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Theorem 2.60. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and consider a relatively closed subset \( F \) of \( \Omega \) satisfying \( \operatorname{supp}u \subseteq F \) . Set\n\n\[ \n{\mathcal{M}}_{F} \mathrel{\text{:=}} \left\{ {\varphi \in {C}^{\infty }\left( \Omega \right) : \operatorname{supp}\v... | Proof of Theorem 2.60. Fix a relatively closed subset \( F \) of \( \Omega \) satisfying \( \operatorname{supp}u \subseteq F \) . First we prove the uniqueness statement in the first part of the theorem. Suppose \( {\widetilde{u}}_{1} \) , \( {\widetilde{u}}_{2} : {\mathcal{M}}_{F} \rightarrow \mathbb{C} \) satisfy \( ... | Yes |
Proposition 2.64. Let \( v : \mathcal{E}\left( \Omega \right) \rightarrow \mathbb{C} \) be a linear map. Then \( v \) is continuous if and only if \( v \) is sequentially continuous. | Proof. The general fact that any linear and continuous functional on topological vector spaces is sequentially continuous gives the left-to-right implication. To prove the converse implication, it suffices to check continuity at zero. This is done reasoning by contradiction. Assume that\n\n\[ v\left( {\varphi }_{j}\rig... | Yes |
Proposition 2.69. Let \( \omega \) and \( \Omega \) be open subsets of \( {\mathbb{R}}^{n} \) such that \( \omega \subseteq \Omega \) . Then every \( u \in {\mathcal{E}}^{\prime }\left( \omega \right) \) extends to a functional \( \widetilde{u} \in {\mathcal{E}}^{\prime }\left( \Omega \right) \) by setting\n\n\[ \widet... | Proof. We first claim that the mapping in (2.6.18) is well defined. To see why this is the case, suppose \( {\psi }_{j} \in {C}_{0}^{\infty }\left( \omega \right) ,{\psi }_{j} \equiv 1 \) in a neighborhood of supp \( u \), for \( j = 1,2 \) . Then for each function \( \varphi \in {C}^{\infty }\left( \Omega \right) \) w... | Yes |
Proposition 2.72. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( \psi \in {C}_{0}^{\infty }\left( \Omega \right) \) . Then \( {\psi u} \in {\mathcal{E}}^{\prime }\left( \Omega \right) \) and\n\n\[ \n{}_{{\mathcal{E}}^{\prime }}\langle {\psi u},\varphi {\rangle }_{\mathcal{E}} = {}_{{\mathcal{D}}^{\... | Proof. Since \( {\psi u} \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( \operatorname{supp}{\psi u} \subseteq \operatorname{supp}\psi \), we have \( {\psi u} \in {\mathcal{D}}_{c}^{\prime }\left( \Omega \right) \), thus \( {\psi u} \in \) \( {\mathcal{E}}^{\prime }\left( \Omega \right) \) (i.e., \( {\psi u}... | Yes |
Let \( m \in \mathbb{N} \) . We are interested in solving the equation\n\n\[ {x}^{m}u = 0\;\text{ in }\;{\mathcal{D}}^{\prime }\left( \mathbb{R}\right) . \] | In this regard, assume that \( u \in {\mathcal{D}}^{\prime }\left( \mathbb{R}\right) \) solves (2.6.27) and note that if \( \varphi \) belongs to \( {C}_{0}^{\infty }\left( {\mathbb{R}\smallsetminus \{ 0\} }\right) \), then \( \frac{1}{{x}^{m}}\varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}\smallsetminus \{ 0\} }\right... | Yes |
Let \( N \in \mathbb{N} \) and \( {a}_{j} \in {\mathbb{R}}^{n}, j \in \{ 1,\ldots, N\} \), be a finite family of distinct points. If \( u \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) is such that \( \operatorname{supp}u \subseteq \left\{ {{a}_{1},{a}_{2},\ldots ,{a}_{N}}\right\} \), then \( u \) has a u... | To justify formula (2.6.32), fix a family of pairwise disjoint balls \( {B}_{j} \mathrel{\text{:=}} B\left( {{a}_{j},{r}_{j}}\right) \) , \( j \in \{ 1,\ldots, N\} \), and for each \( j \) select a function \( {\psi }_{j} \in {C}_{0}^{\infty }\left( {B}_{j}\right) \) satisfying \( {\psi }_{j} \equiv 1 \) in a neighborh... | Yes |
Lemma 2.82. Suppose that the sequence \( {\left\{ {f}_{j}\right\} }_{j \in \mathbb{N}} \subset \mathcal{E}\left( {\mathbb{R}}^{n}\right) \) and \( f \in \mathcal{E}\left( {\mathbb{R}}^{n}\right) \) are such that\n\n\[ \n{\begin{Vmatrix}{\partial }^{\alpha }{f}_{j} - {\partial }^{\alpha }f\end{Vmatrix}}_{{L}^{\infty }\l... | Proof. Suppose \( {\left\{ {f}_{j}\right\} }_{j \in \mathbb{N}} \) and \( f \) satisfy the current hypotheses. Pick an arbitrary \( \varepsilon > 0 \), a multi-index \( \alpha \in {\mathbb{N}}_{0}^{n} \), and fix a compact subset \( K \) of \( {\mathbb{R}}^{n} \) . Then there exists \( {j}_{0} \in \mathbb{N} \) such th... | Yes |
Proposition 2.84. Fix \( m, n \in \mathbb{N} \), let \( U \) be an open subset of \( {\mathbb{R}}^{m} \), and let \( V \) be an open subset of \( {\mathbb{R}}^{n} \). Then for each distribution \( u \in {\mathcal{D}}^{\prime }\left( U\right) \) the following properties hold.\n\n(a) If for each \( \varphi \in {C}_{0}^{\... | Proof of Proposition 2.84. Fix \( \varphi \in {C}_{0}^{\infty }\left( {U \times V}\right) \) and let \( K \mathrel{\text{:=}} \operatorname{supp}\varphi \) that is a compact subset of \( U \times V \). Also, consider \( \psi \) as in (2.7.28) and recall the projections \( {\pi }_{U},{\pi }_{V} \) from (2.7.24). Then cl... | Yes |
Theorem 2.87. Let \( m, n \in \mathbb{N} \) , \( U \) be an open subset of \( {\mathbb{R}}^{m} \), and \( V \) be an open subset of \( {\mathbb{R}}^{n} \) . Consider \( u \in {\mathcal{D}}^{\prime }\left( U\right) \) and \( v \in {\mathcal{D}}^{\prime }\left( V\right) \) . Then the following statements are true.\n\n(i)... | Proof. For each \( \varphi \in {C}_{0}^{\infty }\left( {U \times V}\right) \) consider the function\n\n\[ \psi \left( y\right) \mathrel{\text{:=}} \langle u\left( x\right) ,\varphi \left( {x, y}\right) \rangle \;\text{ for }\;y \in V. \]\n\nBy Proposition 2.84, we have \( \psi \in {C}_{0}^{\infty }\left( V\right) \) an... | Yes |
Theorem 2.89. Let \( m, n \in \mathbb{N} \) , \( U \) be an open subset of \( {\mathbb{R}}^{m} \), and \( V \) be an open subset of \( {\mathbb{R}}^{n} \) . Assume that \( u \in {\mathcal{D}}^{\prime }\left( U\right) \) and \( v \in {\mathcal{D}}^{\prime }\left( V\right) \) . Then the following properties hold.\n\n(a) ... | Proof. We start by proving the set theoretic equality from (a). For the right-to-left inclusion, fix \( \left( {{x}_{0},{y}_{0}}\right) \in \operatorname{supp}u \times \operatorname{supp}v \) . If \( C \subseteq U \times V \) is an open neighborhood of \( \left( {{x}_{0},{y}_{0}}\right) \), then there exists an open se... | Yes |
If \( u, v \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) are two distributions with the property that (2.8.9) is satisfied, then \( \operatorname{supp}\left( {u * v}\right) \subseteq \operatorname{supp}u + \operatorname{supp}v \) . | Let \( u, v \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) be such that (2.8.9) holds. Since \( \operatorname{supp}u + \operatorname{supp}v \) is closed, the inclusion in (a) will follow as soon as we show that\n\n\[{\left. u * v\right| }_{{\mathbb{R}}^{n} \smallsetminus \left( {\operatorname{supp}u + \op... | Yes |
Proposition 2.98. For each \( {x}_{0} \in {\mathbb{R}}^{n} \) and each \( u \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) fixed, the translation mapping \( \mathcal{D}\left( {\mathbb{R}}^{n}\right) \ni \varphi \mapsto \left\langle {u,{t}_{-{x}_{0}}\left( \varphi \right) }\right\rangle \in \mathbb{C} \) i... | Proof. This follows by observing that the mapping in question is the composition \( u \circ {t}_{-{x}_{0}} \) where the latter translation operator is consider in the sense of Exercise 1.19. | No |
