because of Lemma 5.9, since \[ \left\langle {{\Lambda }^{\prime },\varphi }\right\rangle = 0\;\text{ for all }\varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \] implies that \( \Lambda \) is 0 on a dense subset of \( \mathcal{S} \), ... | Yes |
For \( 1 \leq p \leq \infty \) and \( f \) in \( {L}_{p}\left( {\mathbb{R}}^{n}\right) \), the map \( \varphi \mapsto {\int }_{{\mathbb{R}}^{n}}{f\varphi dx} \) , \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) defines a temperate distribution. In this way one has for each \( p \) a continuous injection of ... | Denote the map \( f \mapsto {\int }_{{\mathbb{R}}^{n}}{f\varphi dx} \) by \( {\Lambda }_{f} \) . Let as usual \( {p}^{\prime } \) be given by \( \frac{1}{p} + \frac{1}{{p}^{\prime }} = 1 \), with \( {1}^{\prime } = \infty \) and \( {\infty }^{\prime } = 1 \) . According to the Hölder inequality,\n\n\[ \left| {{\Lambda ... | Yes |
Besides the already mentioned functions \( u \in {L}_{p}\left( {\mathbb{R}}^{n}\right), p \in \left\lbrack {1,\infty }\right\rbrack \), all functions \( v \in {L}_{1,\operatorname{loc}}\left( {\mathbb{R}}^{n}\right) \) with \( \left| {v\left( x\right) }\right| \leq C\langle x{\rangle }^{N} \) for some \( N \) are in \(... | We see this by observing that for such a function \( v \) ,\n\n\[ \left| {\langle v,\varphi \rangle }\right| = \left| {\int {v\varphi dx}}\right| \leq C\int \langle x{\rangle }^{-n - 1}{dx}\sup \left\{ {{\left\langle x\right\rangle }^{N + n + 1}\left| {\varphi \left( x\right) }\right| \mid x \in {\mathbb{R}}^{n}}\right... | Yes |
Lemma 5.13. \( {1}^{ \circ }{D}^{\alpha } \) maps \( {\mathcal{S}}^{\prime } \) continuously into \( {\mathcal{S}}^{\prime } \) for all \( \alpha \in {\mathbb{N}}_{0}^{n} \) . | Proof. For \( \alpha \in {\mathbb{N}}_{0}^{n}, p \in {\mathcal{O}}_{M} \), we set\n\n\[ \left\langle {{D}^{\alpha }u,\varphi }\right\rangle = \left\langle {u,{\left( -D\right) }^{\alpha }\varphi }\right\rangle \]\n\n(5.30)\n\n\[ \langle {pu},\varphi \rangle = \langle u,{p\varphi }\rangle ,\text{ for }\varphi \in \mathc... | Yes |
Consider the operator \( P = 1 - \Delta \) on \( {\mathbb{R}}^{n} \) . By Fourier transformation, the equation\n\n\[ \left( {1 - \Delta }\right) u = f\text{ on }{\mathbb{R}}^{n} \] | is carried into the equation\n\n\[ \left( {1 + {\left| \xi \right| }^{2}}\right) \widehat{u} = \widehat{f}\text{ on }{\mathbb{R}}^{n}, \]\n\nand this leads by division with \( 1 + {\left| \xi \right| }^{2} = \langle \xi {\rangle }^{2} \) to\n\n\[ \widehat{u} = \langle \xi {\rangle }^{-2}\widehat{f} \]\n\nThus (5.46) ha... | Yes |
Theorem 5.21. Let \( O \) be an orthogonal transformation in \( {\mathbb{R}}^{n} \) and let \( {\mu }_{\lambda } \) be the multiplication by the scalar \( \lambda \in \mathbb{R} \smallsetminus \{ 0\} \). The associated coordinate change maps \( T\left( O\right) \) and \( T\left( {\mu }_{\lambda }\right) \) in \( {\math... | \[ \mathcal{F}\left\lbrack {T\left( O\right) u}\right\rbrack = T\left( {\left( {O}^{ * }\right) }^{-1}\right) \mathcal{F}u = T\left( O\right) \mathcal{F}u, \] (5.54) \[ \mathcal{F}\left\lbrack {T\left( {\mu }_{\lambda }\right) u}\right\rbrack = \left| {\lambda }^{n}\right| T\left( {\mu }_{1/\lambda }\right) \mathcal{F}... | Yes |
Corollary 5.22. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), and let \( r \in \mathbb{R} \) . \( {1}^{ \circ } \) If \( u \) only depends on the distance \( \left| x\right| \) to 0, then the same holds for \( \widehat{u} \) . \( {2}^{ \circ } \) If \( u \) is homogeneous of degree \( r \), the... | Proof. \( {1}^{ \circ } \) . The identities (5.56) carry over to similar identities for \( \widehat{u} \) according to (5.54). \( {2}^{ \circ } \) . When \( u \) is homogeneous of degree \( r \), then we have according to (5.55) and (5.57): \[ T\left( {\mu }_{1/a}\right) \mathcal{F}u = {a}^{-n}\mathcal{F}\left\lbrack {... | Yes |
Theorem 6.3. Let \( p\left( \xi \right) \in {\mathcal{O}}_{M} \) and let \( P\left( D\right) \) be the associated pseudodifferential operator \( \mathrm{{Op}}\left( p\right) \). The maximal realization \( P{\left( D\right) }_{\max } \) of \( P\left( D\right) \) in \( {L}_{2}\left( {\mathbb{R}}^{n}\right) \) with domain... | Proof. We write \( P \) for \( P\left( D\right) \) and \( {P}^{\prime } \) for \( {P}^{\prime }\left( D\right) \). It follows immediately from the Parseval-Plancherel theorem (Theorem 5.5) that\n\n\[ {P}_{\max } = {\mathcal{F}}^{-1}{M}_{p}\mathcal{F};\text{ with } \]\n\n\[ D\left( {P}_{\max }\right) = {\mathcal{F}}^{-1... | Yes |
