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Proposition 18.2 Let \( {H}_{o} \) be a closed, proper subspace of \( H \) . Then \( H = {H}_{o} \oplus {H}_{o}^{ \bot } \) , i.e., every \( x \in H \) can be represented in a unique way as\n\n\[ x = {x}_{o} + \eta \;\text{ for some }{x}_{o} \in {H}_{o}\;\text{ and }\eta \in {H}_{o}^{ \bot }.\] | Proof If \( x \in {H}_{o} \) it suffices to take \( {x}_{o} = x \) and \( y = \Theta \) . If \( x \in H - {H}_{o} \) let \( {x}_{o} \) be the unique element claimed by Proposition 18.1 and let \( \delta \) be defined as in (18.2). Set now \( x = {x}_{o} + \eta \) where \( \eta = x - {x}_{o} \) . To prove that \( \eta \... | Yes |
Proposition 18.3 For every \( T \in {H}^{ * } \) there exists a unique \( y \in H \), such that \( T \) can be represented as in (18.4). | Proof The conclusion is trivial if \( T \equiv 0 \) . If \( T ≢ 0 \) its kernel \( {H}_{o} \) is a closed, proper subspace of \( H \) . Select a nontrivial element \( \eta \in {H}_{o}^{ \bot } \) and observe that, for all \( x \in H \)\n\n\[ T\left( \eta \right) x - T\left( x\right) \eta \in {H}_{o}\;\text{ i.e.,}\;T\l... | Yes |
Lemma 19.1 Let \( \mathcal{S} \) be an orthonormal system in \( H \) . Any two distinct elements \( x \) and \( y \) in \( \mathcal{S} \) are at mutual distance \( \sqrt{2} \) . | Proof For any \( x, y \in \mathcal{S} \) and \( x \neq y \), compute\n\n\[ \parallel x - y{\parallel }^{2} = \langle x - y, x - y\rangle = \parallel x{\parallel }^{2} - 2\langle x, y\rangle + \parallel y{\parallel }^{2}. \]\n\nThe conclusion follows since \( \parallel x\parallel = \parallel y\parallel = 1 \) . | Yes |
Proposition 19.1 Let \( H \) be a Hilbert space and let \( \mathcal{S} \) be an orthonormal system in H. Then for any \( n \) -tuple \( \left\{ {{\mathbf{u}}_{1},\ldots ,{\mathbf{u}}_{n}}\right\} \) of elements of \( {\mathcal{S}}^{3} \)\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{\left\langle {\mathbf{u}}_{i}, x\right\r... | Proof For any such \( n \) -tuple and any \( x \in H \)\n\n\[ 0 \leq {\begin{Vmatrix}x - \mathop{\sum }\limits_{{i = 1}}^{n}\left\langle {\mathbf{u}}_{i}, x\right\rangle {\mathbf{u}}_{\mathbf{i}}\end{Vmatrix}}^{2} = \left\langle {x - \mathop{\sum }\limits_{{i = 1}}^{n}\left\langle {{\mathbf{u}}_{i}, x}\right\rangle {\m... | Yes |
Proposition 19.2 Let \( H \) be a separable Hilbert space. Then any orthonormal system \( \mathcal{S} \) in \( H \) is countable. | Proof Let \( {H}_{o} \) be a countable subset of \( H \) dense in \( H \) and let \( \mathcal{S} \) be an orthonormal system in \( H \) . For every \( \mathbf{u} \in \mathcal{S} \) there exists \( x\left( \mathbf{u}\right) \in {H}_{o} \) such that\n\n\[ \parallel \mathbf{u} - x\left( \mathbf{u}\right) \parallel < \frac... | Yes |
Proposition 20.1 Let \( \mathcal{S} \) be a complete orthonormal system in \( H \) . Then for every \( x \in H \)\n\n\[ x = \mathop{\sum }\limits_{{\mathbf{u} \in \mathcal{S}}}\langle x,\mathbf{u}\rangle \mathbf{u}\;\text{ (representation of }x\text{ ). } \]\n\n(20.2)\n\nMoreover\n\n\[ \parallel x{\parallel }^{2} = \ma... | Proof As \( \mathbf{u} \) ranges over \( \mathcal{S} \), only countably many of the numbers \( \langle x,\mathbf{u}\rangle \) are not zero and we order them in some fashion \( \left\{ \left\langle {x,{\mathbf{u}}_{n}}\right\rangle \right\} \) . By the Bessel inequality the series \( \sum {\left\langle x,{\mathbf{u}}_{n... | Yes |
Proposition 1.1 A positive, linear functional \( T \) on \( {C}_{o}\left( {\mathbb{R}}^{N}\right) \) is locally bounded. | Proof For a compact set \( K \subset {\mathbb{R}}^{N} \) choose \( \varphi \in {C}_{o}\left( {\mathbb{R}}^{N}\right) \) such that \( 0 \leq \varphi \leq 1 \), and \( \varphi = 1 \) on \( K \) . Given \( f \in {C}_{o}\left( {\mathbb{R}}^{N}\right) \) with \( \operatorname{supp}\{ f\} \subset K \), the two functions \( \... | Yes |