If \( u \in {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) and \( {v}_{j}\xrightarrow[{j \rightarrow \infty }]{{\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) }v \), then \( u * {v}_{j}\xrightarrow[{j \rightarrow \infty }]{{\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) }u * v \). | To see why (1) is true, fix \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Then by definition, for each \( j \in \mathbb{N} \) we have \( \left\langle {u * {v}_{j},\varphi }\right\rangle = \left\langle {u \otimes {v}_{j},{\psi }_{j}{\varphi }^{A}}\right\rangle \) for any smooth compactly supported ... | Yes |
Theorem 2.106. The set \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is sequentially dense in \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \). | Proof. First we will show that \( {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is sequentially dense in \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \). Let \( \phi \) be as in (1.2.3) and recall the sequence of functions \( {\left\{ {\phi }_{j}\right\} }_{j \in \mathbb{N}} \) from (1.3.7). In particular,... | Yes |
Proposition 2.108. Suppose \( {\Omega }_{1},{\Omega }_{2} \) are open sets in \( {\mathbb{R}}^{n} \) and let \( F : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be a \( {C}^{\infty } \) diffeomorphism. For each \( u \in {\mathcal{D}}^{\prime }\left( {\Omega }_{2}\right) \), define the mapping \( u \circ F : \mathcal{D}\l... | Proof. Fix \( u \in {\mathcal{D}}^{\prime }\left( {\Omega }_{2}\right) \) . The fact that \( u \circ F \in {\mathcal{D}}^{\prime }\left( {\Omega }_{1}\right) \) is an immediate consequence of Exercise 1.25. Also, if \( {u}_{j}\xrightarrow[{j \rightarrow \infty }]{{\mathcal{D}}^{\prime }\left( {\Omega }_{2}\right) }u \)... | No |
Lemma 2.110. Suppose the functions \( {\left\{ {u}_{j}\right\} }_{j \in \mathbb{N}} \) and \( u \) are such that:\n\n(i) \( {u}_{j} \in {C}^{1}\left( \Omega \right) \) for every \( j \in \mathbb{N} \),\n\n(ii) \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j} = u \) uniformly on compact subsets of \( {\mathbb{R... | Proof. From the start, since uniform convergence on compact sets preserves continuity, we have that \( u \in {C}^{0}\left( \Omega \right) \). Fix \( x \in \Omega \) and \( k \in \{ 1,2,\ldots, n\} \) and let \( {t}_{0} > 0 \) be such that \( x + t{\mathbf{e}}_{k} \in \Omega \) whenever \( t \in \left\lbrack {-{t}_{0},{... | Yes |
Lemma 2.111. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) be such that for each \( j \in \{ 1,\ldots, n\} \), the distributional derivatives \( {\partial }_{j}u \) are of function type and belong to \( {C}^{0}\left( \Omega \right) \) . Then \( u \in {C}^{0}\left( \Omega \right) \) . | Proof. Since \( \nabla u \in {\left\lbrack {C}^{0}\left( \Omega \right) \right\rbrack }^{n} \) the function \( v\left( x\right) \mathrel{\text{:=}} {\int }_{0}^{1}\left( {\nabla u}\right) \left( {tx}\right) \cdot x\mathrm{\;d}t \) for \( x \in \Omega \) (where \ | No |
Theorem 2.112. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and suppose that there exists \( m \in {\mathbb{N}}_{0} \) such that for each \( \alpha \in {\mathbb{N}}_{0}^{n} \) satisfying \( \left| \alpha \right| = m \) the distributional derivative \( {\partial }^{\alpha }u \) is continuous on \( \Omega... | Proof. We prove the theorem by induction on \( m \) . For \( m = 0 \) there is nothing to prove. Suppose \( m = 1 \) . Applying Lemma 2.111, we obtain that \( u \in {C}^{0}\left( \Omega \right) \) . To prove that \( u \in {C}^{1}\left( \Omega \right) \), it suffices to show that \( u \) is of class \( {C}^{1} \) in a n... | Yes |
Proposition 2.121. Let \( \Omega \) be an open subset in \( {\mathbb{R}}^{n} \) and let \( f \in {\operatorname{Lip}}_{\text{loc }}\left( \Omega \right) \) . Then for each \( j \in \{ 1,\ldots, n\} \) the pointwise partial derivative \( {\partial }_{j}^{\mathrm{{pw}}}f \) exists at almost every point in \( \Omega \) an... | Proof. If for each \( k \in \mathbb{N} \) we introduce\n\n\[ \n{\Omega }_{k} \mathrel{\text{:=}} \{ x \in \Omega : \operatorname{dist}\left( {x,\partial \Omega }\right) > 1/k\text{ and }\left| x\right| < k\} ,\n\]\n\nthen\neach \( {\Omega }_{k} \) is an open, relatively compact, subset of \( \Omega \) ,\n\n\[ \n\overli... | No |
(a) For each \( a \in \mathcal{L}\left( {\mathbb{R}}^{n}\right) \), the mapping \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \ni \varphi \mapsto {a\varphi } \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) is well defined, linear, and continuous. | Proof. Clearly, the mappings in (a) and (b) are linear. By Fact 3.6 and Theorem 14.1, their continuity is equivalent with sequential continuity at 0 , something that can be easily checked using Fact 3.7. | No |
Proposition 3.16. Let \( m, n \in \mathbb{N} \) . Then \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{m}\right) \otimes {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is sequentially dense in \( \mathcal{S}\left( {{\mathbb{R}}^{m} \times {\mathbb{R}}^{n}}\right) \) . | Proof. Since the topology on \( \mathcal{S}\left( {{\mathbb{R}}^{m} \times {\mathbb{R}}^{n}}\right) \) is metrizable (recall Fact 3.6), there exists a distance function \( d : \mathcal{S}\left( {{\mathbb{R}}^{m} \times {\mathbb{R}}^{n}}\right) \times \mathcal{S}\left( {{\mathbb{R}}^{m} \times {\mathbb{R}}^{n}}\right) \... | Yes |
For every function \( f \in \mathcal{L}\left( {\mathbb{R}}^{n}\right) \) and every function \( g \) in \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) one has \( {\int }_{{\mathbb{R}}^{n}}\left| {f\left( {x - y}\right) \parallel g\left( y\right) }\right| \mathrm{d}y < \infty \) for each \( x \in {\mathbb{R}}^{n} \), an... | Proof. If \( f, g \) are as in the statement, then from (3.1.23) and Exercise 3.5 it follows that there exists \( M \in \mathbb{N} \) such that for every \( N \in \mathbb{N} \) there exists \( C \in \left( {0,\infty }\right) \) such that\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}\left| {f\left( {x - y}\right) }\right| \left| {... | No |
Lemma 3.20. If \( f \in \mathcal{L}\left( {\mathbb{R}}^{n}\right) \) and \( g \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), then for every \( \alpha \in {\mathbb{N}}_{0}^{n} \) the following integration by parts formula holds:\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}\left( {{\partial }^{\alpha }g}\right) \left( x\right) ... | Proof. Fix \( f \in \mathcal{L}\left( {\mathbb{R}}^{n}\right) \) and \( g \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . Since the classes \( \mathcal{L}\left( {\mathbb{R}}^{n}\right) \) and \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) are stable under differentiation, it suffices to show that for each \( j \in ... | Yes |