One has for the operators introduced in Theorem 6.3:\n\n\\( {1}^{ \\circ }P{\\left( D\\right) }_{\\max } \\) is a bounded operator in \\( {L}_{2}\\left( {\\mathbb{R}}^{n}\\right) \\) if and only if \\( p\\left( \\xi \\right) \\) is bounded, and the norm satisfies\n\n\\[ \n\\begin{Vmatrix}{P{\\left( D\\right) }_{\\max }... | Proof. \\( {1}^{ \\circ } \\) . We have from Theorem 12.13 and the subsequent remarks that \\( {M}_{p} \\) is a bounded operator in \\( {L}_{2}\\left( {\\mathbb{R}}^{n}\\right) \\) when \\( p \\) is a bounded function on \\( {\\mathbb{R}}^{n} \\), and that the norm in that case is precisely \\( \\sup \\left\\{ {\\left|... | Yes |
Lemma 6.7 (The Peetre inequality). For any \( s \in \mathbb{R} \) ,\n\n\[ \langle x - y{\rangle }^{s} \leq {c}_{s}\langle x{\rangle }^{s}\langle y{\rangle }^{\left| s\right| }\;\text{ for }s \in \mathbb{R}, \] | Proof. First observe that\n\n\[ 1 + {\left| x - y\right| }^{2} \leq 1 + {\left( \left| x\right| + \left| y\right| \right) }^{2} \leq c\left( {1 + {\left| x\right| }^{2}}\right) \left( {1 + {\left| y\right| }^{2}}\right) \]\n\nthis is easily seen to hold with \( c = 2 \), and with a little more care one can show it with... | Yes |
Lemma 6.8. For \( m \in {\mathbb{N}}_{0}, u \) belongs to \( {H}^{m}\left( {\mathbb{R}}^{n}\right) \) if and only if \( \widehat{u} \) belongs to \( {L}_{2, m}\left( {\mathbb{R}}^{n}\right) \) . The scalar product\n\n\[{\left( u, v\right) }_{m, \land } = {\left( 2\pi \right) }^{-n}{\int }_{{\mathbb{R}}^{n}}\widehat{u}\... | Proof. In view of the inequalities (5.2) and the Parseval-Plancherel theorem,\n\n\[ u \in {H}^{m}\left( {\mathbb{R}}^{n}\right) \Leftrightarrow \mathop{\sum }\limits_{{\left| \alpha \right| \leq m}}{\left| {\xi }^{\alpha }\widehat{u}\left( \xi \right) \right| }^{2} \in {L}_{1}\left( {\mathbb{R}}^{n}\right) \]\n\n\[ \Le... | Yes |
Lemma 6.10. Let \( s \in \mathbb{R} \). \( {1}^{ \circ }{\Xi }^{s} \) defines a homeomorphism of \( \mathcal{S} \) onto \( \mathcal{S} \), and of \( {\mathcal{S}}^{\prime } \) onto \( {\mathcal{S}}^{\prime } \), with inverse \( {\Xi }^{-s} \). | Proof. As noted earlier, \( \mathcal{S} \) is dense in \( {L}_{2}\left( {\mathbb{R}}^{n}\right) \), since \( {C}_{0}^{\infty } \) is so. Since \( \langle \xi {\rangle }^{s} \in \) \( {\mathcal{O}}_{M},{M}_{\langle \xi {\rangle }^{s}} \) maps \( \mathcal{S} \) continuously into \( \mathcal{S} \), and \( {\mathcal{S}}^{\... | Yes |
Theorem 6.11 (THE SOBOLEV THEOREM). Let \( m \) be an integer \( \geq 0 \), and let \( s > m + n/2 \) . Then (cf. (C.10)) \[ {H}^{s}\left( {\mathbb{R}}^{n}\right) \subset {C}_{{L}_{\infty }}^{m}\left( {\mathbb{R}}^{n}\right) \] with continuous injection, i.e., there is a constant \( C > 0 \) such that for \( u \in \) \... | Proof. For \( \varphi \in \mathcal{S} \) one has for \( s = m + t, t > n/2 \) and \( \left| \alpha \right| \leq m \), cf. (5.2), \[ \mathop{\sup }\limits_{{x \in {\mathbb{R}}^{n}}}\left| {{D}^{\alpha }\varphi \left( x\right) }\right| = \sup \left| {{\left( 2\pi \right) }^{-n}{\int }_{{\mathbb{R}}^{n}}{e}^{{ix} \cdot \x... | Yes |
Theorem 6.12. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) with \( \widehat{u} \in {L}_{2,\operatorname{loc}}\left( {\mathbb{R}}^{n}\right) \) . Then one has for \( s \in \mathbb{R} \) ,\n\n\[ \n{\Delta u} \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \Leftrightarrow u \in {H}^{s + 2}\left( {\mathb... | Proof. We start by showing the first line in (6.25). When \( u \in {H}^{s + 2} \), then \( {\Delta u} \in {H}^{s} \), since \( \langle \xi {\rangle }^{s}{\left| \xi \right| }^{2} \leq \langle \xi {\rangle }^{s + 2} \) . Conversely, when \( {\Delta u} \in {H}^{s} \) and \( \widehat{u} \in {L}_{2,\text{loc }} \) , then\n... | Yes |
Corollary 6.14. When \( \Omega = {\mathbb{R}}_{ + }^{n} \), or \( \Omega \) is bounded, smooth and open, then one has for integer \( m \) and \( l \geq 0 \), with \( l > m + n/2 \): \[ {H}^{l}\left( \Omega \right) \subset {C}_{{L}_{\infty }}^{m}\left( \bar{\Omega }\right) ,\text{ with }\sup \left\{ {\left| {{D}^{\alpha... | Proof. Here we use Theorem 4.12, which shows the existence of a continuous map \( E : {H}^{l}\left( \Omega \right) \rightarrow {H}^{l}\left( {\mathbb{R}}^{n}\right) \) such that \( u = {\left. \left( Eu\right) \right| }_{\Omega } \). When \( u \in {H}^{l}\left( \Omega \right) ,{Eu} \) is in \( {H}^{l}\left( {\mathbb{R}... | Yes |
Theorem 6.15. Let \( s \in \mathbb{R} \) . \( {1}^{ \circ }{L}_{2, - s} \) can be identified with the dual space of \( {L}_{2, s} \) by an isometric isomorphism, such that the function \( u \in {L}_{2, - s} \) is identified with the functional \( \Lambda \in {\left( {L}_{2, s}\right) }^{ * } \) precisely when \[ \int u... | Proof. \( {1}^{ \circ } \) . When \( u \in {L}_{2, - s} \), it defines a continuous antilinear functional \( {\Lambda }_{u} \) on \( {L}_{2, s} \) by \[ {\Lambda }_{u}\left( v\right) = \int u\left( \xi \right) \bar{v}\left( \xi \right) {d\xi }\;\text{ for }v \in {L}_{2, s}, \] since \[ \left| {{\Lambda }_{u}\left( v\ri... | Yes |
