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Lemma 26.7.8 Suppose \( \mu \) is a Borel measure on \( {\mathbb{R}}^{n} \) having values in \( \lbrack 0,\infty ) \) . Then there exists a Radon measure, \( {\mu }_{1} \) such that \( {\mu }_{1} = \mu \) on all Borel sets. | Proof: By assumption, \( \mu \left( {\mathbb{R}}^{n}\right) < \infty \) and so it is possible to define a positive linear functional, \( L \) on \( {C}_{c}\left( {\mathbb{R}}^{n}\right) \) by\n\n\[ \n{Lf} \equiv \int {fd\mu } \n\]\n\nBy the Riesz representation theorem for positive linear functionals of this sort, ther... | Yes |
Corollary 26.7.9 Suppose \( \mu \) is a complex Borel measure defined on \( {\mathbb{R}}^{n} \) for which there exists a \( \mu \) measurable set, \( N \) such that for all Borel sets, \( E,\mu \left( E\right) = \mu \left( {E \cap N}\right) \) where \( \bar{m}\left( N\right) = 0 \) . Then\n\n\[ \frac{d\mu }{dm}\left( \... | Proof: Each of \( \operatorname{Re}{\mu }^{ + },\operatorname{Re}{\mu }^{ - },\operatorname{Im}{\mu }^{ + } \), and \( \operatorname{Im}{\mu }^{ - } \) are real measures having values in \( \lbrack 0,\infty ) \) and so by Lemma 26.7.8 each is a Radon measure having the same property that \( \mu \) has in terms of being... | Yes |
Lemma 27.1.2 Let \( \phi : \mathbb{R} \rightarrow \mathbb{R} \) be a convex function. Then \( \phi \) is Lipschitz continuous on \( \left\lbrack {a, b}\right\rbrack \) . | Proof: Since it is convex, the difference quotients,\n\n\[ \frac{\phi \left( t\right) - \phi \left( a\right) }{t - a} \]\n\nare increasing because by convexity, if \( a < t < x \)\n\n\[ \frac{t - a}{x - a}\phi \left( x\right) + \left( {1 - \frac{t - a}{x - a}}\right) \phi \left( a\right) \geq \phi \left( t\right) \]\n\... | Yes |
Proposition 27.1.11 \( {E}_{A}\left( \Omega \right) \) is the maximal linear subspace of \( {K}_{A}\left( \Omega \right) \) . | Proof: Let \( M \) be a subspace of \( {K}_{A}\left( \Omega \right) \) . Is \( M \subseteq {E}_{A}\left( \Omega \right) \) ? For \( f \in M, f/\varepsilon \in \) \( {K}_{A}\left( \Omega \right) \) for all \( \varepsilon > 0 \) because of the fact that \( M \) is a subspace and \( f \in M \) . Thus \( A\left( {\left| f\... | Yes |
Proposition 27.1.12 \( {L}_{B}\left( \Omega \right) \hookrightarrow {L}_{A}\left( \Omega \right) \) if either\n\n\[ B\left( t\right) \geq A\left( t\right) \text{for all}t \geq 0 \]\n\n\( \left( {27.1.12}\right) \)\n\nor if\n\n\[ B\left( t\right) \geq A\left( t\right) \text{ for all }t > M \]\n\n\( \left( {27.1.13}\righ... | Proof: Let \( f \in {L}_{B}\left( \Omega \right) \) and let\n\n\[ {\int }_{\Omega }B\left( \frac{\left| f\right| }{t}\right) {d\mu } \leq 1 \]\n\nThen if 27.1.12 holds, it follows\n\n\[ {\int }_{\Omega }A\left( \frac{\left| f\right| }{t}\right) {d\mu } \leq 1 \]\n\nThus if \( t \geq \parallel f{\parallel }_{B} \) then ... | Yes |
Corollary 27.1.13 Suppose there exists \( C > 0 \), a constant such that either\n\n\[ \n{CB}\left( t\right) \geq A\left( t\right) \n\]\n\nfor all \( t \geq 0 \) or\n\n\[ \n{CB}\left( t\right) \geq A\left( t\right) \n\]\n\nfor all \( t > M \) and \( \mu \left( \Omega \right) < \infty \) . Then\n\n\[ \n{L}_{B}\left( \Ome... | Proof: If \( f \in {L}_{B}\left( \Omega \right) \) then \( f = {\lambda u} \) where \( u \in {K}_{B}\left( \Omega \right) = {K}_{CB}\left( \Omega \right) \) . Hence \( {L}_{CB}\left( \Omega \right) = {L}_{B}\left( \Omega \right) \) and the two norms on \( {L}_{B}\left( \Omega \right) \) ,\n\n\[ \n\parallel {\parallel }... | Yes |
Theorem 27.1.15 Suppose \( \mu \left( \Omega \right) < \infty \) and \( A \) increases essentially more slowly than B. Then\n\n\[ \n{L}_{B}\left( \Omega \right) \hookrightarrow {E}_{A}\left( \Omega \right) \n\] | Proof: Let \( f \in {L}_{B}\left( \Omega \right) \) . Then there exists \( \lambda > 0 \) such that\n\n\[ \n{\int }_{\Omega }B\left( \frac{\left| f\right| }{\lambda }\right) {d\mu } \leq 1 \n\]\n\nLet \( r \) be such that for \( t \geq r \) ,\n\n\[ \nA\left( {\left| \lambda \right| t}\right) \leq B\left( t\right) \n\]\... | Yes |
Theorem 27.2.2 Suppose \( \mu \left( \Omega \right) < \infty \) and suppose \( L \in {E}_{A}{\left( \Omega \right) }^{\prime } \) . Then the map \( v \rightarrow {L}_{v} \) from \( {L}_{\widetilde{A}}\left( \Omega \right) \) to \( {E}_{A}{\left( \Omega \right) }^{\prime } \) is one to one continuous, linear, and onto. ... | Proof: It is obvious this map is linear. From Proposition 27.2.1 it is continuous and one to one. It remains only to verify that it is onto. Let \( L \in {E}_{A}{\left( \Omega \right) }^{\prime } \) and define a complex valued function, \( \lambda \), mapping the measurable sets to \( \mathbb{C} \) as follows.\n\n\[ \l... | Yes |
Lemma 28.1.2 \( {\mathcal{H}}^{s} \) and \( {\mathcal{H}}_{\delta }^{s} \) are outer measures. | Proof: It is clear that \( {\mathcal{H}}^{s}\left( \varnothing \right) = 0 \) and if \( A \subseteq B \), then \( {\mathcal{H}}^{s}\left( A\right) \leq {\mathcal{H}}^{s}\left( B\right) \) with similar assertions valid for \( {\mathcal{H}}_{\delta }^{s} \) . Suppose \( E = { \cup }_{i = 1}^{\infty }{E}_{i} \) and \( {\m... | Yes |
If \( {m}_{n}\left( S\right) = 0 \) then \( {\mathcal{H}}^{n}\left( S\right) = {\mathcal{H}}_{\delta }^{n}\left( S\right) = 0 \) . | Suppose first \( {m}_{n}\left( S\right) = 0 \) . Without loss of generality, \( S \) is bounded. Then by outer regularity, there exists a bounded open \( V \) containing \( S \) and \( {m}_{n}\left( V\right) < \) \( \varepsilon \) . For each \( \mathbf{x} \in S \), there exists a ball \( {B}_{\mathbf{x}} \) such that \... | Yes |
Lemma 28.3.1 Let \( f : {\mathbb{R}}^{n - 1} \rightarrow \lbrack 0,\infty ) \) be Borel measurable and let\n\n\[ S = \{ \left( {\mathbf{x}, y}\right) : \left| y\right| < f\left( \mathbf{x}\right) \} .\n\]\n\nThen \( S \) is a Borel set in \( {\mathbb{R}}^{n} \) . | Proof: Set \( {s}_{k} \) be an increasing sequence of Borel measurable functions converging pointwise to \( f \) .\n\n\[ {s}_{k}\left( \mathbf{x}\right) = \mathop{\sum }\limits_{{m = 1}}^{{N}_{k}}{c}_{m}^{k}{\mathcal{X}}_{{E}_{m}^{k}}\left( \mathbf{x}\right) .\n\]\nLet\n\n\[ {S}_{k} = { \cup }_{m = 1}^{{N}_{k}}{E}_{m}^... | Yes |
Lemma 28.3.2 Let \( \bar{A} \subseteq {\mathbb{R}}^{2n} \) be a Borelisét. \( {}^{\cdots } \) Then \( {}^{\prime }{P}_{i}\mathbf{x} \rightarrow m\left( {A}_{{P}_{i}\mathbf{x}}\right) \) is a Borel measurable function defined on \( {P}_{i}\left( {\mathbb{R}}^{n}\right) \) . | Proof: Let \( \mathcal{K} \) be the \( \pi \) system consisting of sets of the form \( \mathop{\prod }\limits_{{j = 1}}^{n}{A}_{j} \) where \( {A}_{i} \) is Borel. Also let \( \mathcal{G} \) denote those Borel sets of \( {\mathbb{R}}^{n} \) such that if \( A \in \mathcal{G} \) then\n\n\[ \n{P}_{i}\mathbf{x} \rightarrow... | Yes |
Lemma 28.3.4 Suppose \( A \) is a Borel set in \( {\mathbb{R}}^{n} \) such that \( {P}_{j}\mathbf{x} + {\mathbf{e}}_{j}{x}_{j} \in A \) if and only if \( {P}_{j}\mathbf{x} + \left( {-{x}_{j}}\right) {\mathbf{e}}_{j} \in A \) . Then if \( i \neq j,{P}_{j}\mathbf{x} + {\mathbf{e}}_{j}{x}_{j} \in S\left( {A,{\mathbf{e}}_{... | Proof: By definition,\n\n\[ \n{P}_{j}\mathbf{x} + {\mathbf{e}}_{j}{x}_{j} \in S\left( {A,{\mathbf{e}}_{i}}\right) \n\]\n\nif and only if\n\n\[ \n\left| {x}_{i}\right| < {2}^{-1}m\left( {A}_{{P}_{i}\left( {{P}_{j}\mathbf{x} + {\mathbf{e}}_{j}{x}_{j}}\right) }\right) . \n\]\n\nNow\n\n\[ \n{x}_{i} \in {A}_{{P}_{i}\left( {... | Yes |