Example 3.22. Suppose \( \lambda \in \mathbb{C} \) satisfies \( \operatorname{Re}\left( \lambda \right) > 0 \) and if \( \lambda = r{\mathrm{e}}^{\mathrm{i}\theta } \) for \( r > 0 \) and \( - \pi /2 < \theta < \pi /2 \), set \( {\lambda }^{\frac{1}{2}} \mathrel{\text{:=}} \sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2} \) ... | Proof. Fix \( \lambda \in \mathbb{C} \) satisfying the given hypotheses. Then Exercise 3.3 ensures that \( f \) is a Schwartz function. Also, \( f\left( x\right) = {\mathrm{e}}^{-\lambda {x}_{1}^{2}} \otimes \cdots \otimes {\mathrm{e}}^{-\lambda {x}_{n}^{2}} \) for each point \( x = \left( {{x}_{1}\ldots ,{x}_{n}}\righ... | No |
Theorem 3.25. The mapping \( \mathcal{F} : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) is an algebraic and topologic isomorphism, that is, it is bijective, continuous, and its inverse is also continuous. In addition, its inverse is the operator \( {\mathcal{F}}^{-1... | Proof. The proof of the fact that the mapping \( {\mathcal{F}}^{-1} : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) defined as in (3.2.10) is well defined, linear, and continuous is similar to the proof of part \( \left( c\right) \) in Theorem 3.21. There remains to ... | No |
Proposition 3.28. Let \( f, g \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . Then the following identities hold:\n\n(a) \( {\int }_{{\mathbb{R}}^{n}}f\left( x\right) \widehat{g}\left( x\right) \mathrm{d}x = {\int }_{{\mathbb{R}}^{n}}\widehat{f}\left( \xi \right) g\left( \xi \right) \mathrm{d}\xi \) ; | Proof. The identity in (a) follows via a direct computation using Fubini's theorem. | No |
Example 4.4. Let \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) be such that there exist some \( m \in \lbrack 0,\infty ) \) and some \( R \in \left( {0,\infty }\right) \) with the property that\n\n\[ \n{\int }_{\left| x\right| \geq R}{\left| x\right| }^{-m}\left| {f\left( x\right) }\right| \mathrm{d}x < \inft... | To see that this is the case, pick \( N \in {\mathbb{N}}_{0} \) such that \( N \geq m \) and, for an arbitrary \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), estimate\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}\left| {f\varphi }\right| \mathrm{d}x \leq {\int }_{\left| x\right| < R}\left| {f\left( x\right) \varphi ... | Yes |
Let \( p \in \left\lbrack {1,\infty }\right\rbrack \) and \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) be arbitrary. We claim that \( {u}_{f} \) defined in (4.1.5) is a tempered distribution. | To prove this claim, since \( f \) is measurable, by Exercise 4.6, it suffices to show that \( f \) satisfies (4.1.8). If \( p = 1 \), then (4.1.8) holds for \( m = 0 \), while if \( p = \infty \) ,(4.1.8) holds for any \( m < - n \) . If \( p \in \left( {1,\infty }\right) \), by applying Hölder's inequality, we have\n... | Yes |
We claim that the distribution \( \ln \left| x\right| \) is, in fact, a tempered distribution. | Indeed, this follows from Exercise 4.6, since (4.1.8) holds for any \( m < - n \) (seen by using estimate (2.1.9) with \( 0 < \varepsilon < \min \{ n, - m - n\} \) ). | No |
Theorem 4.14. For each \( n, m \in \mathbb{N} \) the following statements are true:\n\n(a) \( {\mathcal{E}}^{\prime }\left( {\mathbb{R}}^{n}\right) \hookrightarrow {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \hookrightarrow {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), where all the embeddings are... | Proof. The statement in (a) is a consequence of parts (c) and (d) in Theorem 3.14 and duality (cf. Proposition 14.4). | Yes |
Proposition 4.20. The space \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is sequentially dense in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . In particular, the Schwartz class \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) is sequentially dense in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}... | Proof. Pick an arbitrary \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Fix a function \( \psi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) such that \( \psi \equiv 1 \) on \( B\left( {0,1}\right) \) and, for each \( j \in \mathbb{N} \), define \( {\psi }_{j}\left( x\right) \mathrel{\text... | No |
Proposition 4.21. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Then the mapping\n\n\[ \widehat{u} : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{C},\;\widehat{u}\left( \varphi \right) \mathrel{\text{:=}} \langle u,\widehat{\varphi }\rangle ,\;\forall \varphi \in \mathcal{S}\l... | Proof. This is an immediate consequence of the fact that \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), part \( \left( c\right) \) in Theorem 3.21, and the identity \( \widehat{u} = u \circ \mathcal{F} \), where \( \mathcal{F} \) is the Fourier transform on \( \mathcal{S}\left( {\mathbb{R}}^{n}\righ... | Yes |
Since \( \delta \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) we may write | \[ \langle \widehat{\delta },\varphi \rangle = \langle \delta ,\widehat{\varphi }\rangle = \widehat{\varphi }\left( 0\right) = {\int }_{{\mathbb{R}}^{n}}\varphi \left( x\right) \mathrm{d}x = \langle 1,\varphi \rangle ,\;\forall \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) ,\] thus \[ \widehat{\delta } = 1\;\te... | Yes |
We claim that \(\widehat{{\mathrm{e}}^{-\mathrm{i}a{\left| x\right| }^{2}}} = {\left( \frac{\pi }{\mathrm{i}a}\right) }^{\frac{n}{2}}{\mathrm{e}}^{\mathrm{i}\frac{{\left| \xi \right| }^{2}}{4a}}\;\text{ in }\;{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) . | To prove (4.2.22), fix \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) and starting with the definition of the Fourier transform on \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) write\n\n\[ \left\langle {\widehat{{\mathrm{e}}^{-\mathrm{i}a{\left| x\right| }^{2}}},\varphi }\right\rangle = \left... | Yes |
Lemma 4.28 (Riemann-Lebesgue Lemma). If \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \), then the tempered distribution \( {u}_{f} \) satisfies \( \widehat{{u}_{f}} \in {C}^{0}\left( {\mathbb{R}}^{n}\right) \) . | Proof This is a consequence of Exercise 4.27 and (3.1.3). | No |
Let \( a \in \left( {0,\infty }\right) \) . We are interested in computing the Fourier transform of the bounded function \( \frac{x}{{x}^{2} + {a}^{2}} \), viewed as a distribution in \( {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \) . | In this vein, consider the auxiliary function \( f\left( x\right) \mathrel{\text{:=}} \frac{1}{{x}^{2} + {a}^{2}} \) for \( x \in \mathbb{R} \), and recall from (4.2.21) that \( \widehat{f}\left( \xi \right) = \frac{\pi }{a}{\mathrm{e}}^{-a\left| \xi \right| } \) in \( {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \)... | Yes |
Proposition 4.32. For each \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) define the mapping\n\n\[ \n{u}^{ \vee } : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{C},\;{u}^{ \vee }\left( \varphi \right) \mathrel{\text{:=}} \left\langle {u,{\varphi }^{ \vee }}\right\rangle ,\;\forall \v... | Proof. Fix \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Then (4.2.33) is simply the composition of \( u \) with the mapping from Exercise 3.26. Since both are linear and continuous, it follows that \( {u}^{ \vee } \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Formula (4.2.34) the... | No |