The \( \delta \) -distribution satisfies\n\n\[ \delta \in {H}^{-s}\left( {\mathbb{R}}^{n}\right) \Leftrightarrow s > n/2, \]\n\nand its \( \alpha \) -th derivative \( {D}^{\alpha }\delta \) is in \( {H}^{-s} \) precisely when \( s > \left| \alpha \right| + n/2 \) . | This follows from the fact that \( \mathcal{F}\left( {{D}^{\alpha }\delta }\right) = {\xi }^{\alpha } \) (cf. (5.39)) is in \( {L}_{2, - s} \) if and only if \( \left| \alpha \right| - s < - n/2 \) . | Yes |
Theorem 6.19. Let \( u \in {\mathcal{E}}^{\prime }\left( \Omega \right) \), identified with a subspace of \( {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) by extension by 0, and let \( N \) be such that for some \( {C}_{N} \) ,\n\n\[ \left| {\langle u,\varphi \rangle }\right| \leq {C}_{N}\sup \left\{ {\left|... | Proof. We have by Theorem 3.12 and its proof that when \( u \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), then \( u \) is of some finite order \( N \), for which there exists a constant \( {C}_{N} \) such that (6.37) holds (regardless of the location of the support of \( \varphi \) ), cf. (3.35). By (6.... | Yes |
Theorem 6.20 (THE STRUCTURE THEOREM). Let \( \Omega \) be open \( \subset {\mathbb{R}}^{n} \) and let \( u \in {\mathcal{E}}^{\prime }\left( \Omega \right) \) . Let \( V \) be an open neighborhood of \( \operatorname{supp}u \) with \( \bar{V} \) compact \( \subset \Omega \), and let \( M \) be an integer \( > \left( {N... | Proof. We have according to Theorem 6.19 that \( u \in {H}^{-s} \) for \( s = N + n/2 + \varepsilon \) (for any \( \varepsilon \in \rbrack 0,1\lbrack \) ). Now \( {H}^{-s} = {\Xi }^{t}{H}^{t - s} \) for all \( t \) . Taking \( t = {2M} > N + n \) , we have that \( t - s \geq N + n + 1 - N - n/2 - \varepsilon = n/2 + 1 ... | Yes |
Corollary 6.21. Let \( \\Omega \) be open \( \\subset {\\mathbb{R}}^{n} \), let \( u \\in {\\mathcal{D}}^{\\prime }\\left( \\Omega \\right) \) and let \( {\\Omega }^{\\prime } \) be an open subset of \( \\Omega \) with \( \\overline{{\\Omega }^{\\prime }} \) compact \( \\subset \\Omega \) . Let \( \\zeta \\in {C}_{0}^{... | \[ u = \\mathop{\\sum }\\limits_{{\\left| \\alpha \\right| \\leq {2M}}}{D}^{\\alpha }{f}_{\\alpha }\\text{ on }{\\Omega }^{\\prime }.\] | Yes |
Theorem 6.22. \( {1}^{ \circ } \) Let \( P\left( D\right) = \operatorname{Op}\left( {p\left( \xi \right) }\right) \), where \( p\left( \xi \right) \in {\mathcal{O}}_{M} \) and there exist \( m \in \mathbb{R}, c > 0 \) and \( r \geq 0 \) such that\n\n\[ \left| {p\left( \xi \right) }\right| \geq c\langle \xi {\rangle }^{... | Proof. \( {1}^{ \circ } \) . That \( P\left( D\right) u \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \) means that \( \langle \xi {\rangle }^{s}p\left( \xi \right) \widehat{u}\left( \xi \right) \in {L}_{2}\left( {\mathbb{R}}^{n}\right) \) . Therefore we have when \( \widehat{u}\left( \xi \right) \in {L}_{2,\operatorname{l... | Yes |
Corollary 6.23. When \( P\left( D\right) \) is an elliptic differential operator of order \( m \) with constant coefficients, one has for each \( s \in \mathbb{R} \), when \( u \in {\mathcal{S}}^{\prime } \) with \( \widehat{u} \in \) \( {L}_{2,\text{ loc }} \) :\n\n\[ P\left( D\right) u \in {H}^{s}\left( {\mathbb{R}}^... | Proof. The implication \( \Leftarrow \) is an immediate consequence of Lemma 6.17, while \( \Rightarrow \) follows from Theorem 6.22. | No |
Theorem 6.24. Let \( P\left( D\right) \) be elliptic of order \( m \) on \( {\mathbb{R}}^{n} \), with constant coefficients. Let \( \Omega \) be an open subset of \( {\mathbb{R}}^{n} \) . The minimal realization \( {P}_{\min } \) of \( P\left( D\right) \) in \( {L}_{2}\left( \Omega \right) \) satisfies\n\n\[ D\left( {P... | Proof. For \( \Omega = {\mathbb{R}}^{n} \) we have already shown in Theorem 6.3 that \( D\left( {P}_{\min }\right) = \) \( D\left( {P}_{\max }\right) \), and the identification of this set with \( {H}^{m}\left( {\mathbb{R}}^{n}\right) \) follows from Corollary 6.23. That the graph-norm and the \( {H}^{m} \) -norm are e... | Yes |
Lemma 6.27. Let \( s \in \mathbb{R} \) . When \( f \in {C}^{\infty }\left( \Omega \right) \) and \( \alpha \in {\mathbb{N}}_{0}^{n} \), then the operator \( u \mapsto f{D}^{\alpha }u \) is a continuous mapping of \( {H}_{\text{loc }}^{s}\left( \Omega \right) \) into \( {H}_{\text{loc }}^{s - \left| \alpha \right| }\lef... | Proof. When \( u \in {H}_{\text{loc }}^{s}\left( \Omega \right) \), one has for each \( j = 1,\ldots, n \), each \( \varphi \in {C}_{0}^{\infty }\left( \Omega \right) \), that \[ \varphi \left( {{D}_{j}u}\right) = {D}_{j}\left( {\varphi u}\right) - \left( {{D}_{j}\varphi }\right) u \in {H}^{s - 1}\left( {\mathbb{R}}^{n... | Yes |