Theorem 28.3.5 Let \( A \) be any Lebesgue measurable set in \( {\mathbb{R}}^{n} \). Then\n\n\[ \n{m}_{n}\left( A\right) \leq \alpha \left( n\right) {\left( r\left( A\right) \right) }^{n}.\n\] | Proof: Suppose first that \( A \) is Borel. Let \( {A}_{1} = S\left( {A,{\mathbf{e}}_{1}}\right) \) and let \( {A}_{k} = \) \( S\left( {{A}_{k - 1},{\mathbf{e}}_{k}}\right) \). Then by the preceding lemmas, \( {A}_{n} \) is a Borel set, \( \operatorname{diam}\left( {A}_{n}\right) \leq \) \( \operatorname{diam}\left( A\... | Yes |
Lemma 28.4.1 The following identities hold.\n\n\[ \n{p\Gamma }\left( p\right) = \Gamma \left( {p + 1}\right) \n\] | Proof: Using integration by parts,\n\n\[ \n\Gamma \left( {p + 1}\right) = {\int }_{0}^{\infty }{e}^{-t}{t}^{p}{dt} = {\left. -{e}^{-t}{t}^{p}\right| }_{0}^{\infty } + p{\int }_{0}^{\infty }{e}^{-t}{t}^{p - 1}{dt} \n\]\n\n\[ \n= {p\Gamma }\left( p\right) \n\] | Yes |
Theorem 28.4.2 \( \alpha \left( n\right) = {\pi }^{n/2}{\left( \Gamma \left( n/2 + 1\right) \right) }^{-1} \) where \( \Gamma \left( s\right) \) is the gamma function | Proof: First let \( n = 1 \) .\n\n\[ \Gamma \left( \frac{3}{2}\right) = \frac{1}{2}\Gamma \left( \frac{1}{2}\right) = \frac{\sqrt{\pi }}{2}. \]\n\nThus\n\n\[ {\pi }^{1/2}{\left( \Gamma \left( 1/2 + 1\right) \right) }^{-1} = \frac{2}{\sqrt{\pi }}\sqrt{\pi } = 2 = \alpha \left( 1\right) . \]\n\nand this shows the theorem... | Yes |
Lemma 28.4.4 Let \( R \in \mathcal{L}\left( {{\mathbb{R}}^{n},{\mathbb{R}}^{m}}\right), n \leq m \), and \( {R}^{ * }R = I \) . Then if \( A \subseteq {\mathbb{R}}^{n} \) , \[ {\mathcal{H}}^{n}\left( {RA}\right) = {\mathcal{H}}^{n}\left( A\right) \] In fact, if \( P : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) sa... | Proof: Note that \[ {\left| R\left( \mathbf{x} - \mathbf{y}\right) \right| }^{2} = \left( {R\left( {\mathbf{x} - \mathbf{y}}\right), R\left( {\mathbf{x} - \mathbf{y}}\right) }\right) = \left( {{R}^{ * }R\left( {\mathbf{x} - \mathbf{y}}\right) ,\mathbf{x} - \mathbf{y}}\right) = {\left| \mathbf{x} - \mathbf{y}\right| }^{... | Yes |
Lemma 28.4.5 Let \( F \in \mathcal{L}\left( {{\mathbb{R}}^{n},{\mathbb{R}}^{m}}\right), n \leq m \), and let \( F = {RU} \) where \( R \) and \( U \) are described in Theorem 5.9.6 on Page 97. Then if \( A \subseteq {\mathbb{R}}^{n} \) is Lebesgue measurable, | Proof: Using Theorem 13.5.7 on Page 373 and Theorem 28.2.3,\n\n\[ \n{\mathcal{H}}^{n}\left( {FA}\right) = {\mathcal{H}}^{n}\left( {RUA}\right) \n\]\n\n\[ \n= {\mathcal{H}}^{n}\left( {UA}\right) = {m}_{n}\left( {UA}\right) = \det \left( U\right) {m}_{n}\left( A\right) .\blacksquare \n\] | Yes |
Theorem 29.0.1 If \( \\mathbf{h} : \\Omega \\rightarrow {\\mathbb{R}}^{m} \) is Lipschitz, then there exists \( \\overline{\\mathbf{h}} : {\\mathbb{R}}^{p} \\rightarrow {\\mathbb{R}}^{m} \) which extends \( \\mathbf{h} \) and is also Lipschitz. | Proof: It suffices to assume \( m = 1 \) because if this is shown, it may be applied to the components of \( \\mathbf{h} \) to get the desired result. Suppose\n\n\[ \n\\left| {h\\left( \\mathbf{x}\\right) - h\\left( \\mathbf{y}\\right) }\\right| \\leq K\\left| {\\mathbf{x} - \\mathbf{y}}\\right| .\n\]\n\n\( \\left( {29... | Yes |
Lemma 29.1.1 If \( \mathbf{h} \) is Lipschitz with Lipschitz constant \( K \) then\n\n\[{\mathcal{H}}^{n}\left( {\mathbf{h}\left( B\right) }\right) \leq {K}^{n}{\mathcal{H}}^{n}\left( B\right)\]\n\nAlso, if \( T \) is a set in \( {\mathbb{R}}^{n},{m}_{n}\left( T\right) = 0 \), then \( {\mathcal{H}}^{n}\left( {\mathbf{h... | Proof: Let \( {\left\{ {C}_{i}\right\} }_{i = 1}^{\infty } \) cover \( B \) with each having diameter less than \( \delta \) and let this cover be such that\n\n\[ \mathop{\sum }\limits_{i}\beta \left( n\right) \frac{1}{2}\operatorname{diam}{\left( {C}_{i}\right) }^{n} < {\mathcal{H}}_{\delta }^{n}\left( B\right) + \var... | Yes |
If \( S \) is a Lebesgue measurable set and \( \mathbf{h} \) is Lipschitz then \( \mathbf{h}\left( S\right) \) is \( {\mathcal{H}}^{n} \) measurable. Also, if \( \mathbf{h} \) is Lipschitz with constant \( K \) , \[ {\mathcal{H}}^{n}\left( {\mathbf{h}\left( S\right) }\right) \leq {K}^{n}{m}_{n}\left( S\right) \] | The estimate follows from Lemma 29.1.1 and the observation that, as shown before, Theorem 28.2.3, if \( S \) is Lebesgue measurable in \( {\mathbb{R}}^{n} \), then \( {\mathcal{H}}^{n}\left( S\right) = \) \( {m}_{n}\left( S\right) \) . The estimate also shows that \( \mathbf{h} \) maps sets of Lebesgue measure zero to ... | Yes |
Lemma 29.1.3 In this situation where \( {R}^{ * }R = I,\left| {{R}^{ * }\mathbf{u}}\right| \leq \left| \mathbf{u}\right| \) . | Proof: First note that\n\n\[ \left( {\mathbf{u} - R{R}^{ * }\mathbf{u}, R{R}^{ * }\mathbf{u}}\right) = \left( {\mathbf{u}, R{R}^{ * }\mathbf{u}}\right) - {\left| R{R}^{ * }\mathbf{u}\right| }^{2} \]\n\n\[ = {\left| {R}^{ * }\mathbf{u}\right| }^{2} - {\left| {R}^{ * }\mathbf{u}\right| }^{2} = 0 \]\n\nand so\n\n\[ {\left... | Yes |
Lemma 29.2.1 Suppose \( S, T \) are linear defined on a finite dimensional normed linear space, \( {S}^{-1} \) exists and let \( \delta \in \left( {0,1}\right) \) . Then whenever \( \parallel S - T\parallel \) is small enough, it follows that\n\n\[ \frac{\left| T\mathbf{v}\right| }{\left| S\mathbf{v}\right| } \in \left... | Proof: Say \( {S}^{-1} \) exists. Then \( \mathbf{v} \rightarrow \left| {S\mathbf{v}}\right| \) is a norm. Then by equivalence of norms, Theorem 8.4.9, there exists \( \eta > 0 \) such that for all \( \mathbf{v},\left| {S\mathbf{v}}\right| \geq \eta \left| \mathbf{v}\right| \) . Say \( \parallel T - S\parallel < r < {\... | Yes |
Lemma 29.2.2 Let \( S, T \) be \( n \times n \) matrices which are invertible. Then\n\n\[ \mathbf{o}\left( {T\mathbf{v}}\right) = \mathbf{o}\left( {S\mathbf{v}}\right) = \mathbf{o}\left( \mathbf{v}\right) \]\n\nand if \( L \) is a continuous linear transformation such that for \( a < b \) ,\n\n\[ \mathop{\sup }\limits_... | Proof: Consider the first claim. For\n\n\[ \frac{\left| o\left( T\mathbf{v}\right) \right| }{\left| \mathbf{v}\right| } = \frac{\left| o\left( T\mathbf{v}\right) \right| }{\left| T\mathbf{v}\right| }\frac{\left| T\mathbf{v}\right| }{\left| \mathbf{v}\right| } \leq \frac{\left| o\left( T\mathbf{v}\right) \right| }{\left... | Yes |
Lemma 29.4.1 Let \( \mathbf{h} \) be differentiable on \( A \subseteq G \) and let \( {A}^{ + } \) consist of those points \( \mathbf{x} \) where \( \det \left( {D\mathbf{h}{\left( \mathbf{x}\right) }^{ * }D\mathbf{h}\left( \mathbf{x}\right) }\right) > 0 \) . Then if \( B \) is any Borel subset of \( {A}^{ + } \) , the... | Proof: It follows from 29.3.10 that for \( \mathbf{x},\mathbf{y} \in T\left( {E\left( {T,\mathbf{c}, i}\right) }\right) \n\n\[ \left| {\mathbf{h}\left( {{T}^{-1}\left( \mathbf{x}\right) }\right) - \mathbf{h}\left( {{T}^{-1}\left( \mathbf{y}\right) }\right) }\right| \leq \left( {1 + {2\varepsilon }}\right) \left| {\math... | Yes |
Lemma 29.4.4 Let \( \mathbf{k} \) be as defined above. Let \( A \) be the set of points where \( D\mathbf{h} \) exists so \( A = {A}^{ + } \) relative to \( \mathbf{k} \) . Then if \( F \) is Lebesgue measurable, \( \mathbf{h}\left( {F \cap A}\right) \) is \( {\mathcal{H}}^{n} \) measurable. Also \( {\mathcal{H}}^{n}\l... | Proof: By Lemma 29.4.1, there are disjoint Borel sets \( {E}_{k} \) such that \( \mathbf{k} \) is Lipschitz on each \( {E}_{k} \) and \( { \cup }_{k}{E}_{k} = A = {A}^{ + } \) where \( {A}^{ + } \) refers to \( \mathbf{k} \) . Thus\n\n\[ P\mathbf{k}\left( {{E}_{k} \cap F \cap A}\right) = \mathbf{h}\left( {{E}_{k} \cap ... | Yes |