Proposition 2.1 For every open covering \( \mathcal{U} \) of \( E \), there exists a partition of unity for \( E \), subordinate to \( \mathcal{U} \) . | Proof Consider the collection of balls \( {B}_{{r}_{i}}\left( {x}_{j}\right) \) centered at points \( {x}_{j} \in E \) of rational coordinates and rational radii \( {r}_{i} \), and contained in some \( \mathcal{O} \in \mathcal{U} \) . The union of \( {B}_{\frac{1}{2}{r}_{i}}\left( {x}_{j}\right) \) covers \( E \) . For... | Yes |
Lemma 3.1 The set function \( \lambda : \mathcal{Q} \rightarrow {\mathbb{R}}^{ * } \) is monotone, countably sub-additive, and it coincides with \( {\mu }_{e} \) on the open sets. | Proof The monotonicity follows from the definition. Let \( \left\{ {\mathcal{O}}_{n}\right\} \) be a countable collection of open sets in \( {\mathbb{R}}^{N} \) and let \( \mathcal{O} = \cup {\mathcal{O}}_{n} \) . For \( f \in {\Gamma }_{\mathcal{O}} \), the collection \( \left\{ {\mathcal{O}}_{n}\right\} \) is an open... | Yes |
Proposition 3.1 The open sets are \( \mu \) -measurable. | Proof An open set \( \mathcal{O} \) is \( \mu \) -measurable if it satisfies the Carathéodory condition, for all sets \( A \subset {\mathbb{R}}^{N} \) of finite outer measure. Assume first that \( A \) is itself open. Then \( A \cap \mathcal{O} \) is open and from the definition of \( \lambda \left( \cdot \right) \), f... | Yes |
Proposition 4.1 The functional \( {T}_{ + } : {C}_{o}{\left( {\mathbb{R}}^{N}\right) }^{ + } \rightarrow {\mathbb{R}}^{ + } \) is positive and linear. | Proof One verifies that \( {T}_{ + }\left( {\alpha f}\right) = \alpha {T}_{ + }\left( f\right) \), for all \( f \in {C}_{o}{\left( {\mathbb{R}}^{N}\right) }^{ + } \) and \( \alpha > 0 \) . To show that \( {T}_{ + } \) is linear, fix two nonnegative functions \( {f}_{1} \) and \( {f}_{2} \) in \( {C}_{o}\left( {\mathbb{... | Yes |
Proposition 4.2 Let \( T \) be a locally bounded linear functional on \( {C}_{o}\left( {\mathbb{R}}^{N}\right) \) and let \( \mu \) be its corresponding, previously constructed Radon measure. Then for every compact set \( K \subset {\mathbb{R}}^{N} \)\n\n\[ \mu \left( K\right) = \mathop{\inf }\limits_{{f \in {\Gamma }_... | Proof Since \( K \) is a Borel set, \( {\mu }_{e}\left( K\right) = \mu \left( K\right) \) and\n\n\[ \mu \left( K\right) \leq \inf \{ \lambda \left( \mathcal{O}\right) \mid \mathcal{O}\text{ open and }K \subset \mathcal{O}\} . \]\n\nFix \( f \in {\Gamma }_{K} \) and \( \varepsilon \in \left( {0,1}\right) \), and conside... | Yes |
Corollary 6.1 Let \( T : {C}_{o}\left( {\mathbb{R}}^{N}\right) \rightarrow \mathbb{R} \) be linear and locally bounded. There exist two Radon measures \( {\mu }_{1} \) and \( {\mu }_{2} \) in \( {\mathbb{R}}^{N} \), such that \[ T\left( f\right) = {\int }_{{\mathbb{R}}^{N}}{fd}{\mu }_{1} - {\int }_{{\mathbb{R}}^{N}}{fd... | Moreover the restrictions of \( {\mu }_{i} \) to the Borel \( \sigma \) -algebra \( \mathcal{B} \) is unique. The two Radon measures need not be finite. However they are finite on bounded sets and the representation formula is well defined since \( f \in {C}_{o}\left( {\mathbb{R}}^{N}\right) \) . If \( T \) is linear a... | Yes |
Lemma 7.1 For any \( f \in {\mathcal{O}}_{j} \), there exists an index \( {\ell }_{j} \) such that \( f + {\mathcal{O}}_{\ell } \subset {\mathcal{O}}_{j} \) for all \( \ell \geq {\ell }_{j} \) . | Proof Having fixed \( f \in {\mathcal{O}}_{j} \), there exists \( \varepsilon > 0 \) such that\n\n\[ \n{p}_{j}\left( f\right) \leq \frac{1}{j + 1} - \varepsilon \n\]\n\nLet \( {\ell }_{j} \) be a positive integer satisfying \( {\ell }_{j} \geq \max \left\{ {j + 1;{\varepsilon }^{-1}}\right\} \) . Then for every \( g \i... | Yes |
Proposition 7.1 The collection \( \mathcal{B} = \left\{ {B}_{\varphi, j}\right\} \) as \( \varphi \) ranges over \( {C}_{o}^{\infty }\left( E\right) \) and \( j = \) \( 0,1,\ldots \), forms a base for a topology \( \mathcal{U} \) of \( {C}_{o}^{\infty }\left( E\right) \) . The topology \( \mathcal{U} \) satisfies the f... | Proof We verify that \( \mathcal{B} \) satisfies the requirements (i)-(ii) of \( §4 \) of Chap. 2 to be a base for a topology. Let \( {B}_{\varphi, i} \) and \( {B}_{\psi, j} \) be out of \( \mathcal{B} \), and with non-empty intersection. Fix \( \eta \in {B}_{\varphi, i} \cap {B}_{\psi, j} \), so that \( \eta - \varph... | Yes |