Corollary 6.30. When \( P \) is an elliptic differential operator on \( \Omega \) of order \( m \), with constant coefficients in the principal symbol, then\n\n\[ D\left( {P}_{\max }\right) \subset {H}_{\text{loc }}^{m}\left( \Omega \right) . \] | The corollary implies that the realizations \( T \) and \( {T}_{1} \) of \( - \Delta \) introduced in Theorems 4.27 and 4.28 have domains contained in \( {H}_{\text{loc }}^{2}\left( \Omega \right) \) ; the so-called \ | No |
For any sequence of symbols \( {p}_{{m}_{j}}\left( {X,\xi }\right) \) in \( {S}_{1,0}^{{m}_{j}}\left( {\sum ,{\mathbb{R}}^{n}}\right) ,{m}_{j} \searrow \) \( - \infty \), there exists a function \( p\left( {X,\xi }\right) \) such that \( p \sim \mathop{\sum }\limits_{j}{p}_{{m}_{j}} \) in \( {S}_{1,0}^{{m}_{0}}\left( {... | For the proof, one takes\n\n\[ p\left( {X,\xi }\right) = \mathop{\sum }\limits_{{j \in {\mathbb{N}}_{0}}}{p}_{{m}_{j}}\left( {X,\xi }\right) \left( {1 - \chi \left( {{\varepsilon }_{j}\xi }\right) }\right) ,\]\n\n(7.10)\n\nwhere \( \chi \) is our usual cut-off function, and \( {\varepsilon }_{j} \) goes to zero suffici... | Yes |
Lemma 7.9. Let \( p\left( {x, y,\xi }\right) \in {S}_{1,0}^{d}\left( {\Omega \times \Omega ,{\mathbb{R}}^{n}}\right) \) . When \( \varphi \left( {x, y}\right) \in {C}^{\infty }\left( {\Omega \times \Omega }\right) \) with \( \operatorname{supp}\varphi \subset \left( {\Omega \times \Omega }\right) \smallsetminus \operat... | Proof. Since \( \varphi \left( {x, y}\right) \) vanishes on a neighborhood of the diagonal \( \operatorname{diag}\left( {\Omega \times \Omega }\right) \) , \( \varphi \left( {x, y}\right) /{\left| y - x\right| }^{2N} \) is \( {C}^{\infty } \) for any \( N \in {\mathbb{N}}_{0} \), so we may write \( \varphi \left( {x, y... | Yes |
Theorem 7.10. Any \( P = \operatorname{Op}\left( {p\left( {x, y,\xi }\right) }\right) \) with \( p \in {S}_{1,0}^{d}\left( {\Omega \times \Omega }\right) \) can be written as the sum of a properly supported operator \( {P}^{\prime } \) and a negligible operator \( \mathcal{R} \) . | Proof. The basic idea is to obtain the situation of Lemma 7.9 with \( \varphi \left( {x, y}\right) = \) \( 1 - \varrho \left( {x, y}\right) \), where \( \varrho \) has the following property: Whenever \( {M}_{1} \) and \( {M}_{2} \) are compact \( \subset \Omega \), then the sets\n\n\[ \n{M}_{12} = \left\{ {y \in \Omeg... | Yes |
Proposition 7.11. A \( \psi \) do \( P \) preserves singular supports:\n\n\[ \text{sing supp}{Pu} \subset \operatorname{sing}\operatorname{supp}u\text{.} \] | Proof. Let \( u \in {\mathcal{E}}^{\prime }\left( \Omega \right) \) and write \( u = {u}_{\varepsilon } + {v}_{\varepsilon } \) where \( \operatorname{supp}{u}_{\varepsilon } \subset \operatorname{sing}\operatorname{supp}u + \) \( B\left( {0,\varepsilon }\right) \), and \( {v}_{\varepsilon } \in {C}_{0}^{\infty }\left(... | Yes |
Lemma 7.12. When \( P = \operatorname{Op}\left( {p\left( {x,\xi }\right) }\right) \) is properly supported and \( \mathcal{R} \) is negligible, then \( P\mathcal{R} \) and \( \mathcal{R}P \) are negligible. | Proof (sketch). Let \( K\left( {x, y}\right) \) be the kernel of \( \mathcal{R} \) . Then for \( u \in {C}_{0}^{\infty }\left( \Omega \right) \), \[ P\mathcal{R}u = {\int }_{\Omega \times \Omega \times {\mathbb{R}}^{n}}{e}^{i\left( {x - z}\right) \cdot \xi }p\left( {x,\xi }\right) K\left( {z, y}\right) u\left( y\right)... | Yes |
Let \( p\left( {x, y,\xi }\right) \in {S}_{1,0}^{d}\left( {\Omega \times \Omega }\right) \). Then | \[ \operatorname{Op}\left( {p\left( {x, y,\xi }\right) }\right) \sim \operatorname{Op}\left( {{p}_{1}\left( {x,\xi }\right) }\right) \sim \operatorname{Op}\left( {{p}_{2}\left( {y,\xi }\right) }\right) ,\text{ where } \] \[ {p}_{1}\left( {x,\xi }\right) \sim \mathop{\sum }\limits_{{\alpha \in {\mathbb{N}}_{0}^{n}}}\fra... | Yes |
Lemma 7.14 (Backwards Leibniz formula). For \( u, v \in {C}^{\infty }\left( \Omega \right) \), \[ \left( {{D}^{\theta }u}\right) v = \mathop{\sum }\limits_{{\alpha ,\beta \in {\mathbb{N}}_{0},\alpha + \beta = \theta }}\frac{\theta !}{\alpha !\beta !}{D}^{\beta }\left( {u{\bar{D}}^{\alpha }v}\right) . \] | Proof. This is deduced from the usual Leibniz formula by noting that \[ \left\langle {\left( {{D}^{\theta }u}\right) v,\varphi }\right\rangle = {\left( -1\right) }^{\left| \theta \right| }\left\langle {u,{D}^{\theta }\left( {v\varphi }\right) }\right\rangle = {\left( -1\right) }^{\left| \theta \right| }\left\langle {u,... | Yes |
Corollary 7.19. \( {1}^{ \circ } \) When \( P \) is a square matrix-formed \( \psi \) do on \( \Omega \) that is elliptic of order \( d \), then it has a properly supported parametrix \( Q \) that is an elliptic \( \psi \) do of order \( - d \) . The parametrix \( Q \) is unique up to a negligible term. | Proof. That \( P \) is surjectively/injectively elliptic of order \( d \) means that \( P = \) \( \operatorname{Op}\left( {p\left( {x,\xi }\right) }\right) + \mathcal{R} \), where \( \mathcal{R} \) is negligible and \( p \in {S}^{d}\left( \Omega \right) \otimes \mathcal{L}\left( {{\mathbb{C}}^{N},{\mathbb{C}}^{{N}^{\pr... | Yes |