Lemma 29.5.1 Let \( \mathbf{h} \) be Lipschitz. Let \( \mathbf{h} \) be one to one and differentiable on \( A \) with \( {m}_{n}\left( {G \smallsetminus A}\right) = 0 \) . If \( N \subseteq G \) has measure zero, then \( \mathbf{h}\left( N\right) \) has \( {\mathcal{H}}^{n} \) measure zero and if \( E \) is Lebesgue me... | Proof: Lemma 29.1.2 implies \( \mathbf{h}\left( N\right) = 0 \) if \( {m}_{n}\left( N\right) = 0 \) .Also from this lemma, \( \mathbf{h}\left( E\right) \) is \( {\mathcal{H}}^{n} \) measurable if \( E \) is. Is \( \nu \) a measure? Suppose \( \left\{ {E}_{i}\right\} \) are disjoint Lebesgue measurable subsets of \( G \... | Yes |
Lemma 29.5.2 Whenever \( E \) is Lebesgue measurable,\n\n\[ \n{\int }_{\mathbf{h}\left( A\right) }{\mathcal{X}}_{E}\left( \mathbf{y}\right) d{\mathcal{H}}^{n} = {\int }_{A}{\mathcal{X}}_{E}\left( {\mathbf{h}\left( \mathbf{x}\right) }\right) {J}_{ * }\left( \mathbf{x}\right) d{m}_{n} \n\] | From this, it follows that if \( s \) is a nonnegative, \( {\mathcal{H}}^{n} \) measurable simple function,29.5.23 continues to be valid with \( s \) in place of \( {\mathcal{X}}_{E} \) . Then approximating an arbitrary nonnegative \( {\mathcal{H}}^{n} \) measurable function \( g \) by an increasing sequence of simple ... | No |
Theorem 29.5.4 Let \( \mathbf{h} : G \subseteq {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) be continuous where \( G \) is an open set and let \( A \subseteq G \) where \( A \) is the Borel measurable set consisting of \( \mathbf{x} \) where \( D\mathbf{h}\left( \mathbf{x}\right) \) exists. Suppose \( \mathbf{h} \)... | Proof: By Lemma 29.4.4, \( \nu \left( E\right) \equiv {\mathcal{H}}^{n}\left( {\mathbf{h}\left( {E \cap A}\right) }\right) \) is a measure defined on the Lebesgue measurable sets contained in \( G \) and \( \nu \ll {m}_{n} \) . The reason it is a measure is\n\n\[ \nu \left( {{ \cup }_{i}{E}_{i}}\right) \equiv {\mathcal... | Yes |
Lemma 29.6.1 For \( S \) defined above, \( {\mathcal{H}}^{n}\left( {\mathbf{h}\left( S\right) }\right) = 0 \) . | Thus \( {m}_{n}\left( N\right) = 0 \) where \( N \) is the set where \( D\mathbf{h}\left( \mathbf{x}\right) \) does not exist. Then by Lemma 29.1.2\n\n\[ \n{\mathcal{H}}^{n}\left( {\mathbf{h}\left( {S \cup N}\right) }\right) \leq {\mathcal{H}}^{n}\left( {\mathbf{h}\left( S\right) }\right) + {\mathcal{H}}^{n}\left( {\ma... | Yes |
Theorem 29.6.4 Let \( \mathbf{h} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) be Lipschitz. Then the function \( \mathbf{y} \rightarrow \# \left( \mathbf{y}\right) \) is \( {\mathcal{H}}^{n} \) measurable and if\n\n\[ g : \mathbf{h}\left( {\mathbb{R}}^{n}\right) \rightarrow \left\lbrack {0,\infty }\right\rbrack \... | Proof: If \( \mathbf{y} \notin \mathbf{h}\left( {S \cup N}\right) \), then \( \mathfrak{n}\left( \mathbf{y}\right) = \# \left( \mathbf{y}\right) \) . By 29.6.25\n\n\[ {\mathcal{H}}^{n}\left( {\mathbf{h}\left( {S \cup N}\right) }\right) = 0 \]\n\nand so \( \mathfrak{n}\left( \mathbf{y}\right) = \# \left( \mathbf{y}\righ... | Yes |
Lemma 29.7.3 Let \( \mathfrak{C} \) be a set whose elements are open subsets of \( {\mathbb{R}}^{p} \) and suppose \( \cup \mathfrak{C} \supseteq H \), a closed set. Then there exists a countable list of open sets, \( {\left\{ {U}_{i}\right\} }_{i = 1}^{\infty } \) such that each \( {U}_{i} \) is bounded, each \( {U}_{... | Proof: The first part was proved earlier. Since \( {\mathbb{R}}^{p} \) is separable, it is completely separable with a countable basis of balls called \( \mathcal{B} \) . For each \( \mathbf{x} \in H \), let \( U \) be a ball from \( \mathcal{B} \) having diameter no more than 1 which is contained in some set of \( \ma... | Yes |
Lemma 29.7.5 Let \( V \) be a bounded open set and let \( X \) be the closed subspace of \( C\left( \bar{V}\right) \), the space of continuous functions defined on \( \bar{V} \), which is given by the following.\n\n\[ \nX = \{ u \in C\left( \bar{V}\right) : u\left( \mathbf{x}\right) = 0\text{ on }\partial V\} .\n\]\n\n... | Proof: Let \( O \subseteq \bar{O} \subseteq W \subseteq \bar{W} \subseteq V \) be such that \( \operatorname{dist}\left( {\bar{O},{V}^{C}}\right) < \eta \) and let \( {\psi }_{\delta }\left( \cdot \right) \) be a mollifier. Let \( u \in X \) and consider \( {\mathcal{X}}_{W}u * {\psi }_{\delta } \) . Let \( \varepsilon... | Yes |
Lemma 29.7.6 Let \( {\alpha }_{1},\cdots ,{\alpha }_{p} \) be real numbers and let \( A\left( {{\alpha }_{1},\cdots ,{\alpha }_{p}}\right) \) be the matrix which has \( 1 + {\alpha }_{i}^{2} \) in the \( i{i}^{\text{th }} \) slot and \( {\alpha }_{i}{\alpha }_{j} \) in the \( i{j}^{\text{th }} \) slot when \( i \neq j ... | \[ \det A = 1 + \mathop{\sum }\limits_{{i = 1}}^{p}{\alpha }_{i}^{2} \] Proof of the claim: The matrix, \( A\left( {{\alpha }_{1},\cdots ,{\alpha }_{p}}\right) \) is of the form \[ A\left( {{\alpha }_{1},\cdots ,{\alpha }_{p}}\right) = \left( \begin{matrix} 1 + {\alpha }_{1}^{2} & {\alpha }_{1}{\alpha }_{2} & \cdots & ... | Yes |
Lemma 29.7.9\n\n\[ \det {\left( D{\mathbf{h}}_{i}{\left( {x}_{1},\cdots ,{x}_{p - 1}\right) }^{ * }D{\mathbf{h}}_{i}\left( {x}_{1},\cdots ,{x}_{p - 1}\right) \right) }^{1/2} \]\n\n\[ = \sqrt{1 + \mathop{\sum }\limits_{{j - 1}}^{{p - 1}}{g}_{i, j}{\left( {x}_{1},\cdots ,{x}_{p - 1}\right) }^{2}} \equiv {J}_{*i}\left( {{... | For\n\n\[ \mathbf{y} = \left( {{x}_{1},\cdots ,{x}_{p - 1},{g}_{i}\left( {{x}_{1},\cdots ,{x}_{p - 1}}\right) }\right) \in \partial U \cap {Q}_{i} \]\n\nand \( \mathbf{n} \) defined by\n\n\[ {\mathbf{n}}_{i}\left( \mathbf{y}\right) = \frac{1}{{J}_{*i}\left( {{x}_{1},\cdots ,{x}_{p - 1}}\right) }{\mathbf{N}}_{i}\left( \... | No |
Theorem 29.7.10 Let \( U \) be a bounded open set with a Lipschitz boundary which lies on one side of its boundary. Then if \( f \in {C}_{c}^{1}\left( {\mathbb{R}}^{p}\right) \) , \[ {\int }_{U}{f}_{, k}\left( \mathbf{x}\right) d{m}_{p} = {\int }_{\partial U}f{n}_{k}d{\mathcal{H}}^{p - 1} \] where \( \mathbf{n} = \left... | Proof: To obtain 29.7.38 apply Lemma 29.7.8 to \( \mathbf{w} = {\mathbf{e}}_{k} \) . Then to obtain 29.7.39 from this, \[ {\int }_{U}\nabla \cdot \mathbf{F}\left( \mathbf{x}\right) d{m}_{p} \] \[ = \mathop{\sum }\limits_{{j = 1}}^{p}{\int }_{U}{F}_{j, j}d{m}_{p} = \mathop{\sum }\limits_{{j = 1}}^{p}{\int }_{\partial U}... | Yes |
Lemma 29.8.2 If \( \mathbf{h} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is Lipschitz, then if \( \mathbf{h}\left( \mathbf{x}\right) = \mathbf{0} \) for all \( \mathbf{x} \in A \), then \( \det \left( {D\mathbf{h}\left( \mathbf{x}\right) }\right) = 0 \) a.e. \( \mathbf{x} \in A \) . | Proof: By the Area formula, \( 0 = {\int }_{\{ \mathbf{0}\} }\# \left( \mathbf{y}\right) {dy} = {\int }_{A}\left| {\det \left( {D\mathbf{h}\left( \mathbf{x}\right) }\right) }\right| {dx} \) and so \( \det \left( {D\mathbf{h}\left( \mathbf{x}\right) }\right) = 0 \) a.e. \( \blacksquare \) | Yes |
Lemma 29.8.3 \( \det \left( {I + U}\right) = 1 + \operatorname{trace}\left( U\right) + o\left( U\right) \) where \( o\left( U\right) \) is defined in terms of the Frobenius norm for \( p \times p \) matrices. | Proof: This is obvious if \( p = 1 \) or 2 . Assume true for \( n - 1 \) . Then for \( U \) an \( n \times n \), expand the matrix along the last column and use induction on the cofactor of \( 1 + {U}_{nn} \) . ∎ | No |