Theorem 4.35. The following statements are true:\n\n(a) If \( a \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) and \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), then \( \widehat{u * a} = \widehat{au} \) in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), where \( \widehat{a} \) is vie... | Proof. By \( \left( d\right) \) in Theorem 4.19 we have \( u * a \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), hence \( \widehat{u * a} \) exists and belongs to \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Also, since \( \widehat{a} \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), by (... | Yes |
If \( a \in \left( {0,\infty }\right) \) then \( {\chi }_{\left\lbrack -a, a\right\rbrack } \), the characteristic function of the interval \( \left\lbrack {-a, a}\right\rbrack \), belongs to \( {\mathcal{E}}^{\prime }\left( \mathbb{R}\right) \) and by statement \( \left( b\right) \) in Theorem 4.35 we have | \[ \widehat{{\chi }_{\left\lbrack -a, a\right\rbrack }}\left( \xi \right) = {\int }_{-a}^{a}{\mathrm{e}}^{-\mathrm{i}{x\xi }}\mathrm{d}x = \left\{ \begin{array}{l} 2\frac{\sin \left( {a\xi }\right) }{\xi }\text{ for }\xi \in \mathbb{R} \smallsetminus \{ 0\} , \\ {2a}\;\text{ for }\xi = 0. \end{array}\right. \] | Yes |
Let \( a \in \mathbb{R} \) and consider the function \( f\left( x\right) \mathrel{\text{:=}} \sin \left( {ax}\right) \) for \( x \in \mathbb{R} \) . Then \( f \in {C}^{\infty }\left( \mathbb{R}\right) \cap {L}^{\infty }\left( \mathbb{R}\right) \), hence \( f \in {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \) . Supp... | More precisely, since \( {f}^{\prime \prime } + {a}^{2}f = 0 \) in \( \mathbb{R} \) , the same equation holds in \( {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \), thus by \( \left( a\right) \) and \( \left( b\right) \) in Theorem 4.26 we have \( \left( {{\xi }^{2} - {a}^{2}}\right) \widehat{f} = 0 \) in \( {\mathc... | Yes |
Let \( a, b \in \mathbb{R} \) . Then the function \( g\left( x\right) \mathrel{\text{:=}} \sin \left( {ax}\right) \sin \left( {bx}\right) \) for \( x \in \mathbb{R} \) satisfies \( g \in {L}^{\infty }\left( \mathbb{R}\right) \), thus \( g \in {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \) (cf. (4.1.9)). | Applying the Fourier transform to the identity in (4.2.48) and using (4.2.34) we obtain\n\n\[ \sin \left( {ax}\right) = \frac{\mathrm{i}}{2}\widehat{{\delta }_{a}} - \frac{\mathrm{i}}{2}\widehat{{\delta }_{-a}}\;\text{ in }{\mathcal{S}}^{\prime }\left( \mathbb{R}\right) . \]\n\n(4.2.49)\n\nAlso, making use of (4.2.49),... | Yes |
Proposition 4.43. Let \( A \in {\mathcal{M}}_{n \times n}\left( \mathbb{R}\right) \) be such that \( \det A \neq 0 \) . For each \( u \) in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) , define the mapping \( u \circ A : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{C} \) by setting\n\n\[... | Proof. This is an immediate consequence of (3.1.41). | No |
Proposition 4.45. Assume that \( A \in {\mathcal{M}}_{n \times n}\left( \mathbb{R}\right) \) is such that \( \det A \neq 0 \) . Then for each \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \)\n\n\[ \widehat{u \circ A} = {\left| \det A\right| }^{-1}\widehat{u} \circ {\left( {A}^{\top }\right) }^{-1}. \]\... | Proof. For each \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \), based on (4.2.2),(4.3.2), and (3.2.9), we may write\n\n\[ \langle \widehat{u \circ A},\varphi \rangle = \langle u \circ A,\widehat{\varphi }\rangle = {\left| \det A\right| }^{-1}\langle u,\widehat{\varphi } \circ {A}^{-1}\rangle \]\n\n\[ = \le... | Yes |
Proposition 4.49. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Then \( u \) is invariant under orthogonal transformations if and only if \( \widehat{u} \) is invariant under orthogonal transformations. | Proof. This is a direct consequence of (4.3.3) and the fact that any orthogonal matrix A satisfies (4.3.9). | No |
Proposition 4.59. Let \( k \in \mathbb{R} \) . If \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) is positive homogeneous of degree \( k \) , then \( \widehat{u} \) is positive homogeneous of degree \( - n - k \) . | Proof. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) be positive homogeneous of degree \( k \), and fix \( t > 0 \) . Then (4.3.8) and the assumption on \( u \) give\n\n\[ \n{\tau }_{t}\widehat{u} = {t}^{-n}\mathcal{F}\left( {{\tau }_{\frac{1}{t}}u}\right) = {t}^{-n}\mathcal{F}\left( {{t}^{-k}u}... | Yes |
Corollary 4.65. Assume that \( n \in \mathbb{N}, n \geq 2 \), and fix \( \lambda \in \lbrack 0, n - 1) \) . Then for each \( j \in \{ 1,\ldots, n\} \), we have\n\n\[ \mathcal{F}\left( \frac{{x}_{j}}{{\left| x\right| }^{\lambda + 2}}\right) = - \mathrm{i}{2}^{n - \lambda - 1}{\pi }^{\frac{n}{2}}\frac{\Gamma \left( \frac... | Proof. Fix an integer \( n \geq 2 \), and suppose first that \( \lambda \in \left( {0, n - 1}\right) \) . In this regime, both (4.3.25) and (4.1.40) hold. In concert with part (b) in Theorem 4.26, these give\n\n\[ - \lambda \mathcal{F}\left( \frac{{x}_{j}}{{\left| x\right| }^{\lambda + 2}}\right) = \mathcal{F}\left( {{... | Yes |
Proposition 4.66. Let \( \Theta \) be a function satisfying (4.4.1). Then the map P.V. \( \Theta \) considered in (4.4.2) is well defined and is a tempered distribution in \( {\mathbb{R}}^{n} \). In addition, \( {\left. \left( \text{ P.V. }\Theta \right) \right| }_{{\mathbb{R}}^{n}\smallsetminus \{ 0\} } = {\left. \The... | Proof of Proposition 4.66. Fix an arbitrary \( \psi \) satisfying (4.4.3). Then, making use of formula (14.9.9) and the properties of \( \Theta \) and \( \psi \), for each \( \varepsilon \in \left( {0,\infty }\right) \) we have\n\n\[ {\int }_{\left| x\right| \geq \varepsilon }\Theta \left( x\right) \psi \left( x\right)... | Yes |
If \( j \in \{ 1,\ldots, n\} \), the function \( \Theta \) defined by \( \Theta \left( x\right) \mathrel{\text{:=}} \frac{{x}_{j}}{{\left| x\right| }^{n + 1}} \) for each \( x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) satisfies (4.4.1). | By Proposition 4.66 we have P.V. \( \frac{{x}_{j}}{{\left| x\right| }^{n + 1}} \) belongs to \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) and part (2) in Remark 4.68 gives that for every \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \)\n\n\[ \left\langle {\text{ P.V. }\frac{{x}_{j}}{{\left| x\... | Yes |
Let \( \Phi \in {C}^{1}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) be positive homogeneous of degree \( 1 - n \) . Then for each \( j \in \{ 1,\ldots, n\} \) it follows that \( {\partial }_{j}\Phi \) satisfies the conditions in (4.4.1). Consequently, P.V. \( \left( {{\partial }_{j}\Phi }\right) \) is a we... | To see why this is true fix \( j \in \{ 1,\ldots, n\} \) and note that \( {\partial }_{j}\Phi \in {C}^{0}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) and \( {\partial }_{j}\Phi \) is positive homogeneous of degree \( - n \) (cf. Exercise 4.51). Moreover, using Exercise 4.52, then integrating by parts based... | No |
Proposition 4.73. For each \( k \in \{ 1,\ldots, n\} \) consider the function\n\n\[ \n{\Phi }_{k}\left( x\right) \mathrel{\text{:=}} \frac{{x}_{k}}{{\left| x\right| }^{n}},\;\forall x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} .\n\]\n\n(4.5.1)\n\nThen for each \( j \in \{ 1,\ldots, n\} \) the function \( {\partial }_{j... | Proof. Fix \( j, k \in \{ 1,\ldots, n\} \) . From Example 4.71 it is clear that the function \( {\partial }_{j}{\Phi }_{k} \) is as in (4.4.1). Moreover, Theorem 4.72 gives that\n\n\[ \n{\partial }_{j}{\Phi }_{k} = \left( {{\int }_{{S}^{n - 1}}{\omega }_{k}{\omega }_{j}\mathrm{\;d}\sigma \left( \omega \right) }\right) ... | Yes |