Corollary 7.20. When \( P \) is injectively elliptic of order \( d \) on \( \Omega \), and properly supported, then any solution \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) of the equation \[ {Pu} = f\;\text{ in }\Omega ,\;\text{ with }f \in {H}_{\text{loc }}^{s}\left( \Omega \right) , \] satisfies \( u \i... | Proof. Let \( Q \) be a properly supported left parametrix of \( P \), it is of order \( - d \) . Then \( {QP} = I - \mathcal{R} \), with \( \mathcal{R} \) negligible, so \( u \) satisfies \[ u = {QPu} + \mathcal{R}u = {Qf} + \mathcal{R}u. \] Here \( Q \) maps \( {H}_{\text{loc }}^{s}\left( \Omega \right) \) into \( {H... | Yes |
Theorem 7.22. Let \( s \) and \( t \in \mathbb{R} \) . \n\n\( {1}^{ \circ } \) For any \( \theta \in \left\lbrack {0,1}\right\rbrack \) one has for all \( u \in {H}^{\max \{ s, t\} }\left( {\mathbb{R}}^{n}\right) \) : \n\n\[ \parallel u{\parallel }_{{\theta s} + \left( {1 - \theta }\right) t, \land } \leq \parallel u{\... | Proof. \( {1}^{ \circ } \) . When \( \theta = 0 \) or 1, the inequality is trivial, so let \( \theta \in \rbrack 0,1\lbrack \) . Then we use the Hölder inequality with \( p = 1/\theta ,{p}^{\prime } = 1/\left( {1 - \theta }\right) \) : \n\n\[ \parallel u{\parallel }_{{\theta s} + \left( {1 - \theta }\right) t, \land }^... | Yes |
Theorem 7.24. Let \( \Omega \subset {\mathbb{R}}^{n} \) be bounded and open, and let \( A = \mathop{\sum }\limits_{{\left| \alpha \right| \leq d}}{a}_{\alpha }{D}^{\alpha } \) be strongly elliptic on a neighborhood \( {\Omega }_{1} \) of \( \bar{\Omega } \) (with \( \left( {N \times N}\right) \) -matrix-formed \( {C}^{... | Proof. The order \( d \) is even, because \( \operatorname{Re}{a}^{0}\left( {x,\xi }\right) \) and \( \operatorname{Re}{a}^{0}\left( {x, - \xi }\right) \) are both positive definite. Theorem 7.23 assures that \[ \operatorname{Re}\left( {{Au}, u}\right) \geq {c}_{0}\parallel u{\parallel }_{m}^{2} - k\parallel u{\paralle... | Yes |
Theorem 8.1. \( {1}^{ \circ } \) Let \( P = \operatorname{Op}\left( {q\left( {x, y,\xi }\right) }\right) \) be a \( \psi \) do with \( q\left( {x, y,\xi }\right) \in \) \( {S}_{1,0}^{m}\left( {\Omega \times \Omega ,{\mathbb{R}}^{n}}\right) \) and let \( K \) be a compact subset of \( \Omega \) . With \( {\kappa }^{\pri... | \n\[ \underline{q}\left( {\underline{x},\underline{y},\underline{\xi }}\right) = q\left( {x, y,{}^{t}M\left( {x, y}\right) \underline{\xi }}\right) \cdot \left| {\det {}^{t}M\left( {x, y}\right) }\right| \left| {\det {\kappa }^{\prime }{\left( y\right) }^{-1}}\right| . \]\n\n(8.3)\n\nIn particular, \( \underline{q}\lef... | Yes |
Theorem 8.2 (Rellich’s Theorem). The injection of \( {H}^{s}\left( X\right) \) into \( {H}^{{s}^{\prime }}\left( X\right) \) is compact when \( s > {s}^{\prime } \) . | This can be proved in the following steps: 1) A reduction to compactly supported distributions in each coordinate patch by a partition of unity, 2) an embedding of a compact subset of a coordinate patch into \( {\mathbb{T}}^{n} \) (the \( n \) - dimensional torus),3) a proof of the property for \( {\mathbb{T}}^{n} \) b... | No |
Theorem 8.3. \( {1}^{ \circ } \) When \( s > 0,{\Lambda }_{-s} \) defines a bounded selfadjoint operator in \( {L}_{2}\left( {\mathbb{T}}^{n}\right) \), which is a compact operator in \( {L}_{2}\left( {\mathbb{T}}^{n}\right) \) . | Proof. Let \( s > 0 \) . Since \( \parallel u{\parallel }_{s, \land } \geq \parallel u{\parallel }_{0. \land },{\Lambda }_{-s} \) defines a bounded operator \( T \) in \( {L}_{2}\left( {\mathbb{T}}^{n}\right) \), and it is clearly symmetric, hence selfadjoint. Moreover, the orthonormal basis \( {\left( {e}^{{ik} \cdot ... | Yes |
Lemma 8.4. Let \( X \) be a compact \( {C}^{\infty } \) manifold.\n\n\( {1}^{ \circ } \) To every finite open cover \( \left\{ {{U}_{1},\ldots ,{U}_{J}}\right\} \) of \( X \) there exists an associated partition of unity \( \left\{ {{\psi }_{1},\ldots ,{\psi }_{J}}\right\} \), that is, a family of nonnegative functions... | Proof. The statement in \( {1}^{ \circ } \) is a simple generalization of the well-known statement for compact sets in \( {\mathbb{R}}^{n} \) (Theorem 2.17): We can choose compact subsets \( {K}_{j},{K}_{j}^{\prime } \) of the \( {U}_{j} \) such that \( {K}_{j}^{\prime } \subset {K}_{j}^{ \circ }, X = \bigcup {K}_{j}^{... | Yes |