Theorem 29.9.1 Let \( A \) be an \( m \times n \) matrix and let \( B \) be an \( n \times m \) matrix for \( m \leq n \). Then for \( I \) an appropriate size identity matrix,\n\n\[ \n\det \left( {I + {AB}}\right) = \det \left( {I + {BA}}\right) \n\] | Proof: Use block multiplication to write\n\n\[ \n\left( \begin{matrix} I + {AB} & 0 \\ B & I \end{matrix}\right) \left( \begin{array}{ll} I & A \\ 0 & I \end{array}\right) = \left( \begin{matrix} I + {AB} & A + {ABA} \\ B & {BA} + I \end{matrix}\right) \n\]\n\n\[ \n\left( \begin{matrix} I & A \\ 0 & I \end{matrix}\righ... | Yes |
Lemma 29.9.6 If \( A \subseteq {\mathbb{R}}^{n} \) is Lebesgue measurable, then\n\n\[ \n{\int }_{{\mathbb{R}}^{m}}{\mathcal{H}}^{n - m}\left( {A \cap {\mathbf{f}}^{-1}\left( \mathbf{y}\right) }\right) {dy} \]\n\n\[ \n\leq C\left( {n, m}\right) {\left( \operatorname{Lip}\left( \mathbf{f}\right) \right) }^{m}{m}_{n}\left... | Proof: This follows from Lemma 29.9.4 and Lemma 29.9.5. Since\n\n\[ \n\mathbf{y} \rightarrow {\mathcal{H}}^{n - m}\left( {A \cap {\mathbf{f}}^{-1}\left( \mathbf{y}\right) }\right) \]\n\nis measurable,\n\n\[ \n{\int }_{{\mathbb{R}}^{m}}{\mathcal{H}}^{n - m}\left( {A \cap {\mathbf{f}}^{-1}\left( \mathbf{y}\right) }\right... | Yes |
Theorem 29.9.7 Let \( A \) be a measurable set in \( {\mathbb{R}}^{n} \) and let \( \mathbf{f} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) be a Lipschitz map. Then the following formula holds along with all measurability assertions needed for it to make sense. | Proof: Let \( S \equiv \left\{ {\mathbf{x} : {J}^{ * }\left( \mathbf{x}\right) = 0}\right\} \) . Thus on \( S \) , \( \det D{\mathbf{f}}^{\mathbf{i}}\left( \mathbf{x}\right) = 0 \) for each \( \mathbf{i} \in \) \( \Lambda \left( {n, m}\right) \) and \( N \equiv \{ \mathbf{x} : D\mathbf{f}\left( \mathbf{x}\right) \) doe... | No |
Corollary 29.9.8 Let \( \mathbf{f} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) where \( m \leq n \) and \( \mathbf{f} \) is locally Lipschitz. This means that for each \( r > 0,\mathbf{f} \) is Lipschitz on \( B\left( {\mathbf{0}, r}\right) \) . Then the coarea formula, 29.9.47, holds for \( \mathbf{f} \) . | Proof: Let \( A \subseteq B\left( {\mathbf{0}, r}\right) \) and let \( {\mathbf{f}}_{r} \) be Lipschitz with \( \mathbf{f}\left( \mathbf{x}\right) = {\mathbf{f}}_{r}\left( \mathbf{x}\right) \) for \( \mathbf{x} \in \) \( B\left( {\mathbf{0}, r + 1}\right) \) . Then\n\n\[ \n{\int }_{A}{J}^{ * }\mathbf{f}\left( \mathbf{x... | Yes |
Theorem 29.10.1 Let \( g \geq 0 \) be Lebesgue measurable and let\n\n\[ \mathbf{f} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m}, n \geq m \]\n\nsatisfy the Coarea formula. Then\n\n\[ {\int }_{{\mathbb{R}}^{n}}g\left( \mathbf{x}\right) {J}^{ * }\mathbf{f}\left( \mathbf{x}\right) {dx} = {\int }_{\mathbf{f}\left( {\mat... | Proof: Let \( {s}_{i} \uparrow g \) where \( {s}_{i} \) is a simple function satisfying 29.10.58. Then let \( i \rightarrow \infty \) and use the monotone convergence theorem to replace \( {s}_{i} \) with \( g \) . This proves the change of variables formula. | No |
Example 29.10.2 Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be given by \( f\left( \mathbf{x}\right) \equiv \left| \mathbf{x}\right| \) . Then \( {J}^{ * }\left( \mathbf{x}\right) \) ends up being 1. Then by the coarea formula, | \[ {\int }_{B\left( {\mathbf{0}, r}\right) }d{m}_{n} = {\int }_{0}^{r}{\mathcal{H}}^{n - 1}\left( {B\left( {\mathbf{0}, r}\right) \cap {f}^{-1}\left( y\right) }\right) {dy} = {\int }_{0}^{r}{\mathcal{H}}^{n - 1}\left( {\partial B\left( {\mathbf{0}, y}\right) }\right) {dy} \] Then \( {m}_{n}\left( {B\left( {\mathbf{0}, ... | Yes |
Lemma 29.11.2 Let \( f \in {C}_{c}\left( {\mathbf{h}{\left( \partial \Omega \right) }^{C}}\right) \) for \( \Omega \) a bounded open set and let \( \mathbf{h} \) be Lipschitz on \( {\mathbb{R}}^{n} \) . Say \( \partial \Omega \) has measure zero so that \( \mathbf{h}\left( {\partial \Omega }\right) \) has measure zero.... | \[ \int f\left( \mathbf{y}\right) d\left( {\mathbf{y},\Omega ,\mathbf{h}}\right) {dy} = {\int }_{\Omega }\det \left( {D\mathbf{h}\left( \mathbf{x}\right) }\right) f\left( {\mathbf{h}\left( \mathbf{x}\right) }\right) {dx}. \] | Yes |
Proposition 29.12.3 Let \( \Omega \) be an open connected bounded set in \( {\mathbb{R}}^{n}, n \geq 1 \) such that \( {\mathbb{R}}^{n} \smallsetminus \partial \Omega \) consists of two, three if \( n = 1 \), connected components. Let \( \mathbf{f} \in C\left( {\bar{\Omega };{\mathbb{R}}^{n}}\right) \) be continuous an... | Proof: First suppose \( n \geq 2 \) . By the Jordan separation theorem, \( {\mathbb{R}}^{n} \smallsetminus \mathbf{f}\left( {\partial \Omega }\right) \) consists of two components, a bounded component \( B \) and an unbounded component \( U \) . Using the Tietze extention theorem, there exists \( \mathbf{g} \) defined ... | Yes |
Lemma 29.12.4 Let \( \mathbf{h} \in {W}^{1, p}\left( {{\mathbb{R}}^{n};{\mathbb{R}}^{n}}\right), p > n \) where \( \mathbf{h} \) is one to one, \( \mathbf{h}\left( {\partial \Omega }\right) ,\partial \Omega \) have measure zero for \( \Omega \) a bounded open connected set in \( {\mathbb{R}}^{n} \) . Then \( \mathbf{h}... | Proof: Consider the first claim. Let \( \delta \) be such that \( \overline{B\left( {{\mathbf{x}}_{1},\delta }\right) } \subseteq \Omega \) and let \( {\left\{ {f}_{j}\left( \mathbf{y}\right) \right\} }_{j = 1}^{\infty } \) be nonnegative, increasing in \( j \) and converging pointwise to \( {\mathcal{X}}_{\mathbf{h}\l... | Yes |
Theorem 30.2.1 Let \( A \) be an \( n \times m \) matrix with \( n \geq m \) and let \( B \) be a \( m \times n \) matrix. Also let \( {A}_{i} \)\n\n\[ i = 1,\cdots, C\left( {n, m}\right) \]\n\nbe the \( m \times m \) submatrices of \( A \) which are obtained by deleting \( n - m \) rows and let \( {B}_{i} \) be the \(... | Proof: This follows from a computation. By Corollary 5.4.5 on Page 73, \( \det \left( {BA}\right) = \)\n\n\[ \frac{1}{m!}\mathop{\sum }\limits_{\left( {i}_{1}\cdots {i}_{m}\right) }\mathop{\sum }\limits_{\left( {j}_{1}\cdots {j}_{m}\right) }\operatorname{sgn}\left( {{i}_{1}\cdots {i}_{m}}\right) \operatorname{sgn}\left... | Yes |
Theorem 30.4.1 Let \( \Omega \) be an oriented Lipschitz manifold and let\n\n\[ \omega = \mathop{\sum }\limits_{I}{a}_{I}\left( \mathbf{x}\right) d{x}_{{i}_{1}} \land \cdots \land d{x}_{{i}_{n - 1}}. \]\n\nwhere each \( {a}_{I} \) is \( {C}^{1}\left( \bar{\Omega }\right) \) . For \( {\left\{ {U}_{j},{\mathbf{R}}_{j}\ri... | What if \( {a}_{I} \) is only the restriction to \( \Omega \) of a function in \( {W}^{1, p}\left( {\mathbb{R}}^{m}\right), p > 1 \) ? Would the same formula still hold? Let \( {\phi }_{\varepsilon } \) be a mollifier and let \( {a}_{I\varepsilon } \equiv {a}_{I} * {\phi }_{\varepsilon } \) . Then Stoke's theorem appli... | Yes |
Theorem 30.5.1 Let \( \Omega \) be a bounded open set having Lipschitz boundary as described above. Also let\n\n\[ \omega = \mathop{\sum }\limits_{I}{a}_{I}\left( \mathbf{x}\right) d{x}_{{i}_{1}} \land \cdots \land d{x}_{{i}_{n - 1}} \]\n\nbe a differential form where \( {a}_{I} \) is assumed to be the restriction to \... | It can be shown that, since the boundary is Lipschitz, it would have sufficed to assume \( u \in {W}^{1, p}\left( \Omega \right) \) and then it is automatically the restriction of one in \( {W}^{1, p}\left( {\mathbb{R}}^{n}\right) \) . However, these terms have not all been defined and the necessary results are not pro... | No |