Proposition 7.2 \( D\left( E\right) \) is a topological vector space. | Proof Let \( {\varphi }_{1},{\varphi }_{2} \in D\left( E\right) \) and let \( U \) be an open set containing \( {\varphi }_{1} + {\varphi }_{2} \) . There exists an open neighborhood of the origin \( {\mathcal{O}}_{j} \) such that \( {\varphi }_{1} + {\varphi }_{2} + {\mathcal{O}}_{j} \subset U \) . Since \( {\mathcal{... | Yes |
Proposition 10.1 (Schwartz [141]) The collection \( \mathcal{V} = \{ \varphi + V\} \) as \( \varphi \) ranges over \( {C}_{o}^{\infty }\left( E\right) \) and \( V \) ranges over \( {\mathcal{V}}_{o} \), forms a base for a topology \( \mathcal{W} \) on \( {C}_{o}^{\infty }\left( E\right) \) . | Proof Let \( {\varphi }_{1} + {V}_{1} \) and \( {\varphi }_{2} + {V}_{2} \) be any two elements of \( \mathcal{V} \) with non empty intersection. Choose\n\n\[ \eta \in \left( {{\varphi }_{1} + {V}_{1}}\right) \cap \left( {{\varphi }_{2} + {V}_{2}}\right) \]\n\nand let \( K \) be a compact subset of \( E \) such that\n\... | Yes |
Proposition 10.2 The restriction of \( \mathcal{W} \) to \( \mathcal{D}\left( K\right) \) is \( {\mathcal{U}}_{K} \) . | Proof The inclusion \( \mathcal{U} \subset \mathcal{W} \), implies the inclusion \( {\mathcal{U}}_{K} \subset \mathcal{W} \cap \mathcal{D}\left( K\right) \) . For the converse inclusion, let \( W \in \mathcal{W} \) and select \( \varphi \in W \cap \mathcal{D}\left( K\right) \) . There exists \( V \in {\mathcal{V}}_{o} ... | Yes |
Proposition 11.1 Let \( B \) be a bounded subset of \( \mathcal{D}\left( E\right) \). There exists a compact subset \( K \subset E \), such that \( B \subset \mathcal{D}\left( K\right) \). | Proof Assuming that such a \( K \) does not exists, there exists be a countable collection \( \left\{ {K}_{n}\right\} \) of compact, nested subsets of \( E \), exhausting \( E \), a sequence of functions \( \left\{ {f}_{n}\right\} \subset \) \( B \), and a sequence of points \( \left\{ {x}_{n}\right\} \), such that \( ... | Yes |
Corollary 11.3 The inclusion \( \mathcal{U} \subset \mathcal{W} \) is strict. | Proof If \( \mathcal{W} = \mathcal{U} \), the topology \( \mathcal{W} \) would be metric. | No |
Proposition 12.2 A linear map \( T : \mathcal{D}\left( E\right) \rightarrow \mathcal{D}\left( E\right) \) is continuous if and only if is bounded. | The statement holds true for maps \( T \) between metric spaces. The proof consists in establishing that if \( T \) is bounded, it is restricted to some \( \mathcal{D}\left( K\right) \), which is a metric space.\n\nProof Continuity of \( T \) implies \( T \) is bounded. To prove the converse, if \( B \subset \mathcal{D... | Yes |
Corollary 12.2 The differentiation maps \( {D}^{\alpha } : \mathcal{D}\left( E\right) \rightarrow \mathcal{D}\left( E\right) \) are continuous for every multi-index \( \alpha \) . | Proof Let \( B \subset \mathcal{D}\left( E\right) \) be bounded and let \( {\lambda }_{j} \) be the positive numbers claimed by the characterization (12.1) of the bounded subsets of \( \mathcal{D}\left( E\right) \) . Then\n\n\[ \n{p}_{K;j}\left( {{D}^{\alpha }f}\right) < {\lambda }_{j + \left| \alpha \right| }.\n\]\n\n... | Yes |
Proposition 14.1 (Stokes Formula) \( {}^{1} \) For all \( \varphi \in {C}_{o}^{\infty }\left( {\mathbb{R}}^{N}\right) \) and all \( x \in {\mathbb{R}}^{N} \)\n\n\[ \varphi \left( x\right) = - {\int }_{{\mathbb{R}}^{N}}F\left( {x;y}\right) {\Delta \varphi dy}. \] | Proof For a fixed \( x \in {\mathbb{R}}^{N} \), the function \( y \rightarrow F\left( {x;y}\right) \) is integrable about \( x \) . Therefore\n\n\[ {\int }_{{\mathbb{R}}^{N}}F\left( {x;y}\right) {\Delta \varphi dy} = \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{\int }_{{\mathbb{R}}^{N} - {B}_{\varepsilon }\left(... | Yes |