Theorem 8.5. Let \( P \) be a pseudodifferential operator on \( X \) of order \( d \) . Then \( P \) is continuous from \( {H}^{s}\left( X\right) \) to \( {H}^{s - d}\left( X\right) \) for all \( s \in \mathbb{R} \) . | Proof. Let \( {\varrho }_{k} \) be a partition of unity as in Lemma \( {8.42}^{ \circ } \) . Then \( P = \) \( \mathop{\sum }\limits_{{j, k < {J}_{0}}}{\varrho }_{j}P{\varrho }_{k} \) . For each \( j, k \) there is an \( i \leq {I}_{1} \) such that \( {\varrho }_{j} \) and \( {\varrho }_{k} \) are supported in \( {U}_{... | Yes |
Theorem 8.7. Let \( E \) and \( {E}^{\prime } \) be complex vector bundles over \( X \) of fiber dimensions \( N \) resp. \( {N}^{\prime } \), and let \( P \) be a pseudodifferential operator of order \( d \) from the sections of \( E \) to the sections of \( {E}^{\prime } \) . \( {1}^{ \circ } \) If \( {N}^{\prime } \... | In the case of trivial bundles over \( X \), the proof can be left to the reader (to deduce it from Theorem 7.18 by the method in Theorem 8.6). In the situation of general bundles one should replace the coordinate changes \( {\kappa }_{j} \) by the local trivializations \( {\Psi }_{j} \), with an appropriate notation f... | No |
Proposition 8.8. When \( T \in \mathbf{B}\left( {{H}_{1},{H}_{2}}\right) \) is bijective, and \( K \in \mathbf{B}\left( {{H}_{1},{H}_{2}}\right) \) is compact, then \( T + K \) is a Fredholm operator. | Proof. Recall that a compact operator maps any bounded sequence into a sequence that has a convergent subsequence. We first show why \( Z\left( {T + K}\right) \) is finite dimensional: \( Z\left( {T + K}\right) \) is a linear space, and for \( x \in Z\left( {T + K}\right) \) ,\n\n\[ \n{Tx} = - {Kx}. \n\] \n\nLet \( {x}... | Yes |
An operator \( T \in \mathbf{B}\left( {{H}_{1},{H}_{2}}\right) \) is Fredholm if and only if there exist \( {S}_{1},{S}_{2} \in \mathbf{B}\left( {{H}_{2},{H}_{1}}\right) ,{K}_{1} \) compact in \( {H}_{1} \) and \( {K}_{2} \) compact in \( {H}_{2} \), such that\n\n\[ \n{S}_{1}T = I + {K}_{1},\;T{S}_{2} = I + {K}_{2}.\n\... | Proof. When \( X \) is a closed subspace of a Hilbert space \( H \), we denote by \( {\operatorname{pr}}_{X} \) the orthogonal projection onto \( X \), and by \( {\mathrm{i}}_{X} \) the injection of \( X \) into \( H \) . (When \( X \subset {X}_{1} \subset H \), we denote the injection of \( X \) into \( {X}_{1} \) by ... | Yes |
Theorem 8.10. \( {1}^{ \circ } \) Multiplicative property of the index. When \( {T}_{1} \in \) \( \mathbf{B}\left( {{H}_{1},{H}_{2}}\right) \) and \( {T}_{2} \in \mathbf{B}\left( {{H}_{2},{H}_{3}}\right) \) are Fredholm operators, then \( {T}_{2}{T}_{1} \in \mathbf{B}\left( {{H}_{1},{H}_{3}}\right) \) is also Fredholm,... | Proof. For \( {1}^{ \circ } \), we give a brief indication of the proof, found with more details in [C90, IX §3]. When \( T \) is an operator of the form\n\n\[ \nT = \left( \begin{matrix} {T}^{\prime } & R \\ 0 & {T}^{\prime \prime } \end{matrix}\right) : \begin{matrix} {H}_{1}^{\prime } \\ \oplus \\ {H}_{1}^{\prime \p... | Yes |
Corollary 8.12. The index of \( P \) depends only on the principal symbol of \( P \) . | Proof. Let \( {P}_{1} \) be a polyhomogeneous \( \psi \) do with the same principal symbol as \( P \), i.e., \( {P}_{1} - P \) is of order \( d - 1 \) . Then\n\n\[ \n{P}_{1}Q = {PQ} + \left( {{P}_{1} - P}\right) Q = I + {\mathcal{R}}_{1}^{\prime }, \n\]\n\n\[ \nQ{P}_{1} = {QP} + Q\left( {{P}_{1} - P}\right) = I + {\mat... | Yes |
For \( m \in {\mathbb{N}}_{0} \), there are inequalities \[ \parallel u{\parallel }_{m}^{2} \leq \parallel u{\parallel }_{m,\prime }^{2} \leq {C}_{m}^{\prime }\parallel u{\parallel }_{m}^{2} \] valid for all \( u \in \mathcal{S}\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), and hence \( \parallel u{\parallel }_{m,... | Proof. We have as in (5.2), for \( k \in \mathbb{N} \) : \[ \mathop{\sum }\limits_{{\beta \in {\mathbb{N}}_{0}^{n - 1},\left| \beta \right| \leq k}}{\left( {\xi }^{\prime }\right) }^{2\beta } \leq {\left\langle {\xi }^{\prime }\right\rangle }^{2k} = \mathop{\sum }\limits_{{\left| \beta \right| \leq k}}{C}_{k,\beta }{\l... | Yes |
Theorem 9.2. Let \( m \) be an integer \( > 0 \) . For \( 0 \leq j \leq m - 1 \), the trace mapping\n\n\[ \n{\gamma }_{j} : u\left( {{x}^{\prime },{x}_{n}}\right) \mapsto {D}_{{x}_{n}}^{j}u\left( {{x}^{\prime },0}\right) \n\]\n\n(9.12)\n\nfrom \( {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\... | Proof. As in Theorem 4.24, we shall use an inequality like (4.47), but now in a slightly different and more precise way. For \( v\left( t\right) \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) one has\n\n\[ \n{\left| v\left( 0\right) \right| }^{2} = - {\int }_{0}^{\infty }\frac{d}{dt}\left\lbrack {v\left( t\right) \ba... | Yes |