Theorem 30.6.1 Let \( \Omega \) be a bounded open set having Lipschitz boundary as described above. Also let \( \mathbf{F} \) be a vector field with the property that for each component function of \( \mathbf{F},{F}_{k} \) is the restriction to \( \Omega \) of a function in \( {W}^{1, p}\left( {\mathbb{R}}^{n}\right), ... | It is clear \( \mathbf{n} \) is unique \( {\mathcal{H}}^{n - 1} \) a.e. since if there were two, then a simple manipulation shows for all such \( \mathbf{F} \) ,\n\n\[{\int }_{\partial \Omega }\mathbf{F} \cdot \left( {\mathbf{n} - {\mathbf{n}}_{1}}\right) d{\mathcal{H}}^{n - 1} = 0\]\n\nThus \( \mathbf{n} - {\mathbf{n}... | Yes |
Lemma 31.1.1 \( Z \) is measurable and \( \mu \left( Z\right) = 0 \) . | Proof: For each \( \mathbf{x} \in Z \), there exists a ball \( B\left( {\mathbf{x}, r}\right) \) with \( \mu \left( {B\left( {\mathbf{x}, r}\right) }\right) = 0 \) . Let \( \mathcal{C} \) be the collection of these balls. Since \( {\mathbb{R}}^{n} \) has a countable basis, a countable subset, \( \widetilde{\mathcal{C}}... | Yes |
Theorem 31.1.2 Let \( \mu \) be a Radon measure and let \( f \in {L}^{1}\left( {{\mathbb{R}}^{n},\mu }\right) \). Then for a.e.x, | Proof: First consider the following claim which is a weak type estimate of the same sort used when differentiating with respect to Lebesgue measure.\n\nClaim 1: The following inequality holds for \( {N}_{n} \) the constant of the Besicovitch covering theorem.\n\n\[ \bar{\mu }\left( \left\lbrack {{Mf} > \varepsilon }\ri... | Yes |
Corollary 31.1.3 If \( f \in {L}_{loc}^{1}\left( {{\mathbb{R}}^{n},\mu }\right) \), then for a.e. \( \mathbf{x} \notin Z \) ,\n\n\[ \mathop{\lim }\limits_{{r \rightarrow 0}}\frac{1}{\mu \left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }\left| {f\left( \mathbf{y}\right) - f\left( ... | Proof: If \( f \) is replaced by \( f{\mathcal{X}}_{B\left( {\mathbf{0}, k}\right) } \) then the conclusion 31.1.4 holds for all \( \mathbf{x} \notin {F}_{k} \) where \( {F}_{k} \) is a set of \( \mu \) measure 0 . Letting \( k = 1,2,\cdots \), and \( F \equiv { \cup }_{k = 1}^{\infty }{F}_{k} \), it follows that \( F ... | Yes |
Lemma 31.2.1 The space \( {C}_{c}\left( {\mathbb{R}}^{m}\right) \) with the norm\n\n\[ \parallel f\parallel \equiv \sup \left\{ {\left| {f\left( \mathbf{y}\right) }\right| : \mathbf{y} \in {\mathbb{R}}^{m}}\right\} \]\n\nis separable. | Proof: Let \( {\mathcal{D}}_{l} \) consist of all functions which are of the form\n\n\[ \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}{a}_{\alpha }{\mathbf{y}}^{\alpha }{\left( \operatorname{dist}\left( \mathbf{y}, B{\left( \mathbf{0}, l + 1\right) }^{C}\right) \right) }^{{n}_{\alpha }} \]\n\nwhere \( {a}_{\alp... | Yes |
Lemma 31.2.2 If \( \mu \) and \( \nu \) are two Radon measures defined on \( \sigma \) algebras, \( {\mathcal{S}}_{\mu } \) and \( {\mathcal{S}}_{\nu } \), of subsets of \( {\mathbb{R}}^{n} \) and if \( \mu \left( V\right) = \nu \left( V\right) \) for all \( V \) open, then \( \mu = \nu \) and \( {\mathcal{S}}_{\mu } =... | Proof: Every compact set is a countable intersection of open sets so the two measures agree on every compact set. Hence it is routine that the two measures agree on every \( {G}_{\delta } \) and \( {F}_{\sigma } \) set. (Recall \( {G}_{\delta } \) sets are countable intersections of open sets and \( {F}_{\sigma } \) se... | Yes |
Theorem 31.2.3 Let \( \mu \) be a finite Radon measure on \( {\mathbb{R}}^{n + m} \) defined on a \( \sigma \) algebra, \( \mathcal{F} \). Then there exists a unique finite Radon measure \( \alpha \), defined on a \( \sigma \) algebra \( \mathcal{S} \), of sets of \( {\mathbb{R}}^{n} \) which satisfies\n\n\[ \alpha \le... | Proof:\n\nFirst consider the uniqueness of \( \alpha \). Suppose \( {\alpha }_{1} \) is another Radon measure satisfying 31.2.5. Then in particular, \( {\alpha }_{1} \) and \( \alpha \) agree on open sets and so the two measures are the same by Lemma 31.2.2.\n\nTo establish the existence of \( \alpha \), define \( {\al... | Yes |
Lemma 31.3.2 Let \( \lambda \) and \( \mu \) be Radon measures. If \( A \) is a bounded subset of \( \left\{ {\mathbf{x} \notin Z : {\bar{D}}_{\mu }\lambda \left( \mathbf{x}\right) \geq a}\right\} \), then\n\n\[ \bar{\lambda }\left( A\right) \geq a\bar{\mu }\left( A\right) \]\n\nand if \( A \) is a bounded subset of \(... | Proof: Suppose first that \( A \) is a bounded subset of \( \left\{ {\mathbf{x} \notin Z : {\bar{D}}_{\mu }\lambda \left( \mathbf{x}\right) \geq a}\right\} \), let \( \varepsilon > 0 \), and let \( V \) be a bounded open set with \( V \supseteq A \) and \( \lambda \left( V\right) - \varepsilon < \bar{\lambda }\left( A\... | Yes |
Corollary 31.3.6 Let \( \mu ,\lambda \) be two Radon measures. Then there exist two measures, \( {\lambda }_{\mu },{\lambda }_{ \bot } \) such that\n\n\[ \n{\lambda }_{\mu } \ll \mu ,\lambda = {\lambda }_{\mu } + {\lambda }_{ \bot }\n\]\n\nand a set of \( \mu \) measure zero \( N \) such that\n\n\[ \n{\lambda }_{ \bot ... | Proof: If \( \mathbf{x} \in N \), this could happen two ways, either \( \mathbf{x} \in Z \) or \( {D}_{\mu }\lambda \left( \mathbf{x}\right) \) fails to exist. It only remains to verify that \( {\lambda }_{\mu } \) given above satisfies \( {\lambda }_{\mu } \ll \mu \) . However, this is obvious because if \( \mu \left(... | No |
Lemma 32.1.3 \( \mathcal{G} \) is dense in \( {C}_{0}\left( {\mathbb{R}}^{n}\right) \) with respect to the norm, \[ \parallel f{\parallel }_{\infty } \equiv \sup \left\{ {\left| {f\left( \mathbf{x}\right) }\right| : \mathbf{x} \in {\mathbb{R}}^{n}}\right\} \] | Proof: By the Weierstrass approximation theorem, it suffices to show \( \mathcal{G} \) separates the points and annihilates no point. It was already observed in the above definition that \( \bar{f} \in \mathcal{G} \) whenever \( f \in \mathcal{G} \) . If \( {\mathbf{y}}_{1} \neq {\mathbf{y}}_{2} \) suppose first that \... | Yes |
Theorem 32.1.4 For each \( p \geq 1, p < \infty ,\mathcal{G} \) is dense in \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \) . | Proof: Let \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) . Then there exists \( g \in {C}_{c}\left( {\mathbb{R}}^{n}\right) \) such that \( \parallel f - g{\parallel }_{p} < \varepsilon \) . Now let \( b > 0 \) be large enough that\n\n\[{\int }_{{\mathbb{R}}^{n}}{\left( {e}^{-b{\left| \mathbf{x}\right| }^{2}}\right... | Yes |
Lemma 32.2.2 The following formulas are true. \( \left( {c > 0}\right) \)\n\n\[ \n{\int }_{\mathbb{R}}{e}^{-c{t}^{2}}{e}^{-{ist}}{dt} = {\int }_{\mathbb{R}}{e}^{-c{t}^{2}}{e}^{ist}{dt} = {e}^{-\frac{{s}^{2}}{4c}}\frac{\sqrt{\pi }}{\sqrt{c}}, \]\n\n\( \left( {32.2.1}\right) \)\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}{e}^{-c{\... | Proof: Consider the first one. Let \( h\left( s\right) \) be given by the left side. Then\n\n\[ \nH\left( s\right) \equiv {\int }_{\mathbb{R}}{e}^{-c{t}^{2}}{e}^{-{ist}}{dt} = {\int }_{\mathbb{R}}{e}^{-c{t}^{2}}\cos \left( {st}\right) {dt} \]\n\nThen using the dominated convergence theorem to differentiate,\n\n\[ \n{H}... | Yes |
Lemma 32.3.2 The following is obtained for all \( \phi ,\psi \in \mathcal{G} \) .\n\n\[ \n{F\psi }\left( \phi \right) = \psi \left( {F\phi }\right) ,{F}^{-1}\psi \left( \phi \right) = \psi \left( {{F}^{-1}\phi }\right) \n\]\n\nAlso if \( \psi \in \mathcal{G} \) and \( \psi = 0 \) in \( {\mathcal{G}}^{ * } \) so that \(... | Proof:\n\n\[ \n{F\psi }\left( \phi \right) \equiv {\int }_{{\mathbb{R}}^{n}}{F\psi }\left( \mathbf{t}\right) \phi \left( \mathbf{t}\right) {dt} \n\]\n\n\[ \n= {\int }_{{\mathbb{R}}^{n}}{\left( \frac{1}{2\pi }\right) }^{n/2}{\int }_{{\mathbb{R}}^{n}}{e}^{-i\mathbf{t} \cdot \mathbf{x}}\psi \left( \mathbf{x}\right) {dx\ph... | Yes |