Proposition 15.1 \( {W}^{m, p}\left( E\right) \) is a Banach space. | Proof If \( \left\{ {u}_{n}\right\} \) is a Cauchy sequence in \( {W}^{m, p}\left( E\right) \), the sequences \( \left\{ {{D}^{\alpha }{u}_{n}}\right\} \) are Cauchy in \( {L}^{p}\left( E\right) \) for all multi-indices \( 0 \leq \left| \alpha \right| \leq m \) . By the completeness of \( {L}^{p}\left( E\right) \), the... | Yes |
Theorem 15.1 (Meyers-Serrin [107]) Let \( 1 \leq p < \infty \) . Then \( {C}^{\infty }\left( E\right) \) is dense in \( {W}^{m, p}\left( E\right) \), and as a consequence \( {H}^{m, p}\left( E\right) = {W}^{m, p}\left( E\right) \) . | Proof Having chosen \( u \in {W}^{m, p}\left( E\right) \) and \( \varepsilon \in \left( {0,1}\right) \), we exhibit a function \( \varphi \in \) \( {C}^{\infty }\left( E\right) \) such that \( \parallel u - \varphi {\parallel }_{m, p} < \varepsilon \) . For \( j = 1,2,\ldots \), set\n\n\[ \n{E}_{j} = \left\{ {x \in E \... | Yes |
Proposition 18.1 ([93]) Let \( E \) be a bounded domain in \( {\mathbb{R}}^{N} \) with boundary \( \partial E \) satisfying the segment property and of class \( {C}^{1} \), and let \( {\left\{ {B}_{t}\left( {x}_{j}\right) \right\} }_{j = 1}^{n} \) be a finite open covering of \( \partial E \) with balls of radius \( t ... | Proof By density we may assume \( u \in {C}^{1}\left( \bar{E}\right) \) . Consider first the following special case. For \( \bar{x} = \left( {{x}_{1},\ldots ,{x}_{N - 1}}\right) \) and \( R > 0 \) let \( {\mathcal{B}}_{R} = \left\lbrack {\left| \bar{x}\right| < R}\right\rbrack \) be the \( \left( {N - 1}\right) \) - di... | Yes |
Proposition 19.1 Let \( u \in {W}^{1, p}\left( E\right) \) for some \( 1 \leq p < \infty \), and let \( f \in {C}^{1}\left( \mathbb{R}\right) \) satisfy \( \sup \left| {f}^{\prime }\right| \leq M \) for some positive constant \( M \) . Then the composition \( f\left( u\right) \) belongs to \( {W}^{1, p}\left( E\right) ... | Proof Let \( {C}^{\infty }\left( E\right) \supset \left\{ {u}_{n}\right\} \rightarrow u \) in \( {W}^{1, p}\left( E\right) \) . By possibly passing to a subsequence we may assume that \( \left\{ {u}_{n}\right\} \rightarrow u \) a.e. in \( E \) . Then \( \left\{ {f\left( {u}_{n}\right) }\right\} \rightarrow f\left( u\ri... | Yes |
Proposition 19.2 Let \( u \in {W}^{1, p}\left( E\right) \) for some \( 1 \leq p < \infty \) . Then \( {u}^{ + },{u}^{ - } \) and \( \left| u\right| \) belong to \( {W}^{1, p}\left( E\right) \) and\n\n\[ D{u}^{ + } = \left\{ {\begin{matrix} {Du} & \text{ a.e. }\left\lbrack {u > 0}\right\rbrack \\ 0 & \text{ a.e. }\left\... | Proof For \( \varepsilon > 0 \) let\n\n\[ {f}_{\varepsilon }\left( u\right) = \left\{ \begin{array}{ll} \sqrt{{u}^{2} + {\varepsilon }^{2}} - \varepsilon & \text{ if }u > 0 \\ 0 & \text{ if }u \leq 0. \end{array}\right.\n\]\n\nThen \( {f}_{\varepsilon } \in {C}^{1}\left( \mathbb{R}\right) \) and \( \left| {f}_{\varepsi... | Yes |
Corollary 19.2 Let \( f, g \in {W}^{1, p}\left( E\right) \) . Then \( \max \{ f;g\} \) and \( \min \{ f;g\} \) are in \( {W}^{1, p}\left( E\right) \) | Proof It follows from Proposition 19.2 and the formulae\n\n\[ 2\max \{ f;g\} = \left( {f + g}\right) + \left| {f - g}\right| \]\n\n\[ 2\min \{ f;g\} = \left( {f + g}\right) - \left| {f - g}\right| \text{.} \] | Yes |
Proposition 20.1 Let \( u \in {L}^{p}\left( E\right) \) for some \( p \in \lbrack 1,\infty ) \) . Then \( {u}_{h, i} \rightarrow u \) in \( {L}^{p}\left( E\right) \) as \( h \rightarrow 0 \) . | Proof For almost all \( x \in E \)\n\n\[ \left| {{u}_{h, i}\left( x\right) - u\left( x\right) }\right| = \left| {\frac{1}{h}{\int }_{{x}_{i}}^{{x}_{i} + h}\left\lbrack {u\left( {\ldots ,{\xi }_{i},\ldots }\right) - u\left( x\right) }\right\rbrack d{\xi }_{i}}\right| \]\n\n\[ = \left| {\frac{1}{h}{\int }_{0}^{h}\left\lb... | Yes |