Theorem 4.74. Let \( \Theta \) be a function satisfying the conditions in (4.4.1). Then the function given by the formula\n\n\[ \n{m}_{\Theta }\left( \xi \right) \mathrel{\text{:=}} - {\int }_{{S}^{n - 1}}\Theta \left( \omega \right) \log \left( {\mathrm{i}\left( {\xi \cdot \omega }\right) }\right) \mathrm{d}\sigma \le... | Proof. First, we show that the integral in (4.5.6) is absolutely convergent for each vector \( \xi \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) . To see this, fix an arbitrary \( \xi \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) and observe that for each \( \omega \in {S}^{n - 1} \) we have\n\n\[ \n\log \left( {\mathr... | Yes |
Theorem 4.79. If \( \Phi \in {C}^{4}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) is odd and positive homogeneous of degree \( 1 - n \) , then\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ \pm }}}\Phi \left( {{x}^{\prime },\varepsilon }\right) = \pm \frac{\mathrm{i}}{2}\widehat{\Phi }\left( {{... | Proof of Theorem 4.79. Assume \( \Phi \in {C}^{4}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) is odd and positive homogeneous of degree \( 1 - n \), and let \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) . Then, for any fixed \( \varepsilon > 0 \), write\n\n\[ \mathop{\lim }\limits_{{t \ri... | No |
Corollary 4.81. Let the function \( \Phi \in {C}^{4}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) be odd and positive homogeneous of degree \( 1 - n \), and assume that \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) . Then for every \( {x}^{\prime } \in {\mathbb{R}}^{n - 1} \) one has\n\n\[... | Proof. Given any \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) and any \( {x}^{\prime } \in {\mathbb{R}}^{n - 1} \), write\n\n\[ \mathop{\lim }\limits_{{t \rightarrow {0}^{ \pm }}}{\int }_{{\mathbb{R}}^{n - 1}}\Phi \left( {{x}^{\prime } - {y}^{\prime }, t}\right) \varphi \left( {y}^{\prime }\right) \m... | Yes |
Corollary 4.83. Assume that the function \( \Phi \in {C}^{4}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) is odd and positive homogeneous of degree \( 1 - n \) . Then\n\n\[{\mathcal{F}}^{\prime }\left( {\text{ P.V. }\Phi \left( {\cdot ,0}\right) }\right) = \mp \frac{\mathrm{i}}{2}\widehat{\Phi }\left( {{0}^... | Proof. Formula (4.7.50) is a direct consequence of Theorem 4.79, the continuity of \( {\mathcal{F}}^{\prime } \) on \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n - 1}\right) \), and the fact that \( {\mathcal{F}}^{\prime }\left( {\delta \left( {x}^{\prime }\right) }\right) = 1 \) . | Yes |
Proposition 4.84. For each \( j \in \{ 1,\ldots, n\} \) we have (with \( \mathcal{F} \) denoting the Fourier transform in \( {\mathbb{R}}^{n} \) )\n\n\[ \mathcal{F}\left( {\text{ P.V. }\frac{{x}_{j}}{{\left| x\right| }^{n + 1}}}\right) = - \frac{\mathrm{i}{\omega }_{n}}{2}\frac{{\xi }_{j}}{\left| \xi \right| }\;\text{ ... | Proof. Fix some \( j \in \{ 1,\ldots, n\} \) and consider the function defined by\n\n\[ \Phi \left( {x, t}\right) \mathrel{\text{:=}} \frac{{x}_{j}}{{\left( {\left| x\right| }^{2} + {t}^{2}\right) }^{\frac{n + 1}{2}}},\;\forall \left( {x, t}\right) \in {\mathbb{R}}^{n + 1} \smallsetminus \{ 0\} . \]\n\n(4.7.52)\n\nNote... | Yes |
Corollary 4.86. Suppose that \( \Phi \in {C}^{4}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) is odd and positive homogeneous of degree \( 1 - n \) . Then in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n - 1}\right) \) , \[ \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ \pm }}}{\mathcal{F}}^{\prim... | Proof. This follows from Corollary 4.83, (4.5.11), and (4.5.6). | No |
Proposition 4.88. Given any \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) and any \( x = \left( {{x}^{\prime },{x}_{n}}\right) \in {\mathbb{R}}^{n} \) with \( {x}_{n} \neq 0 \) , define (where \( P \) denotes the harmonic Poisson kernel)\n\n\[ \n\left( {\mathcal{P}\varphi }\right) \left( x\right) \mat... | Proof. The function \( P : {\mathbb{R}}^{n} \smallsetminus \{ 0\} \rightarrow \mathbb{R} \) from (4.8.1) is \( {C}^{\infty } \), odd, and positive homogeneous of degree \( 1 - n \) . Moreover, Corollary 4.65 gives\n\n\[ \n\widehat{P}\left( \xi \right) = - 2\mathrm{i}\frac{{\xi }_{n}}{{\left| \xi \right| }^{2}}\;\text{ ... | Yes |
Proposition 4.94. Let \( \Theta \) be a function satisfying the conditions in (4.4.1) and consider the singular integral operator \( {T}_{\Theta } \) associated with \( \Theta \) as in (4.9.3). Then\n\n\[ \n{T}_{\Theta } : \mathcal{S}\left( {\mathbb{R}}^{n}\right) \rightarrow {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^... | Proof. All claims are consequences of Definition 4.93, Proposition 4.70, part (e) in Theorem 4.19, part (a) in Theorem 4.35, and (4.5.11). | No |
Corollary 4.98. Let \( j \in \{ 1,\ldots, n - 1\} \) and \( t \in \left( {0,\infty }\right) \), and recall the functions \( {p}_{t} \) and \( {\left( {q}_{j}\right) }_{t} \) from (4.8.8) and (4.8.23), respectively. Then\n\n\[ \n{\left( {q}_{j}\right) }_{t} = \frac{2}{{\omega }_{n - 1}}{R}_{j}{p}_{t}\;\text{ in }\;{L}^{... | Proof. By (3.2.31), it suffices to check identity (4.9.29) on the Fourier transform side. That the latter holds is an immediate consequence of part (2) in Proposition 4.92, (4.9.15), and part (2) in Proposition 4.90. | Yes |
Corollary 4.99. The Hilbert transform defined in (4.9.2) satisfies the following properties:\n\n(a) The operator\n\n\[ H : \mathcal{S}\left( \mathbb{R}\right) \rightarrow {\mathcal{S}}^{\prime }\left( \mathbb{R}\right) \;\text{ is well defined,}\]\n\n(4.9.30)\n\n\[ \text{linear, and sequentially continuous.}\]\n\n(b) F... | Proof. This corresponds to Theorem 4.97 in the case when \( n = 1 \) . | Yes |
Theorem 4.100. Assume that \( \Theta \) is a function satisfying the conditions in (4.4.1) and let \( {T}_{\Theta } \) be the singular integral operator associated with \( \Theta \) as in (4.9.3). Then the following statements are true.\n\n(a) The operator\n\n\[ \n{T}_{\Theta } : \mathcal{S}\left( {\mathbb{R}}^{n}\righ... | Proof. Parts part (a)-(b) are contained in Proposition 4.94, while part (c) follows from (4.9.6) and Theorem 4.74. Part \( \left( d\right) \) is justified by reasoning as in the proof of part \( \left( d\right) \) in Theorem 4.97, keeping in mind (4.5.8). As for part \( \left( e\right) \), in a first stage we note that... | Yes |
Theorem 4.104. Let \( \Phi \in {C}^{1}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) be a function that is positive homogeneous of degree \( 1 - n \) . Consider a measurable set \( \Omega \subseteq {\mathbb{R}}^{n} \) and assume that \( f \in {L}^{p}\left( \Omega \right) \) for some \( p \in \left( {1, n}\ri... | Proof. The main claim in the statement follows by applying Theorem 4.103 to the function\n\n\[ \n\widetilde{f} \mathrel{\text{:=}} \left\{ \begin{array}{l} f\text{ in }\Omega , \\ 0\text{ in }{\mathbb{R}}^{n} \smallsetminus \Omega , \end{array}\right.\n\]\n\n(4.10.21)\n\nupon observing that \( \widetilde{f} \in {L}^{p}... | Yes |