Lemma 32.3.4 \( F \) and \( {F}^{-1} \) are both one to one, onto, and are inverses of each other. | Proof: First note \( F \) and \( {F}^{-1} \) are both linear. This follows directly from the definition. Suppose now \( {FT} = 0 \) . Then \( {FT}\left( \phi \right) = T\left( {F\phi }\right) = 0 \) for all \( \phi \in \mathcal{G} \) . But \( F \) and \( {F}^{-1} \) map \( \mathcal{G} \) onto \( \mathcal{G} \) because ... | Yes |
Lemma 32.3.5 If \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) and \( {\int }_{{\mathbb{R}}^{n}}{f\phi dx} = 0 \) for all \( \phi \in {C}_{c}\left( {\mathbb{R}}^{n}\right) \), then \( f = 0 \) a.e. | Proof: For \( r > 0 \), let\n\n\[ E \equiv \{ \mathbf{x} : f\left( \mathbf{x}\right) \geq r\} ,{E}_{R} \equiv E \cap B\left( {\mathbf{0}, R}\right) .\n\]\n\nLet \( {K}_{m} \) be an increasing sequence of compact sets, and let \( {V}_{m} \) be a decreasing sequence of open sets satisfying\n\n\[ {K}_{m} \subseteq {E}_{R}... | Yes |
Corollary 32.3.6 Let \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) and suppose\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}f\left( \mathbf{x}\right) \phi \left( \mathbf{x}\right) {dx} = 0 \n\]\n\nfor all \( \phi \in \mathcal{G} \) . Then \( f = 0 \) a.e. | Proof: Let \( \psi \in {C}_{c}\left( {\mathbb{R}}^{n}\right) \) . Then by the Stone Weierstrass approximation theorem, there exists a sequence of functions, \( \left\{ {\phi }_{k}\right\} \subseteq \mathcal{G} \) such that \( {\phi }_{k} \rightarrow \psi \) uniformly. Then by the dominated convergence theorem,\n\n\[ \n... | Yes |
Theorem 32.3.7 Let \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right), p \geq 1 \), or suppose \( f \) is measurable and has polynomial growth,\n\n\[ \left| {f\left( \mathbf{x}\right) }\right| \leq K{\left( 1 + {\left| \mathbf{x}\right| }^{2}\right) }^{m} \]\n\nfor some \( m \in \mathbb{N} \) . Then if\n\n\[ \int {f\psi dx... | Proof: First note that if \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) or has polynomial growth, then it makes sense to write the integral \( \int {f\psi dx} \) described above. This is obvious in the case of polynomial growth. In the case where \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) it also makes sense... | Yes |
Theorem 32.3.8 Let \( f \) be a measurable function with polynomial growth,\n\n\[ \left| {f\left( \mathbf{x}\right) }\right| \leq C{\left( 1 + {\left| \mathbf{x}\right| }^{2}\right) }^{N}\text{ for some }N, \]\n\nor let \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) for some \( p \in \left\lbrack {1,\infty }\right\r... | Proof: Let \( f \) have polynomial growth first. Then the above integral is clearly well defined and so in this case, \( f \in {\mathcal{G}}^{ * } \) .\n\nNext suppose \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) with \( \infty > p \geq 1 \) . Then it is clear again that the above integral is well defined because ... | Yes |
Theorem 32.3.9 Let \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) . Then \( {Ff}\left( \phi \right) = {\int }_{{\mathbb{R}}^{n}}{g\phi dt} \) where\n\n\[ g\left( \mathbf{t}\right) = {\left( \frac{1}{2\pi }\right) }^{n/2}{\int }_{{\mathbb{R}}^{n}}{e}^{-i\mathbf{t} \cdot \mathbf{x}}f\left( \mathbf{x}\right) {dx} \]\n\... | Proof: From the definition and Fubini's theorem,\n\n\[ {Ff}\left( \phi \right) \equiv {\int }_{{\mathbb{R}}^{n}}f\left( \mathbf{t}\right) {F\phi }\left( \mathbf{t}\right) {dt} = {\int }_{{\mathbb{R}}^{n}}f\left( \mathbf{t}\right) {\left( \frac{1}{2\pi }\right) }^{n/2}{\int }_{{\mathbb{R}}^{n}}{e}^{-i\mathbf{t} \cdot \m... | Yes |
Theorem 32.3.10 If \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) and \( {\begin{Vmatrix}{f}_{k} - f\end{Vmatrix}}_{1} \rightarrow 0 \), then \( F{f}_{k} \) and \( {F}^{-1}{f}_{k} \) converge uniformly to \( {Ff} \) and \( {F}^{-1}f \) respectively. If \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \), then \( {F}^{... | Proof: The first claim follows from the following inequality.\n\n\[ \left| {F{f}_{k}\left( \mathbf{t}\right) - {Ff}\left( \mathbf{t}\right) }\right| \leq {\left( 2\pi \right) }^{-n/2}{\int }_{{\mathbb{R}}^{n}}\left| {{e}^{-i\mathbf{t} \cdot \mathbf{x}}{f}_{k}\left( \mathbf{x}\right) - {e}^{-i\mathbf{t} \cdot \mathbf{x}... | Yes |
Theorem 32.3.11 Let \( f, g \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) . Then \( f * g \in {L}^{1} \) and \( F\left( {f * g}\right) = {\left( 2\pi \right) }^{n/2}{FfFg} \) . | Proof: Consider\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}{\int }_{{\mathbb{R}}^{n}}\left| {f\left( {\mathbf{x} - \mathbf{y}}\right) g\left( \mathbf{y}\right) }\right| {dydx}.\n\]\n\nThe function, \( \left( {\mathbf{x},\mathbf{y}}\right) \rightarrow \left| {f\left( {\mathbf{x} - \mathbf{y}}\right) g\left( \mathbf{y}\right) }\r... | Yes |
Theorem 32.3.12 For \( \phi \in \mathcal{G},\parallel {F\phi }{\parallel }_{2} = {\begin{Vmatrix}{F}^{-1}\phi \end{Vmatrix}}_{2} = \parallel \phi {\parallel }_{2} \) . | Proof: First note that for \( \psi \in \mathcal{G} \) ,\n\n\[ F\left( \bar{\psi }\right) = \overline{{F}^{-1}\left( \psi \right) },{F}^{-1}\left( \bar{\psi }\right) = \overline{F\left( \psi \right) }.\]\n\n\( \left( {32.3.6}\right) \)\n\nThis follows from the definition. For example,\n\n\[ F\bar{\psi }\left( \mathbf{t}... | Yes |
Lemma 32.3.13 Let \( f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) and let \( {\phi }_{k} \rightarrow f \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) where \( {\phi }_{k} \in \mathcal{G} \) . (Such a sequence exists because of density of \( \mathcal{G} \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) .) Then \( {... | Proof: Let \( \psi \in \mathcal{G} \) be given. Then \[ {Ff}\left( \psi \right) \equiv f\left( {F\psi }\right) \equiv {\int }_{{\mathbb{R}}^{n}}f\left( \mathbf{x}\right) {F\psi }\left( \mathbf{x}\right) {dx} \] \[ = \mathop{\lim }\limits_{{k \rightarrow \infty }}{\int }_{{\mathbb{R}}^{n}}{\phi }_{k}\left( \mathbf{x}\ri... | Yes |
Theorem 32.3.15 (Plancherel)\n\n\[ \parallel f{\parallel }_{2} = \parallel {Ff}{\parallel }_{2} = {\begin{Vmatrix}{F}^{-1}f\end{Vmatrix}}_{2} \] | Proof: Use the density of \( \mathcal{G} \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) to obtain a sequence, \( \left\{ {\phi }_{k}\right\} \) converging to \( f \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . Then by Lemma 32.3.13\n\n\[ \parallel {Ff}{\parallel }_{2} = \mathop{\lim }\limits_{{k \rightarrow \in... | Yes |
Lemma 32.3.16 Suppose \( {f}_{k} \rightarrow f \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) and \( {g}_{k} \rightarrow g \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{\int }_{{\mathbb{R}}^{n}}{f}_{k}{g}_{k}{dx} = {\int }_{{\mathbb{R}}^{n}}{fgdx} \]\n | Proof:\n\n\[ \left| {{\int }_{{\mathbb{R}}^{n}}{f}_{k}{g}_{k}{dx} - {\int }_{{\mathbb{R}}^{n}}{fgdx}}\right| \leq \left| {{\int }_{{\mathbb{R}}^{n}}{f}_{k}{g}_{k}{dx} - {\int }_{{\mathbb{R}}^{n}}{f}_{k}{gdx}}\right| +\n\n\[ \left| {{\int }_{{\mathbb{R}}^{n}}{f}_{k}{gdx} - {\int }_{{\mathbb{R}}^{n}}{fgdx}}\right| \]\n\n... | Yes |
Corollary 32.3.17 For \( f, g \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) , \[ {\int }_{{\mathbb{R}}^{n}}f\bar{g}{dx} = {\int }_{{\mathbb{R}}^{n}}{Ff}\overline{Fg}{dx} = {\int }_{{\mathbb{R}}^{n}}{F}^{-1}f\overline{{F}^{-1}g}{dx}. \] | Proof: First note the above formula is obvious if \( f, g \in \mathcal{G} \) . To see this, note \[ {\int }_{{\mathbb{R}}^{n}}{Ff}\overline{Fg}{dx} = {\int }_{{\mathbb{R}}^{n}}{Ff}\left( \mathbf{x}\right) \frac{1}{{\left( 2\pi \right) }^{n/2}}{\int }_{{\mathbb{R}}^{n}}{e}^{-i\mathbf{x} \cdot \mathbf{t}}g\left( \mathbf{... | Yes |