Proposition 20.2 Let \( u \in {L}^{p}\left( E\right) \) for some \( 1 < p < \infty \) and assume that there exists positive constant \( {C}_{p} \) and \( {\delta }_{o} \), such that\n\n\[{\begin{Vmatrix}{\mathbf{w}}_{\mathbf{h}}\end{Vmatrix}}_{p,{E}_{\delta }} \leq {C}_{p}\;\text{ for all }\delta \leq {\delta }_{o}\tex... | Proof Fix \( \delta \in \left( {0,{\delta }_{o}}\right) \) and \( \varphi \in {C}_{o}^{\infty }\left( {E}_{\delta }\right) \) . For fixed \( i \in \{ 1,\ldots, N\} \)\n\n\[ \mathop{\lim }\limits_{{{h}_{i} \rightarrow 0}}{\int }_{E}{w}_{{h}_{i}, i}{\varphi dx} = - \mathop{\lim }\limits_{{{h}_{i} \rightarrow 0}}{\int }_{... | Yes |
Proposition 20.3 A function \( u \) belongs to \( {W}^{1, p}\left( E\right) \) for some \( 1 < p < \infty \), if and only if there exists a positive number \( {\delta }_{o} \) and a vector valued function \( \mathbf{w} \in {L}^{p}\left( E\right) \) such that\n\n\[ \parallel u\left( {\cdot + \mathbf{h}}\right) - u - \ma... | Proof (Sufficient Condition) Let \( u \in {L}^{p}\left( E\right) \) satisfy (20.1) and for \( \left| h\right| > 0 \) let \( {\mathbf{w}}_{\mathbf{h}} \) denote its discrete gradient. Fix \( \delta > 0 \) and choose \( \mathbf{h} = \left( {0,\ldots ,{h}_{i},\ldots 0}\right) \) with \( 0 < \) \( \left| {h}_{i}\right| < \... | Yes |
Proposition 21.1 Let \( f : {\mathbb{R}}^{N} \rightarrow \mathbb{R} \) be locally Lipschitz continuous. Then \( {D}_{\mathbf{u}}f \) exists a.e. in \( {\mathbb{R}}^{N} \). In particular \( \nabla f \) exists a.e. in \( {\mathbb{R}}^{N} \), and \( {D}_{\mathbf{u}}f = \mathbf{u} \cdot \nabla f \), a.e. in \( {\mathbb{R}}... | Proof Let \( {E}_{\mathbf{u}} = \left\lbrack {{D}_{\mathbf{u}}^{\prime }f < {D}_{\mathbf{u}}^{\prime \prime }f}\right\rbrack \), be the set where \( {D}_{\mathbf{u}}f \) does not exist. Since \( {D}_{\mathbf{u}}^{\prime }f \) and \( {D}_{\mathbf{u}}^{\prime \prime }f \) are measurable \( {E}_{\mathbf{u}} \) is measurab... | Yes |
Theorem 21.1 (Rademacher [120]) Let \( f : {\mathbb{R}}^{N} \rightarrow \mathbb{R} \) be locally Lipschitz continuous. Then \( f \) is a.e. differentiable in \( {\mathbb{R}}^{N} \) . | Proof Let \( \left\{ {\mathbf{u}}_{n}\right\} \) be a countable dense subset of the unit sphere in \( {\mathbb{R}}^{N} \), and let \( {E}_{{\mathbf{u}}_{n}} = \) \( \left\lbrack {{D}_{{\mathbf{u}}_{n}}^{\prime }f < {D}^{\prime \prime }{\mathbf{u}}_{n}f}\right\rbrack \), that is the set where the directional derivative ... | Yes |
Corollary 1.1 Let \( E \subset {\mathbb{R}}^{N} \) be measurable and of finite measure, and let \( \mathcal{F} \) be a collection of cubes in \( {\mathbb{R}}^{N} \), with faces parallel to the coordinate planes and covering E. For every \( \varepsilon > 0 \), there exists a finite collection \( \left\{ {{Q}_{1},\ldots ... | \[ \mu \left( E\right) - \varepsilon \leq {5}^{N}\mathop{\bigcup }\limits_{{j = 1}}^{m}\mu \left( {Q}_{j}\right) \] | No |
Proposition 2.1 \( M\left( f\right) \) is measurable and lower semi-continuous. Moreover, if \( f \) is the characteristic function of a bounded measurable set \( E \), there exists positive constants \( {C}_{o},{C}_{1} \), and \( \gamma \), depending only upon \( E \), such that\n\n\[ \frac{{C}_{o}}{{\left| x\right| }... | Proof If \( \left| f\right| = 0 \), also \( M\left( f\right) = 0 \) . Otherwise \( M\left( f\right) \left( x\right) > 0 \) for all \( x \in {\mathbb{R}}^{N} \) . Let \( c > 0 \) and \( x \in \left\lbrack {M\left( f\right) > c}\right\rbrack \) . There exist \( \varepsilon > 0 \) and a cube \( Q \), such that\n\n\[ \frac... | Yes |
Corollary 2.1 Let \( f \in {L}^{1}\left( {\mathbb{R}}^{N}\right) \) be of compact support in \( {\mathbb{R}}^{N} \) and not identically zero. There exists positive constants \( {C}_{o},{C}_{1} \) and \( \gamma \), depending only upon \( f \), such that \[ \frac{{C}_{o}}{{\left| x\right| }^{N}} \leq M\left( f\right) \le... | It follows from (2.1) and (2.2) that \( M\left( f\right) \) is not in \( {L}^{1}\left( {\mathbb{R}}^{N}\right) \) even if \( f \) is bounded and compactly supported, unless \( f = 0 \) . | No |