Theorem 4.105. Let \( \Phi \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) be a function that is positive homogeneous of degree \( m \in \mathbb{R} \) where \( m > - n \) . Define the generalized volume potential associated with \( \Phi \) by setting for each \( f \in {L}_{\text{comp }}^{\inf... | Proof. Fix \( f \in {L}_{\text{comp }}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and let \( K \mathrel{\text{:=}} \operatorname{supp}f \) which is a compact set in \( {\mathbb{R}}^{n} \) . By Exercise 4.51 and Exercise 4.53,\n\n\[ \text{for every}\alpha \in {\mathbb{N}}_{0}^{n}\text{we have that}{\partial }^{\alpha }\... | No |
Proposition 5.2. Suppose \( P\left( D\right) \) is a constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \) with the property that \[ P\left( \xi \right) \neq 0\;\text{ for every }\;\xi \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} . \] Then if \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\r... | Proof. If \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) is such that \( P\left( D\right) u = 0 \) in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), after taking the Fourier transform, we obtain \( P\left( \xi \right) \widehat{u} = 0 \) in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{... | No |
Lemma 5.3. Let \( P\left( x\right) \) be a polynomial in \( {\mathbb{R}}^{n} \) with the property that there exist \( N \in \) \( {\mathbb{N}}_{0} \) and \( C, R \in \lbrack 0,\infty ) \) such that\n\n\[ \left| {P\left( x\right) }\right| \leq C{\left| x\right| }^{N}\;\text{ whenever }\;\left| x\right| \geq R. \]\n\n(5.... | Proof. We begin by noting that the case \( n = 1 \) is easily handled. In the multidimensional setting, assume that \( P\left( x\right) = \mathop{\sum }\limits_{{\left| \alpha \right| \leq m}}{a}_{\alpha }{x}^{\alpha } \) is a polynomial in \( {\mathbb{R}}^{n} \) satisfying (5.1.7)\n\nfor some \( N \in {\mathbb{N}}_{0}... | Yes |
Theorem 5.4 (A general Liouville type theorem). Assume \( P\left( D\right) \) is a constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \) such that (5.1.6) holds. Also, suppose \( u \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) satisfies \( P\left( D\right) u = 0 \) in \( {\mathcal{S}}^{\prime... | Proof. Since (5.1.10) implies that the locally integrable function \( u \) belongs to \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) (cf. Example 4.4), Proposition 5.2 implies that \( u \) is a polynomial in \( {\mathbb{R}}^{n} \) . Moreover, Lemma 5.3 gives that the degree of this polynomial is at most \(... | Yes |
Lemma 5.9. Let \( m \in \mathbb{N} \) and let \( P\left( D\right) \) be a differential operator of order \( m \) as in (5.1.1). In addition, suppose that \( {P}_{m}\left( \eta \right) \neq 0 \) for some \( \eta \in {S}^{n - 1} \) . Then there exist \( \varepsilon > 0,{r}_{0},{r}_{1},\ldots ,{r}_{m} > 0 \) and closed se... | Proof. Choose \( {r}_{j} \mathrel{\text{:=}} \frac{j + 1}{m} \), for \( j \in \{ 0,1,\ldots, m\} \) and set \( \varepsilon \mathrel{\text{:=}} {\left( 2m\right) }^{-m}\left| {{P}_{m}\left( \eta \right) }\right| > 0 \) . Also, for each \( j \in \{ 1,\ldots, m\} \) set\n\n\[ \n{F}_{j} \mathrel{\text{:=}} \left\{ {\xi \in... | Yes |
Corollary 5.11. Let \( P\left( D\right) \) be a constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \) which is not identically zero. Suppose \( \Omega \) is a non-empty, open subset of \( {\mathbb{R}}^{n} \) and that \( f \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) is given. Then for every... | Proof. Let \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) be a fundamental solution of \( P\left( D\right) \), which is known to exist by Theorem 5.10. Also, fix \( \psi \in {C}_{0}^{\infty }\left( \Omega \right) \) such that \( \psi \equiv 1 \) on \( \omega \) . Then \( {\psi f} \in {\mathcal{E}}^{\... | Yes |
Let \( m \in \mathbb{N} \) and let \( P \mathrel{\text{:=}} {\left( \frac{\mathrm{d}}{\mathrm{d}x}\right) }^{m} + \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}{a}_{j}{\left( \frac{\mathrm{d}}{\mathrm{d}x}\right) }^{j} \) be a differential operator of order \( m \) in \( \mathbb{R} \) with constant coefficients. Suppose tha... | We claim that \( {u}_{ + } \) and \( {u}_{ - } \) are fundamental solutions for \( P \) in \( {\mathcal{D}}^{\prime }\left( \mathbb{R}\right) \) . To see that this is the case, express \( P \) as \( P\left( D\right) = {\mathrm{i}}^{m}{\left( \frac{1}{\mathrm{i}}\frac{\mathrm{d}}{\mathrm{d}x}\right) }^{m} + \mathop{\sum... | Yes |
Let \( c \in \mathbb{C} \) and consider the differential operator \( P \mathrel{\text{:=}} \frac{\mathrm{d}}{\mathrm{d}x} + c \) in \( \mathbb{R} \). It is easy to see that \( v\left( x\right) = {\mathrm{e}}^{-{cx}}, x \in \mathbb{R} \) is the solution (in the classical sense) of the initial value problem \( {Pv} = 0 \... | \[ P\left( {vH}\right) = {\left( vH\right) }^{\prime } + {cvH} = {v}^{\prime }H + v{H}^{\prime } + {cvH} = {v\delta } = \delta \;\text{ in }{\mathcal{D}}^{\prime }\left( \mathbb{R}\right) . \] | Yes |
Theorem 5.14. Let \( P\left( D\right) \) be a nonzero constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \) . Then there exists \( E \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) which is a fundamental solution for \( P\left( D\right) \) . In particular, the equality \( P\left( D\ri... | We prove this theorem by relying on a deep result due to L. Hörmander and S. Lojasiewicz. | No |
\[ \text{sing}\operatorname{supp}\delta = \{ 0\} \text{.} \] | To see why this is true, note that clearly sing supp \( \delta \subset \{ 0\} \) and sing supp \( \delta \neq \varnothing \) since, by Example 2.13, the distribution \( \delta \) is not of function type. | No |
Theorem 6.11. Let \( P\left( D\right) \) be a constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \). Then \( P\left( D\right) \) is hypoelliptic in \( {\mathbb{R}}^{n} \) if and only if there exists \( F \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) which is a parametrix of \( P\lef... | Proof. In one direction, if \( P\left( D\right) \) is hypoelliptic in \( {\mathbb{R}}^{n} \), then by Theorem 6.8 there exists a fundamental solution \( E \) of \( P\left( D\right) \) with sing \( \operatorname{supp}E = \{ 0\} \). The desired conclusion follows by taking \( F \mathrel{\text{:=}} E \). Conversely, suppo... | Yes |
Theorem 6.12. Let \( P\left( D\right) \) be a nonzero constant coefficient linear differential operator in \( {\mathbb{R}}^{n} \) . Then \( P\left( D\right) \) is hypoelliptic if and only if there exist constants \( C, R, c \in \) \( \left( {0,\infty }\right) \) with the property that\n\n\[ \left| \frac{{\partial }^{\b... | This characterization is not going to play a significant role for us here. For a proof, the interested reader is referred to [35, Theorem 11.1.3, p. 62]. | No |
Proposition 6.17. The poly-harmonic operator is hypoelliptic. In particular, the operators \( \Delta \) and \( {\Delta }^{2} \) are hypoelliptic. | Proof. This is a consequence of Theorem 6.15 and Remark 6.14. | No |