Theorem 32.3.18 For \( f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \), let \( {f}_{r} = f{\mathcal{X}}_{{E}_{r}} \) where \( {E}_{r} \) is a bounded measurable set with \( {E}_{r} \uparrow {\mathbb{R}}^{n} \). Then the following limits hold in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \).\n\n\[ \n{Ff} = \mathop{\lim }\l... | Proof: \( {\begin{Vmatrix}f - {f}_{r}\end{Vmatrix}}_{2} \rightarrow 0 \) and so \( {\begin{Vmatrix}Ff - F{f}_{r}\end{Vmatrix}}_{2} \rightarrow 0 \) and \( {\begin{Vmatrix}{F}^{-1}f - {F}^{-1}{f}_{r}\end{Vmatrix}}_{2} \rightarrow 0 \) by Plancherel's Theorem. - | Yes |
Theorem 32.3.19 Let \( h \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) and let \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) . Then \( h * f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \), \[ {F}^{-1}\left( {h * f}\right) = {\left( 2\pi \right) }^{n/2}{F}^{-1}h{F}^{-1}f \] \[ F\left( {h * f}\right) = {\left( 2\pi \rig... | Proof: An application of Minkowski's inequality yields \[ {\left( {\int }_{{\mathbb{R}}^{n}}{\left( {\int }_{{\mathbb{R}}^{n}}\left| h\left( \mathbf{x} - \mathbf{y}\right) \right| \left| f\left( \mathbf{y}\right) \right| dy\right) }^{2}dx\right) }^{1/2} \leq \parallel f{\parallel }_{1}\parallel h{\parallel }_{2}. \] He... | Yes |
Theorem 32.3.22 Let \( \psi \in \mathfrak{S} \) . Then \( \left( {F \circ {F}^{-1}}\right) \left( \psi \right) = \psi \) and \( \left( {{F}^{-1} \circ F}\right) \left( \psi \right) = \psi \) whenever \( \psi \in \mathfrak{S} \) . Also \( F \) and \( {F}^{-1} \) map \( \mathfrak{S} \) one to one and onto \( \mathfrak{S}... | Proof: The first claim follows from the fact that \( F \) and \( {F}^{-1} \) are inverses of each other on \( {\mathcal{G}}^{ * } \) which was established above. For the second, let \( \psi \in \mathfrak{S} \) . Then \( \psi = F\left( {{F}^{-1}\psi }\right) \) . Thus \( F \) maps \( \mathfrak{S} \) onto \( \mathfrak{S}... | Yes |
Theorem 32.3.25 Let \( f \in {\mathcal{G}}^{ * } \) and let \( \phi \in \mathcal{G} \) . \[ F\left( {f * \phi }\right) = {\left( 2\pi \right) }^{n/2}{F\phi Ff} \] \[ {F}^{-1}\left( {f * \phi }\right) = {\left( 2\pi \right) }^{n/2}{F}^{-1}\phi {F}^{-1}f. \] | Proof: Note that 32.3.16 follows from Definition 32.3.24 and both assertions hold for \( f \in \mathcal{G} \) . Consider 32.3.17. Here is a simple formula involving a pair of functions in \( \mathcal{G} \) . \[ \left( {\psi * {F}^{-1}{F}^{-1}\phi }\right) \left( \mathbf{x}\right) \] \[ = \left( {\iint \int \int \psi \l... | Yes |
Lemma 33.1.2 If \( p \in \left\lbrack {1, r}\right\rbrack \), then \( {L}^{p}\left( \Omega \right) \subseteq {L}^{1}\left( \Omega \right) + {L}^{r}\left( \Omega \right) \) . | Proof: Let \( \lambda > 0 \) and let \( f \in {L}^{p}\left( \Omega \right) \)\n\n\[ \n{f}_{1}\left( x\right) \equiv \left\{ {\begin{array}{l} f\left( x\right) \text{ if }\left| {f\left( x\right) }\right| \leq \lambda \\ 0\text{ if }\left| {f\left( x\right) }\right| > \lambda \end{array},{f}_{2}\left( x\right) \equiv \l... | Yes |
Theorem 33.3.3 Let \( \rho \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \cap {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and suppose\n\n\[{\int }_{\left| \mathbf{x}\right| \geq 2\left| \mathbf{y}\right| }\left| {{F}^{-1}\rho \left( {\mathbf{x} - \mathbf{y}}\right) - {F}^{-1}\rho \left( \mathbf{x}\right) }\right| {dx} \... | Proof: From Lemma 33.3.2, \( {F}^{-1}\rho * \) is weak \( \left( {1,1}\right) \), weak \( \left( {2,2}\right) \), and maps\n\n\[{L}^{1}\left( {\mathbb{R}}^{n}\right) + {L}^{2}\left( {\mathbb{R}}^{n}\right)\]\n\nto measurable functions. Therefore, by the Marcinkiewicz interpolation theorem, there exists a constant \( {A... | Yes |
Lemma 33.3.4 Let 33.3.25 hold and suppose \( \psi \in {C}_{c}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ \mathbf{0}\} }\right) \) . Then for each \( \alpha ,\left| \alpha \right| \leq L \), there exists a constant \( C \equiv C\left( {\alpha, n,\psi }\right) \) independent of \( k \) such that\n\n\[ \mathop{\su... | Proof:\n\n\[ {\left| \mathbf{x}\right| }^{\left| \alpha \right| }\left| {{D}^{\alpha }\left( {\rho \left( \mathbf{x}\right) \psi \left( {{2}^{k}\mathbf{x}}\right) }\right) }\right| \leq {\left| \mathbf{x}\right| }^{\left| \alpha \right| }\mathop{\sum }\limits_{{\beta + \gamma = \alpha }}\left| {{D}^{\beta }\rho \left( ... | Yes |
Lemma 33.3.5 There exists\n\n\\[ \n\\phi \\in {C}_{c}^{\\infty }\\left( \\left\\lbrack {\\mathbf{x} : {4}^{-1} < \\left| \\mathbf{x}\\right| < 4}\\right\\rbrack \\right) ,\\phi \\left( \\mathbf{x}\\right) \\geq 0,\n\\]\n\nand\n\n\\[ \n\\mathop{\\sum }\\limits_{{k = - \\infty }}^{\\infty }\\phi \\left( {{2}^{k}\\mathbf{... | Proof: Let\n\n\\[ \n\\psi \\geq 0,\\psi = 1\\text{ on }\\left\\lbrack {{2}^{-1} \\leq \\left| \\mathbf{x}\\right| \\leq 2}\\right\\rbrack\n\\]\n\n\\[ \n\\operatorname{spt}\\left( \\psi \\right) \\subseteq \\left\\lbrack {{4}^{-1} < \\left| \\mathbf{x}\\right| < 4}\\right\\rbrack\n\\]\n\nConsider\n\n\\[ \ng\\left( \\mat... | Yes |
Lemma 33.3.6 There exists a constant depending only on the indicated objects, \( {C}_{1} = C\left( {L, n,\phi ,{C}_{0}}\right) \) such that when \( \left| \mathbf{y}\right| \leq t \) , \n\n\[ \n{\int }_{\left| \mathbf{x}\right| \geq {2t}}\left| {{F}^{-1}\rho \left( {\mathbf{x} - \mathbf{y}}\right) - {F}^{-1}\rho \left(... | Proof: \( {F}^{-1}\rho = \mathop{\lim }\limits_{{m \rightarrow \infty }}{F}^{-1}{\rho }_{m} \) in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . Let \( {m}_{k} \rightarrow \infty \) be such that convergence is pointwise a.e. Then if \( \left| \mathbf{y}\right| \leq t \), Fatou’s lemma implies \n\n\[ \n{\int }_{\left| \m... | Yes |
Theorem 33.3.7 (Mihlin’s theorem) Suppose \( \rho \) satisfies\n\n\[ \n{C}_{0} \geq \sup \left\{ {{\left| \mathbf{x}\right| }^{\left| \alpha \right| }\left| {{D}^{\alpha }\rho \left( \mathbf{x}\right) }\right| : \left| \alpha \right| \leq L,\mathbf{x} \in {\mathbb{R}}^{n}\smallsetminus \{ \mathbf{0}\} }\right\} ,\n\]\n... | Proof: Since \( {\rho }_{m} \) satisfies 33.3.35, and is obviously in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \cap {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \), Theorem 33.3.3 implies there exists a constant \( {A}_{p} \) depending only on \( p, n,{\begin{Vmatrix}{\rho }_{m}\end{Vmatrix}}_{\infty } \), and \( {C}_{1... | Yes |
Lemma 33.4.2 Suppose\n\n\\[ \nK \in {L}^{2}\left( {\\mathbb{R}}^{n}\\right) ,\\parallel {FK}{\\parallel }_{\\infty } \leq B < \\infty ,\n\\]\n\n(33.4.39)\n\nand\n\n\\[ \n{\\int }_{\\left| \\mathbf{x}\\right| > 2\\left| \\mathbf{y}\\right| }\\left| {K\\left( {\\mathbf{x} - \\mathbf{y}}\\right) - K\\left( \\mathbf{x}\\ri... | Proof: Let \\( {FK} = \\rho \\) so \\( {F}^{-1}\\rho = K \\) . Then from 33.4.39 \\( \\rho \\in {L}^{2}\left( {\\mathbb{R}}^{n}\\right) \\cap {L}^{\\infty }\\left( {\\mathbb{R}}^{n}\\right) \\) and \\( K = {F}^{-1}\\rho \\) . By Theorem 33.3.3 listed above,\n\n\\[ \n\\parallel K * f{\\parallel }_{p} = {\\left| \\left| ... | Yes |