Proposition 2.2 Let \( f \in {L}^{1}\left( {\mathbb{R}}^{N}\right) \) . Then for all \( t > 0 \)\n\n\[ \mu \left( \left\lbrack {M\left( f\right) > t}\right\rbrack \right) \leq \frac{{5}^{N}}{t}{\int }_{{\mathbb{R}}^{N}}\left| f\right| {dx}. \]\n\n(2.3) | Proof Assume first that \( f \) is of compact support. Then (2.2) implies that \( \left\lbrack {M\left( f\right) > t}\right\rbrack \) is of finite measure. For every \( x \in \left\lbrack {M\left( f\right) > t}\right\rbrack \), there exists a cube \( Q\left( x\right) \) centered at \( x \) and with faces parallel to th... | Yes |
Proposition 3.1 Let \( f \in {L}^{p}\left( {\mathbb{R}}^{N}\right) \) for some \( p \in (1,\infty \rbrack \) . Then the maximal function \( M\left( f\right) \) is in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) and\n\n\[ \parallel M\left( f\right) {\parallel }_{p} \leq {\gamma }_{p}\parallel f{\parallel }_{p}\;\text{ wh... | Proof The estimate is obvious for \( p = \infty \) . Assuming then \( p \in \left( {1,\infty }\right) \) fix \( t > 0 \) and\n\nset\n\[ \mathbf{g}\left( x\right) = \left\{ \begin{array}{l} f\left( x\right) \text{ if }\left| {f\left( x\right) }\right| \geq \frac{1}{2}t \\ 0\;\text{ if }\left| {f\left( x\right) }\right| ... | Yes |
Theorem 4.1 Let \( f \) be a nonnegative function in \( {L}^{1}\left( {\mathbb{R}}^{N}\right) \) . Then, for any fixed \( \alpha > 0 \) , \( {\mathbb{R}}^{N} \) can be partitioned into two disjoint sets \( E \) and \( F \), such that\n\n(i) \( f \leq \alpha \) a.e. in \( E \) ;\n\n(ii) \( F \) is the countable union of... | Proof Let \( \alpha > 0 \) be fixed and partition \( {\mathbb{R}}^{N} \) into closed cubes with pairwise disjoint interior, with faces parallel to the coordinate planes, and of equal edge. Since \( f \in \) \( {L}^{1}\left( {\mathbb{R}}^{N}\right) \), such a partition can be realized so that for every cube \( {Q}^{\pri... | Yes |
Proposition 5.1 If \( f \in {BMO}\left( {Q}_{o}\right) \) then for all cubes \( Q \subset {Q}_{o} \)\n\n\[ f - {f}_{Q} \in {L}^{p}\left( Q\right) \;\text{ and }\;f \in {L}^{p}\left( Q\right) \;\text{ for all }1 \leq p < \infty . \]\n | Proof From (5.2)\n\n\[ {\int }_{Q}{\left| f - {f}_{Q}\right| }^{p}{dy} = p{\int }_{0}^{\infty }{t}^{p - 1}\mu \left( {\left\lbrack {\left| {f - {f}_{Q}}\right| > t}\right\rbrack \cap Q}\right) {dt} \]\n\n\[ \leq p{C}_{1}\mu \left( Q\right) {\int }_{0}^{\infty }{t}^{p - 1}\exp \left\{ {-\frac{{C}_{2}t}{{\left| f\right| ... | Yes |
Proposition 5.2 Let \( f \in {L}^{1}\left( {Q}_{o}\right) \) and assume that for every sub-cube \( Q \subset {Q}_{o} \) there is a constant \( {\gamma }_{Q} \) such that\n\n\[ \mu \left( {\left\lbrack {\left| {f - {\gamma }_{Q}}\right| > t}\right\rbrack \cap Q}\right) \leq {\gamma }_{1}\exp \left\{ {-{\gamma }_{2}t}\ri... | Proof Assume first that \( {\gamma }_{Q} = {f}_{Q} \) . Then\n\n\[ {\int }_{Q}\left| {f - {f}_{Q}}\right| {dy} = {\int }_{0}^{\infty }\mu \left( {\left\lbrack {\left| {f - {f}_{Q}}\right| > t}\right\rbrack \cap Q}\right) {dt} \]\n\n\[ \leq {\gamma }_{1}\mu \left( Q\right) {\int }_{0}^{\infty }{e}^{-{\gamma }_{2}t}{dt} ... | Yes |
Proposition 5.3 The function \( x \rightarrow \ln \left| x\right| \) is of bounded mean oscillation in the unit cube \( {Q}_{o} \), centered at the origin of \( {\mathbb{R}}^{N} \) . | Proof Having fixed \( Q \in {Q}_{o} \), let \( \xi \) be the element of largest Euclidean length in \( Q \) and set \( {\gamma }_{Q} = \ln \left| \xi \right| \) . Then\n\n\[ \n{\sum }_{t} = \left\{ {x \in Q\left| \right| \ln \left| x\right| - {\gamma }_{Q} \mid > t}\right\} \n\]\n\n\[ \n= \left\{ {x \in Q\left| {\;\ln ... | Yes |