Corollary 6.18. Let \( P\left( D\right) \) be a linear, constant coefficient elliptic operator in \( {\mathbb{R}}^{n} \) and let \( \Omega \) be an open subset of \( {\mathbb{R}}^{n} \) . If \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) is such that for some open subset \( \omega \) of \( \Omega \) we have \... | Proof. From Theorem 6.15 and Remark 6.2 we know that the restriction of \( P\left( D\right) \) to \( \omega \) is a hypoelliptic operator. Since \( P\left( D\right) {\left. u\right| }_{\omega } \in {\mathcal{D}}^{\prime }\left( \omega \right) \) has an empty singular support, Proposition 6.6 gives that the singular sup... | Yes |
Theorem 6.19 (A general integral representation formula). Assume that the constant coefficient, linear, differential operator \( P\left( \partial \right) \) is as in (6.3.1) and is hypoelliptic in \( {\mathbb{R}}^{n} \) . Let \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) be a fundamental solution fo... | Proof. The fact that \( u \in {C}^{\infty }\left( \Omega \right) \) is a consequence of the hypoellipticity of the operator \( P\left( \partial \right) \) (cf. Definition 6.1 and Remark 6.2). As regards (6.3.2), pick an arbitrary \( {x}_{0} \in \Omega \), fix \( r \in \left( {0,\operatorname{dist}\left( {{x}_{0},\parti... | Yes |
Theorem 6.20 (Interior estimates). Let \( P\left( \partial \right) \) be a constant coefficient, linear, differential operator, of order \( m \in {\mathbb{N}}_{0} \), which is hypoelliptic in \( {\mathbb{R}}^{n} \) . Also suppose \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) is a fundamental solutio... | Proof. Pick a function \( \phi \in {C}_{0}^{\infty }\left( {B\left( {0,1}\right) }\right) \) such that \( \phi \equiv 1 \) near \( \overline{B\left( {0,3/4}\right) } \), and note that if we set\n\n\[ \psi \left( x\right) \mathrel{\text{:=}} \phi \left( {\left( {x - {x}_{0}}\right) /r}\right) ,\;\forall x \in {\mathbb{R... | Yes |
Lemma 6.21. Let \( \mathcal{A} \subset {C}^{\infty }\left( \Omega \right) \) be a family of functions satisfying the following two conditions:\n\n(1) \( {\partial }^{\alpha }u \in \mathcal{A} \) for every \( u \in \mathcal{A} \) and every \( \alpha \in {\mathbb{N}}_{0}^{n} \) ;\n\n(2) there exists \( C \in \left( {0,\i... | Proof. In a first stage, we propose to prove by induction over \( k \) that, given any \( u \in \mathcal{A} \), for every \( x \in \Omega \), every \( r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \), and every \( k \in \mathbb{N} \) we have (with \( C \) as in (6.3.13))\n\n\[ \left| {{... | Yes |
Lemma 6.24. Suppose \( u \in {C}^{\infty }\left( \Omega \right) \) is a function with the property that for each \( x \in \Omega \) there exist \( r = r\left( x\right) \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right), M = M\left( x\right) \in \left( {0,\infty }\right) \), and \( C = C\left( ... | Proof. Fix \( x \in \Omega \) and let \( r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \) be such that (6.3.23) holds for every \( k \in \mathbb{N} \) . Write Taylor’s formula (14.2.10) for \( u \) at \( x \) to obtain that for each \( N \in \mathbb{N} \) and each \( y \in B\left( {x, r... | Yes |
Theorem 6.25 (Unique continuation). Suppose \( \Omega \subseteq {\mathbb{R}}^{n} \) is an open and connected set. Then any real-analytic function \( u \) in \( \Omega \) with the property that there exists \( {x}_{0} \in \Omega \) such that \( {\partial }^{\alpha }u\left( {x}_{0}\right) = 0 \) for all \( \alpha \in {\m... | Proof. Suppose \( u \) satisfies the hypotheses of the theorem and define the set \[ U \mathrel{\text{:=}} \left\{ {x \in \Omega : {\partial }^{\alpha }u\left( x\right) = 0\text{ for all }\alpha \in {\mathbb{N}}_{0}^{n}}\right\} . \] Since \( {x}_{0} \in U \) we have \( U \neq \varnothing \) . Also, \( U \) is relative... | Yes |
Theorem 6.26. Let \( P\left( \partial \right) \) be a constant coefficient, linear, differential operator in \( {\mathbb{R}}^{n} \), which is homogeneous of degree \( m \in {\mathbb{N}}_{0} \) . Assume that \( P\left( \partial \right) \) has a fundamental solution \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}... | Proof. The fact that \( P\left( \partial \right) \) has a fundamental solution \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) with the property that \( E \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) implies sing supp \( E = \{ 0\} \) ; hence, \( P\left( \partial \right) ... | Yes |
Theorem 7.2. The function \( E \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) defined as\n\n\[ E\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} \frac{-1}{\left( {n - 2}\right) {\omega }_{n - 1}}\frac{1}{{\left| x\right| }^{n - 2}} & \text{ if }x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\}, n \geq 3, \\... | Remark 7.4. Via a direct computation, one may check that \( E \) as in (7.1.12) defines a fundamental solution for the Laplacian. We outline the computation for \( n \geq 3 \) and leave the cases \( n = 2 \) and \( n = 1 \) as an exercise.\n\nFirst, observe that \( E \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetmi... | No |
Proposition 7.5. Let \( E \) be the fundamental solution for \( \Delta \) from (7.1.12) and recall the Riesz transforms in \( {\mathbb{R}}^{n} \) (cf. Theorem 4.97). Then for every function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) and each \( j, k \in \{ 1,\ldots, n\} \) we have\n\n\[ \n{R}_{j}\left(... | Proof. Fix \( j, k \in \{ 1,\ldots, n\} \) along with some \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . Since the Fourier transform is an isomorphism of \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), it suffices to show that (7.1.18) holds on the Fourier transform side. With this in mind,... | Yes |
Proposition 7.6. Let \( E \) be the fundamental solution for \( \Delta \) as in (7.1.12) and recall the Riesz transforms in \( {\mathbb{R}}^{n} \) (cf. Theorem 4.97). Also, fix \( p \in \left( {1,\infty }\right) \) . Then for every \( j, k \in \{ 1,\ldots, n\} \) and \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) we... | Proof. All claims are consequences of Proposition 7.5, Theorem 4.101, and the density of \( \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) in \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \) . | No |
Let \( E \) be the fundamental solution for \( \Delta \) from (7.1.12). Suppose \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Then the function \[ u\left( x\right) \mathrel{\text{:=}} {\int }_{{\mathbb{R}}^{n}}E\left( {x - y}\right) f\left( y\right) \mathrm{d}y,\;\forall x \in {\mathbb{R}}^{n}, \] satis... | Proof. Fix \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and let \( E \) be as in (7.1.12). Define \( u \mathrel{\text{:=}} E * f \) in \( {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . Then Remark 5.6 implies that \( u \) is a solution of the equation \( {\Delta u} = f \) in \( {\mathcal{D}}^... | Yes |
Theorem 7.10. For an open set \( \Omega \subseteq {\mathbb{R}}^{n} \) the following are equivalent:\n\n(i) \( u \) is harmonic in \( \Omega \) ;\n\n(ii) \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( {\Delta u} = 0 \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) ;\n\n(iii) \( u \in {C}^{\infty... | Proof. This is a direct consequence of Theorem 6.8 and Theorem 7.2 (or, alternatively, of Remark 6.14 and Corollary 6.18). | No |