Theorem 9.3. Define the Poisson operator \( {K}_{\gamma } \) from \( \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) to \( \mathcal{S}\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) by\n\n\[ \n{K}_{\gamma } : \varphi \left( {x}^{\prime }\right) \mapsto {\mathcal{F}}_{{\xi }^{\prime } \rightarrow {x}^{\prime }}^{-1}... | Proof. It is easily checked that \( {e}^{-\left\langle {\xi }^{\prime }\right\rangle {x}_{n}} \) belong to \( \mathcal{S}\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) (one needs to check derivatives of \( {e}^{-\left\langle {\xi }^{\prime }\right\rangle {x}_{n}} \), where (5.7) is useful); then also \( {e}^{-\left... | Yes |
Theorem 9.5. Let \( \psi \in \mathcal{S}\left( \mathbb{R}\right) \) with \( \psi \left( t\right) = 1 \) on a neighborhood of 0 . Define the Poisson operator \( {\mathcal{K}}_{\left( m\right) } \) from \( \varphi = \left\{ {{\varphi }_{0},\ldots ,{\varphi }_{m - 1}}\right\} \in \mathcal{S}{\left( {\mathbb{R}}^{n - 1}\ri... | Proof. Since \( \psi \left( 0\right) = 1 \) and \( {D}_{{x}_{n}}^{j}\psi \left( 0\right) = 0 \) for \( j > 0 \), and \( {\left\lbrack {D}_{{x}_{n}}^{k}\left( \frac{1}{j!}{i}^{j}{x}_{n}^{j}\right) \right\rbrack }_{{x}_{n} = 0} = \) \( {\delta }_{kj} \) (the Kronecker delta),\n\n\[ \n{\gamma }_{k}{\mathcal{K}}_{j}{\varph... | Yes |
Theorem 9.6. For all \( m \in \mathbb{N} \) ,\n\n\[ \n{H}_{0}^{m}\left( {\mathbb{R}}_{ + }^{n}\right) = \left\{ {u \in {H}^{m}\left( {\mathbb{R}}_{ + }^{n}\right) \mid {\gamma }_{0}u = \cdots = {\gamma }_{m - 1}u = 0}\right\} .\n\] | This is proved by a variant of the proof of Theorem 4.25, or by the method described after it, and will not be written in detail here. | No |
Theorem 9.7. For each \( m \in {\mathbb{N}}_{0},{H}_{0}^{m}\left( {\mathbb{R}}_{ + }^{n}\right) \) identifies, by extension by zero on \( {\mathbb{R}}_{ - }^{n} \), with the subspace of \( {H}^{m}\left( {\mathbb{R}}^{n}\right) \) consisting of the functions supported in \( {\overline{\mathbb{R}}}_{ + }^{n} \) . | Proof (indications). When \( u \in {H}_{0}^{m}\left( {\mathbb{R}}_{ + }^{n}\right) \), it is the limit of a sequence of function \( {u}_{k} \in {C}_{0}^{\infty }\left( {\mathbb{R}}_{ + }^{n}\right) \) in the \( m \) -norm. Extending the \( {u}_{k} \) by 0 (to \( {e}^{ + }{u}_{k} \) ), we see that \( {e}^{ + }{u}_{k} \r... | No |
Theorem 9.8. The space \( {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) is dense in \( D\left( {A}_{\max }\right) \) . | Proof. This follows if we show that when \( \ell \) is a continuous antilinear (conjugate linear) functional on \( D\left( {A}_{\max }\right) \) which vanishes on \( {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), then \( \ell = 0 \) . So let \( \ell \) be such a functional; it can b... | Yes |
Lemma 9.9. For \( u \) and \( v \) in \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) one has Green’s formula\n\n\[{\left( Au, v\right) }_{{L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) } - {\left( u, Av\right) }_{{L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) } = {\left( \nu u,{\gamma }_{0}v\right) }_{{L}_{2}\left( {\math... | Proof. For functions in \( {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), the formula follows directly from (A.20). (It is easily verified by integration by parts.) Since \( {\gamma }_{0} \) and \( \nu \) by Theorem 9.2 map \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) to spaces... | Yes |
Theorem 9.10. Let \( A = I - \Delta \) on \( {\mathbb{R}}_{ + }^{n} \) . The Cauchy trace operator \( \varrho = \) \( \left\{ {{\gamma }_{0},\nu }\right\} \), defined on \( {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), extends by continuity to a continuous mapping from \( D\left( {... | Proof. Let \( u \in D\left( {A}_{\max }\right) \) . In the following, we write \( {H}^{s}\left( {\mathbb{R}}^{n - 1}\right) \) as \( {H}^{s} \) . We want to define \( \varrho u = \left\{ {{\gamma }_{0}u,{\nu u}}\right\} \) as a continuous antilinear functional on \( {H}^{\frac{1}{2}} \times {H}^{\frac{3}{2}} \) , depen... | Yes |
For any \( \varphi \in {H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) ,{K}_{\gamma }\varphi \) is the unique solution in \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) of (9.35). | Proof. Let \( v = {K}_{\gamma }\varphi \), then \( v \in {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) with \( {Av} = 0 \) and \( {\gamma }_{0}v = \varphi \) . So a function \( z \) solves (9.35) if and only if \( w = z - v \) solves\n\n\[ \n{Aw} = 0,\;{\gamma }_{0}w = 0.\n\]\n\nHere if \( z \in {H}^{2}\left( {\mathbb... | Yes |
Corollary 9.14. The mappings\n\n\[ \n{\gamma }_{0} : {Z}^{2}\left( A\right) \rightarrow {H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \text{and}{K}_{\gamma } : {H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \rightarrow {Z}^{2}\left( A\right) \n\]\n\nare inverses of one another. | Proof. The mappings are well-defined according to Theorem 9.3, which also shows the identity \( {\gamma }_{0}{K}_{\gamma } = I \) on \( {H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \) ; it implies surjectiveness of \( {\gamma }_{0} \) and injectiveness of \( {K}_{\gamma } \) . Corollary 9.13 implies that when \(... | Yes |