Theorem 7.1 (Fefferman-Stein [45]) Let \( f \in {L}^{1}\left( {Q}_{o}\right) \) and assume the corresponding sharp maximal function \( {f}^{\# } \) is in \( {L}^{p}\left( {Q}_{o}\right) \) . Then \( f \in {L}^{p}\left( {Q}_{o}\right) \) and there exists a positive constant \( \gamma = \gamma \left( {N, p}\right) \) dep... | ## 8 Proof of the Fefferman-Stein Theorem\n\nFix \( t > {\left| f\right| }_{{Q}_{o}} \) and apply the Calderón-Zygmund decomposition to the function \( \left| f\right| \) , for \( \alpha = t \) . This generates a countable collection \( \left\{ {Q}_{n}^{t}\right\} \) of closed cubes with faces parallel to the coordinat... | Yes |
Lemma 8.1 Let \( t > {2}^{N + 1}{\left| f\right| }_{{Q}_{o}} \) . Then\n\n\[ m\left( t\right) \leq \mu \left( {\left\lbrack {{f}^{\# } > {t\delta }}\right\rbrack \cap {Q}_{o}}\right) + {2\delta m}\left( {t{2}^{-\left( {N + 1}\right) }}\right) \]\n\nwhere \( \delta \) is an arbitrary positive constant. | Proof Set \( \tau = t{2}^{-\left( {N + 1}\right) } \) and determine the two countable families of cubes \( \left\{ {Q}_{n}^{t}\right\} \) and \( \left\{ {Q}_{j}^{\tau }\right\} \) satisfying \( {\left( {8.1}\right) }_{t} \) and \( {\left( {8.1}\right) }_{\tau } \), respectively. Fix one of the cubes \( {Q}_{j}^{\tau } ... | Yes |
Theorem 9.1 (Marcinkiewicz [103]) Let \( T \) be a quasi-linear map defined both in \( {L}^{p}\left( E\right) \) and \( {L}^{q}\left( E\right) \) for some pair \( 1 \leq p < q \leq \infty \) . Assume that \( T \) is both of weak type \( \left( {p, p}\right) \) and of weak type \( \left( {q, q}\right) \), i.e., there ex... | ## 10 Proof of the Marcinkiewicz Theorem\n\nHaving fixed \( r \in \left( {p, q}\right) \), and some \( f \in {L}^{r}\left( E\right) \), decompose it as \( f = {f}_{1} + {f}_{2} \), where\n\n\[ \n{f}_{1} = \left\{ {\begin{array}{l} f\text{ for }\left| f\right| > {\lambda t}; \\ 0\text{ for }\left| f\right| \leq {\lambda... | Yes |
Proposition 11.1 Let \( f \) and \( g \) be real-valued, nonnegative, measurable functions satisfying (11.4). Then\n\n(i) \( {f}^{ * } \) is nonnegative, radially symmetric and nonincreasing;\n\n(ii) \( f \leq g \) implies \( {f}^{ * } \leq {g}^{ * } \) ;\n\n(iii) Let \( F : {\mathbb{R}}^{ + } \rightarrow {\mathbb{R}}^... | Proof The statements (i)-(vi) follow from the construction of \( {f}^{ * } \) leading to (11.5). As for (vii), the statement is obvious if \( p = \infty \) . If \( 1 \leq p < \infty \), since \( f \) and \( {f}^{ * } \) are equi-measurable\n\n\[ \parallel f{\parallel }_{p}^{p} = {\int }_{0}^{\infty }{t}^{p - 1}\mu \lef... | Yes |
Proposition 12.1 Let \( f \) and \( g \) be real-valued, nonnegative, measurable functions in \( {\mathbb{R}}^{N} \) satisfying (11.4). Then\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{fgd\mu } \leq {\int }_{{\mathbb{R}}^{N}}{f}^{ * }{\mathbf{g}}^{ * }{d\mu } \n\] | Proof Assume first that \( f \) and \( \mathbf{g} \) are simple and both take only the values 0 and 1 . Set\n\n\[ \nE = \left\lbrack {f = 1}\right\rbrack ,{E}^{ * } = \left\lbrack {{f}^{ * } = 1}\right\rbrack \n\]\n\n\[ \nG = \left\lbrack {g = 1}\right\rbrack ,{G}^{ * } = \left\lbrack {{g}^{ * } = 1}\right\rbrack . \n\... | Yes |
Corollary 12.1 Let \( f \) and \( g \) be real-valued, nonnegative, measurable functions in \( {\mathbb{R}}^{N} \) satisfying (11.4). Then\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}f{\chi }_{\left\lbrack g \leq t\right\rbrack }{d\mu } \geq {\int }_{{\mathbb{R}}^{N}}{f}^{ * }{\chi }_{\left\lbrack {g}^{ * } \leq t\right\rbrack }... | Proof Assume first \( f \in {L}^{1}\left( {\mathbb{R}}^{N}\right) \), and apply (12.1) to the pair of functions \( f \) and \( {\chi }_{\left\lbrack g > t\right\rbrack } \). Writing the latter as \( 1 - {\chi }_{\left\lbrack g \leq t\right\rbrack } \) gives\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}f\left( {1 - {\chi }_{\left\... | Yes |