Theorem 7.11. Let \( \Omega \) be an open set in \( {\mathbb{R}}^{n} \) and \( u \) be a harmonic function in \( \Omega \) . Then for every \( x \in \Omega \) and every \( r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \) we have\n\n\[ u\left( x\right) = {\int }_{B\left( {x, r}\right) }u... | Proof. From Theorem 7.10 we know that \( u \in {C}^{\infty }\left( \Omega \right) \) and \( {\Delta u} = 0 \) in a pointwise sense in \( \Omega \) . Fix \( x \in \Omega \) and define the function\n\n\[ \phi : \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \rightarrow \mathbb{R},\]\n\n\[ \phi \... | Yes |
Theorem 7.12 (Interior estimates for the Laplacian). Suppose \( u \) is harmonic in \( \Omega \) . Then for each \( x \in \Omega \), each \( r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \), and each \( k \in \mathbb{N} \), we have\n\n\[ \mathop{\max }\limits_{{y \in \overline{B\left( {... | Proof. Let \( j \in \{ 1,\ldots, n\} \) be arbitrary and note that, by Theorem 7.10, \( u \in {C}^{\infty }\left( \Omega \right) \) and hence, \( {\partial }_{j}u \) is harmonic in \( \Omega \) . Thus, by (7.2.9) and the integration by parts formula (14.8.4), for each \( x \in \Omega \) and each \( r \in \left( {0,\ope... | Yes |
Theorem 7.13. Any harmonic function in \( \Omega \) is real analytic in \( \Omega \) . | Proof. This is an immediate consequence of Theorem 7.12 and Lemma 6.24 (or, alternatively, Theorem 6.26). | No |
Corollary 7.14. Suppose \( u \) is harmonic in an open, connected set \( \Omega \subseteq {\mathbb{R}}^{n} \) with the property that there exists \( {x}_{0} \in \Omega \) such that \( {\partial }^{\alpha }u\left( {x}_{0}\right) = 0 \) for all \( \alpha \in {\mathbb{N}}_{0}^{n} \) . Then \( u \) vanishes identically in ... | Proof. This is an immediate consequence of Theorem 6.25 and Theorem 7.13. | No |
Theorem 7.15 (Liouville’s Theorem for the Laplacian). If \( u \) is a bounded harmonic function in \( {\mathbb{R}}^{n} \) then there exists \( c \in \mathbb{C} \) such that \( u = c \) in \( {\mathbb{R}}^{n} \) . | Proof. This may be justified in several ways. For example, it suffices to note that this is a particular case of Theorem 5.4. Another proof may be given based on interior estimates. Specifically, let \( j \in \{ 1,\ldots, n\} \) and using (7.2.14), for each \( x \in {\mathbb{R}}^{n} \), we may write\n\n\[ \mathop{\lim ... | Yes |
Theorem 7.21. The function \( E \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) defined as\n\n\[ E\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{lll} \frac{1}{2\left( {n - 2}\right) \left( {n - 4}\right) {\omega }_{n - 1}}{\left| x\right| }^{4 - n} & \text{ if }x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\}... | Proof. It is clear from (7.3.8) that \( E \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \cap {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) . The fact that \( {\Delta }^{2}E = \delta \) in \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) has been checked in (7.3.4),(7.3.7), and Exercise 5.16. Fin... | No |
Theorem 7.23. For an open set \( \Omega \subseteq {\mathbb{R}}^{n} \) the following are equivalent:\n\n(i) \( u \) is biharmonic in \( \Omega \) ;\n\n(ii) \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( {\Delta }^{2}u = 0 \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) ;\n\n(iii) \( u \in {C}^{... | Proof. This is a direct consequence of Theorem 6.8 and Theorem 7.21 (or, alternatively, of Remark 6.14 and Corollary 6.18). | No |
Theorem 7.24. Let \( \Omega \) be an open set in \( {\mathbb{R}}^{n} \) and let \( u \) be a biharmonic function in \( \Omega \) . Then for every \( x \in \Omega \) and every \( r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \) the following formulas hold:\n\n\[ u\left( x\right) = {\int ... | Proof. Fix \( x \in \Omega \) and recall the function \( \phi \) defined in (7.2.10). Then (7.2.11) holds. Since \( \Delta \left( {\Delta u}\right) = 0 \) in \( \Omega \) we have that \( {\Delta u} \) is harmonic in \( \Omega \) and we can apply (7.2.9) to obtain that \( {\phi }^{\prime }\left( r\right) = \frac{r}{n}{\... | Yes |
Theorem 7.25 (Liouville’s Theorem for \( {\Delta }^{2} \) ). Any bounded biharmonic function in \( {\mathbb{R}}^{n} \) is constant. | Proof. One way to justify this result is by observing that it is a particular case of Theorem 5.4. Another proof based on interior estimates goes as follows. Let \( u \) be a bounded biharmonic function in \( {\mathbb{R}}^{n} \) . Formula (7.4.2) in the current setting gives that\n\n\[ \n{\Delta u}\left( x\right) = \fr... | Yes |
Theorem 7.26. Suppose \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) satisfies \( {\Delta }^{2}u = 0 \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) . Then \( u \) is real analytic in \( \Omega \) and there exists a dimensional constant \( C \in \left( {0,\infty }\right) \) such that\n\n\[ \mathop{\m... | Proof. In the case when \( n = 1 \) or \( n \geq 3 \), all claims are direct consequences of Theorem 6.26 and Theorem 7.21. To treat the case \( n = 2 \) we shall introduce a \ | No |
Theorem 7.27. Assume \( n \in \mathbb{N} \) satisfies \( n \geq 3 \) and \( n \neq 4 \) . Then for each function \( f \in {L}_{\text{comp }}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and each \( c \in \mathbb{C} \) the Poisson problem for the bi-Laplacian in \( {\mathbb{R}}^{n} \) , \[ \left\{ \begin{array}{l} u \in {... | Proof. That the function \( u \) defined as in (7.4.9) is of class \( {C}^{3}\left( {\mathbb{R}}^{n}\right) \) and formula (7.4.10) holds for each \( \alpha \in {\mathbb{N}}_{0}^{n} \) with \( \left| \alpha \right| \leq 3 \) can be established much as in the proof of Proposition 7.8 keeping in mind that \( {\partial }^... | Yes |
Theorem 7.28. Let \( m, n \in \mathbb{N} \) and consider the function \[ {F}_{m, n}\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} \frac{{\left( -1\right) }^{m}\Gamma \left( {n/2 - m}\right) }{{\pi }^{n/2}{4}^{m}\left( {m - 1}\right) !}{\left| x\right| }^{{2m} - n} & \text{ if }n > {2m}\text{ or }n\text{ ... | Proof. The case \( n = 1 \) is immediate from Exercise 5.16. Assume in what follows that \( n \geq 2 \) . Since \( n - {2m} < n \), we clearly have \( {\left| x\right| }^{{2m} - n} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) and furthermore, by Exercise 4.5, that \( {\left| x\right| }^{{2m} - n} \in {\mathcal{S}... | No |
Theorem 7.30. Whenever \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) satisfies \( {\Delta }^{m}u = 0 \) in \( {\mathcal{D}}^{\prime }\left( \Omega \right) \), it follows that \( u \) is real analytic in \( \Omega \) and there exists a constant \( C = C\left( {n, m}\right) \in \left( {0,\infty }\right) \) wit... | Proof. In the case when either \( n \) is odd, or \( n \geq {2m} \), all claims are consequences of Theorem 6.26 and Theorem 7.28. The remaining cases may be reduced to the ones just treated by introducing \ | No |
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