Corollary 33.4.4 Suppose 33.4.40 - 33.4.42 hold. Then if \( g \in {C}_{c}^{1}\left( {\mathbb{R}}^{n}\right) ,{K}_{\varepsilon } * g \) converges uniformly and in \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \) as \( \varepsilon \rightarrow 0 \) . | Proof:\n\n\[ \n{K}_{\varepsilon } * g\left( \mathbf{x}\right) \equiv \int {K}_{\varepsilon }\left( \mathbf{y}\right) g\left( {\mathbf{x} - \mathbf{y}}\right) {dy}. \]\n\nLet \( 0 < \eta < \varepsilon \) . Then since \( g \in {C}_{c}^{1}\left( {\mathbb{R}}^{n}\right) \), there exists a constant, \( K \) such that \( K\l... | Yes |
Theorem 33.4.5 Suppose 33.4.40 - 33.4.42. Then for \( {K}_{\varepsilon } \) given by 33.4.43 and \( p > 1 \), there exists a constant \( A\left( {p, n, B}\right) \) such that for all \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) , \[ {\begin{Vmatrix}{K}_{\varepsilon } * f\end{Vmatrix}}_{p} \leq A\left( {p, n, B}\ri... | Proof: From 33.4.40 it follows \( {K}_{\varepsilon } \in {L}^{{p}^{\prime }}\left( {\mathbb{R}}^{n}\right) \cap {L}^{2}\left( {\mathbb{R}}^{n}\right) \) where, as usual, \( 1/p + \) \( 1/{p}^{\prime } = 1 \) . By continuity of translation in \( {L}^{{p}^{\prime }}\left( {\mathbb{R}}^{n}\right), x \rightarrow {K}_{\vare... | Yes |
Lemma 33.5.2 For \( n \geq 2 \)\n\n\[ \n{\Phi }_{,{ij}}\left( \mathbf{y}\right) = \frac{{\Omega }_{ij}\left( \mathbf{y}\right) }{{\left| \mathbf{y}\right| }^{n}}\n\]\n\nwhere\n\n\( {\Omega }_{ij} \) is Lipschitz continuous on \( {S}^{n - 1} \),\n\n(33.5.66)\n\n\[ \n{\Omega }_{ij}\left( {\lambda \mathbf{y}}\right) = {\O... | Proof: The case \( n = 2 \) is left to the reader. 33.5.66 and 33.5.67 are obvious from the above descriptions. It remains to verify 33.5.68. If \( n \geq 3 \) and \( i \neq j \), then this formula is also clear from 33.5.64. Thus consider the case when \( n \geq 3 \) and \( i = j \) . By symmetry,\n\n\[ \nI \equiv {\i... | No |
Lemma 33.5.3 Let \( U \) be a bounded open set in \( {\mathbb{R}}^{n} \) with Lipschitz boundary and let \( B \supseteq U - U \) where \( B = B\left( {\mathbf{0}, R}\right) \) . Let \( f \in {C}_{c}^{\infty }\left( U\right) \) . Then for \( \mathbf{x} \in U \) , \[ {\int }_{B}\Phi \left( \mathbf{y}\right) f\left( {\mat... | and it follows that if \( u \) is given by one of the above formulas, then for all \( x \in U \) , \[ - {\Delta u}\left( \mathbf{x}\right) = f\left( \mathbf{x}\right) \] | No |
Corollary 33.5.6 In the situation of Theorem 33.5.4, all weak derivatives of \( u \) of order 2 are in \( {L}^{p}\left( U\right) \) and also \( f \rightarrow {u}_{,{ij}} \) is a continuous map. | Proof:\n\n\[ \n{u}_{, i}\left( \mathbf{x}\right) = {\int }_{U}{\Phi }_{, i}\left( {\mathbf{x} - \mathbf{y}}\right) f\left( \mathbf{y}\right) {dy} \]\n\nand so \( {u}_{,{ij}} \in {L}^{p}\left( U\right) \) and \( f \rightarrow {u}_{,{ij}} \) is continuous by Lemma 33.5.5. | Yes |
Proposition 34.0.1 Suppose \( V \) is reflexive and a subset of \( H \) a separable Hilbert space with the inclusion map continuous. Suppose also that \( V \) is dense in \( H \) . Then identifying \( H \) and \( {H}^{\prime } \), it follows that \( H \) is dense in \( {V}^{\prime } \) and \( V \) is separable. | Proof: If \( H \) is not dense in \( {V}^{\prime } \), then by the Hahn Banach theorem, there exists \( {\phi }^{* * } \in {V}^{\prime \prime } \) such that \( {\phi }^{* * }\left( H\right) = 0 \) but \( {\phi }^{* * }\left( {\phi }^{ * }\right) \neq 0 \) for some \( {\phi }^{ * } \in {V}^{\prime } \smallsetminus \bar{... | Yes |
Proposition 34.0.3 Denote by \( \mathcal{B}\left( X\right) \) the Borel sets of \( X \) where \( X \) is any separable Banach space. Then\n\n\[ \mathcal{B}\left( X\right) = \sigma \left( {X}^{\prime }\right) \]\n\nHere \( \sigma \left( {X}^{\prime }\right) \) is the smallest \( \sigma \) algebra such that each \( \phi ... | Proof: By Lemma 21.1.6 there exists a countable subset of the unit ball in \( {X}^{\prime } \)\n\n\[ {\left\{ {\phi }_{n}\right\} }_{n = 1}^{\infty } = {D}^{\prime } \]\n\nsuch that\n\n\[ \parallel v{\parallel }_{X} = \sup \left\{ {\left| {\phi \left( v\right) }\right| : \phi \in {D}^{\prime }}\right\} .\n\nConsider a ... | Yes |
Proposition 34.0.4 Let \( X \subseteq Y, X \) dense in \( Y \) and suppose \( X, Y \) are Banach spaces and that \( X \) is reflexive. Then \( X \in \mathcal{B}\left( Y\right) \) . | Proof: Define the functional\n\n\[ \phi \left( x\right) \equiv \left\{ \begin{array}{l} \parallel x{\parallel }_{X}\text{ if }x \in X \\ \infty \text{ if }x \in Y \smallsetminus X \end{array}\right. \]\n\nThen \( \phi \) is lower semicontinuous on \( Y \) . Here is why. Suppose \( \left( {x, a}\right) \notin \operatorn... | Yes |
Lemma 34.1.1 It is possible to consider \( {L}^{p}\left( D\right) \equiv V \) as a dense subspace of \( {\left( {H}_{0}^{1}\right) }^{\prime } \equiv H \) as follows. For \( f \in {L}^{p}\left( D\right) \) and \( \phi \in {H}_{0}^{1}\left( D\right) \) , \n\n\[ \n\langle f,\phi \rangle \equiv {\int }_{D}f\left( x\right)... | Proof: First of all, note that by 34.1.1 \n\n\[ \n\left| {\langle f,\phi \rangle }\right| \leq \parallel f{\parallel }_{{L}^{p}}\parallel \phi {\parallel }_{{L}^{{p}^{\prime }}} \leq C\parallel f{\parallel }_{{L}^{p}}\parallel \phi {\parallel }_{{H}_{0}^{1}} \n\] \n\nand so it is certainly possible to consider \( {L}^{... | Yes |
Lemma 34.2.4 Let \( \bar{f} \) be as defined in Definition 34.2.2. Then for \( f \in {L}^{p}\left( {a, b;X}\right) \) for \( p \in \lbrack 1,\infty ) \) , \[ \mathop{\lim }\limits_{{\delta \rightarrow 0}}{\int }_{a}^{b}\parallel \bar{f}\left( {t - \delta }\right) - f\left( t\right) {\parallel }_{X}^{p}{dt} = 0. \] | Proof: Regarding the measure space as \( \left( {a, b}\right) \) with Lebesgue measure, by regularity of the measure, there exists \( g \in {C}_{c}\left( {a, b;X}\right) \) such that \( \parallel f - g{\parallel }_{p} < \varepsilon \) . Here the norm is the norm in \( {L}^{p}\left( {a, b;X}\right) \) . Therefore, \[ {\... | Yes |
Lemma 34.2.6 The above definition is well defined. | Proof: Suppose both \( h \) and \( g \) work in the definition for \( {f}^{\prime } \) . Then for all \( \phi \in \) \( {C}_{c}^{\infty }\left( {a, b}\right) \)\n\n\[ \n{\int }_{a}^{b}\left( {h\left( t\right) - g\left( t\right) }\right) \phi \left( t\right) {dt} = 0.\n\]\n\nTherefore, by Lemma 34.2.1, \( h\left( t\righ... | Yes |
Lemma 34.2.8 Suppose \( f \in {L}^{1}\left( {a, b;X}\right) \) and for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \), \[ {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} = 0 \] Then there exists a constant, \( a \in X \) such that \( f\left( t\right) = a \) a.e. | Proof: Let \( {\phi }_{0} \in {C}_{c}^{\infty }\left( {a, b}\right) ,{\int }_{a}^{b}{\phi }_{0}\left( x\right) {dx} = 1 \) and define for \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) \[ {\psi }_{\phi }\left( x\right) \equiv {\int }_{a}^{x}\left\lbrack {\phi \left( t\right) - \left( {{\int }_{a}^{b}\phi \left( y... | Yes |
Theorem 34.2.9 Suppose \( f,{f}^{\prime } \) both are in \( {L}^{1}\left( {a, b;X}\right) \) where the derivative is taken in the sense of \( X \) valued distributions. Then there exists a unique point of \( X \) , denoted by \( f\left( a\right) \) such that the following formula holds a.e. \( t \) . \[ f\left( t\right... | Proof: \[ {\int }_{a}^{b}\left( {f\left( t\right) - {\int }_{a}^{t}{f}^{\prime }\left( s\right) {ds}}\right) {\phi }^{\prime }\left( t\right) {dt} = {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} - {\int }_{a}^{b}{\int }_{a}^{t}{f}^{\prime }\left( s\right) {\phi }^{\prime }\left( t\right) {dsdt}.... | Yes |
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