Proposition 9.15. \( {Z}^{2}\left( A\right) \) is dense in \( {Z}^{0}\left( A\right) \) . | Proof. Let \( z \in {Z}^{0}\left( A\right) \) . By Theorem 9.8 there exists a sequence \( {u}_{k} \in {C}_{\left( 0\right) }^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) such that \( {u}_{k} \rightarrow z \) in \( {L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right), A{u}_{k} \rightarrow 0 \) . Let \( {v}_{k} = {... | Yes |
Theorem 9.16. The mappings\n\n\[ \n{\gamma }_{0} : {Z}^{0}\left( A\right) \rightarrow {H}^{-\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \text{ and }{K}_{\gamma } : {H}^{-\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \rightarrow {Z}^{0}\left( A\right) \n\]\n\nare inverses of one another. | Proof. We already have the identity \( {\gamma }_{0}{K}_{\gamma }\varphi = \varphi \) for \( \varphi \in {H}^{-\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \) from Theorem 9.10, and the other identity \( {K}_{\gamma }{\gamma }_{0}z = z \) for \( z \in {Z}^{0}\left( A\right) \) now follows from the identity valid on \... | Yes |
Corollary 9.17. Let \( m \in {\mathbb{N}}_{0} \) . The mappings\n\n\[ \n{\gamma }_{0} : {Z}^{m}\left( A\right) \rightarrow {H}^{m - \frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \text{ and }{K}_{\gamma } : {H}^{m - \frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \rightarrow {Z}^{m}\left( A\right) \n\]\n\nare inverses o... | Proof. That the mappings are well-defined follows from Theorem 9.3 and its extension in Theorem 9.10. Then the statements \( {\gamma }_{0}{K}_{\gamma }\varphi = \varphi \) and \( {K}_{\gamma }{\gamma }_{0}z = z \) for the relevant spaces follow by restriction from Theorem 9.16. | Yes |
Theorem 9.18. The Dirichlet problem (9.33) is uniquely solvable in \( D\left( {A}_{\max }\right) \) for \( f \in {L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) ,\varphi \in {H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \) ; the solution belongs to \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) and is defined by t... | \[ \left( \begin{array}{l} A \\ {\gamma }_{0} \end{array}\right) : {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \rightarrow \mathop{\sum }\limits_{{{H}^{\frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) }}^{{{L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) }}\text{ has inverse }\left( {{R}_{\gamma }{K}_{\gamma }}\right) : \math... | Yes |
Theorem 9.19. Define the Poisson operator \( {K}_{\nu } \) from \( \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) to \( \mathcal{S}\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) by\n\n\[ \n{K}_{\nu } : \varphi \left( {x}^{\prime }\right) \mapsto {\mathcal{F}}_{{\xi }^{\prime } \rightarrow {x}^{\prime }}^{-1}\left... | Proof. \( {1}^{ \circ } \) is seen from\n\n\[ \n\nu {K}_{\nu }\varphi = {\left\lbrack {\mathcal{F}}_{{\xi }^{\prime } \rightarrow {x}^{\prime }}^{-1}\left( -{\partial }_{{x}_{n}}\langle {\xi }^{\prime }{\rangle }^{-1}{e}^{-\left\langle {\xi }^{\prime }\right\rangle {x}_{n}}\widehat{\varphi }\left( {\xi }^{\prime }\righ... | Yes |
Theorem 9.20. The solution operator \( {A}_{\nu }^{-1} \) of (9.46) equals\n\n\[ {A}_{\nu }^{-1} = {r}^{ + }Q{e}^{ + } - {K}_{\nu }\nu {r}^{ + }Q{e}^{ + },\text{ also denoted }{R}_{\nu }. \] | It maps \( {L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) continuously into \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \), hence\n\n\[ D\left( {A}_{\nu }\right) = \left\{ {u \in {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \mid {\nu u} = 0}\right\} . \] | No |
Corollary 9.21. For any \( \varphi \in {H}^{\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) ,{K}_{\nu }\varphi \) is the unique solution in \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) of (9.47). | The mappings\n\n\[ \nu : {Z}^{2}\left( A\right) \rightarrow {H}^{\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \text{ and }{K}_{\nu } : {H}^{\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \rightarrow {Z}^{2}\left( A\right) \]\n\nare inverses of one another. | No |
Theorem 9.22. Let \( m \in {\mathbb{N}}_{0} \) . The mappings \[ \nu : {Z}^{m}\left( A\right) \rightarrow {H}^{m - \frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \text{ and }{K}_{\nu } : {H}^{m - \frac{3}{2}}\left( {\mathbb{R}}^{n - 1}\right) \rightarrow {Z}^{m}\left( A\right) \] are inverses of one another. | This is shown just as in the proofs of Theorem 9.16 and Corollary 9.17. | No |
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