Theorem 12.1 (Chiti [27]; also in [29]) Let \( f \) and \( g \) be nonnegative functions in \( {L}^{p}\left( {\mathbb{R}}^{N}\right) \) for \( 1 \leq p \leq \infty \) . If \( p = \infty \) assume also that \( f \) and \( \mathbf{g} \) satisfy (11.4). Then\n\n\[ \n{\begin{Vmatrix}{f}^{ * } - {\mathbf{g}}^{ * }\end{Vmatr... | Proof The inequality is obvious for \( p = \infty \) . Let \( 1 \leq p < \infty \) and assume first that \( f \geq g \) . Compute\n\n\[ \n{\left( f - g\right) }^{p} = - {\int }_{g}^{f}\frac{d}{dt}{\left( f - t\right) }^{p}{dt} \n\]\n\n\[ \n= p{\int }_{g}^{f}{\left( f - t\right) }^{p - 1}{dt} \n\]\n\n\[ \n= p{\int }_{0}... | Yes |
Proposition 12.2 Let \( E \subset {\mathbb{R}}^{N} \) be measurable and of finite measure and let \( f \) be nonnegative, radially symmetric, and strictly decreasing in \( \left| x\right| \) . If\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}f{\chi }_{E}{dx} = {\int }_{{\mathbb{R}}^{N}}{f}^{ * }{\chi }_{{E}^{ * }}{dx}\n\]\n\n(12.4... | Proof The assumptions on \( f \) imply that \( f = {f}^{ * } \) and \( f > 0 \) in \( {\mathbb{R}}^{N} \) . The sets \( \left\lbrack {f > t}\right\rbrack \) are balls of radius \( \rho \left( t\right) \) about the origin, for some positive function \( \rho \left( \cdot \right) \) . While \( f \) may exhibit jump discon... | Yes |
Corollary 12.2 Let \( g \) be a real-valued, nonnegative, measurable function in \( {\mathbb{R}}^{N} \) satisfying (11.4), and let \( f \) be nonnegative, radially symmetric, and strictly decreasing in \( \left| x\right| \) . Then (12.1) holds with equality if and only if \( \mathbf{g} = {\mathbf{g}}^{ * } \) . | Proof For all \( t > 0 \) by (12.1)\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}f{\chi }_{\left\lbrack g > t\right\rbrack }{dx} \leq {\int }_{{\mathbb{R}}^{N}}f{\chi }_{\left\lbrack {g}^{ * } > t\right\rbrack }{dx}.\n\]\n\nIntegrating in \( {dt} \) over \( {\mathbb{R}}^{ + } \) ,\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{fgdx} = {\int ... | Yes |
Theorem 13.1 (Riesz [131]; also Zygmund [178]) Let \( f, g \) and \( h \) be real-valued, nonnegative, measurable functions in \( {\mathbb{R}}^{N} \) satisfying (11.4). Then\n\n\[ \mathcal{I} = {\int }_{{\mathbb{R}}^{N}}{\int }_{{\mathbb{R}}^{N}}f\left( x\right) \mathbf{g}\left( y\right) h\left( {y - x}\right) {dxdy} \... | Since \( f,\mathbf{g} \) and \( h \) are measurable and nonnegative, the integrals in (13.1), finite or infinite are well defined, and the inequality holds with the understanding that if \( \mathcal{I} = \infty \) then also \( {\mathcal{I}}^{ * } = \infty \) . | Yes |
Lemma 14.1 \( 0 \leq \Gamma \left( x\right) - {\Gamma }_{1}\left( x\right) \leq 1 \), for all \( x \in \mathbb{R} \) . | ## 14.4 Proof of the Lemma 14.1\n\nIt is apparent that such a function is affine within any interval of the form \( \left( {n, n + 1}\right) \) for integral \( n \) . Therefore it must take its extrema for some integral value of \( x \) . If \( x \) is an integer, the set \( {H}_{x} \) is the finite union of unit inter... | Yes |
Proposition 15.1 (Hardy [66]) Let \( f \in {L}^{p}\left( {\mathbb{R}}^{ + }\right) \) for some \( p > 1 \), be nonnegative.\n\nThen\n\[{\int }_{0}^{\infty }\frac{1}{{x}^{p}}{\left( {\int }_{0}^{x}f\left( t\right) dt\right) }^{p}{dx} \leq {\left( \frac{p}{p - 1}\right) }^{p}{\int }_{0}^{\infty }{f}^{p}{dx}.\] | Proof Fix \( 0 < \xi < \eta < \infty \) . Then, by integration by parts\n\n\[{\int }_{\xi }^{\eta }\frac{1}{{x}^{p}}{\left( {\int }_{0}^{x}f\left( t\right) dt\right) }^{p}{dx} = \frac{-1}{p - 1}{\int }_{\xi }^{\eta }{\left( {\int }_{0}^{x}f\left( t\right) dt\right) }^{p}\frac{d}{dx}{x}^{1 - p}{dx}\]\n\n\[= \frac{{\xi }... | Yes |
Lemma 17.1 Let \( t \rightarrow u\left( t\right), v\left( t\right) \) be nonnegative and measurable on a measurable subset \( E \subset \mathbb{R} \) of finite measure. Assume in addition that \( u \) is nondecreasing and \( v \) is nonincreasing. Then\n\n\[ \n{\int }_{E}{uvdt} \leq \frac{1}{\mu \left( E\right) }\left(... | Proof By the stated monotonicity of \( u \) and \( v \) ,\n\n\[ \n{\int }_{E}{\int }_{E}\left( {u\left( x\right) - u\left( y\right) }\right) \left( {v\left( y\right) - v\left( x\right) }\right) {dxdy} \geq 0. \n\]\n\nFrom this\n\n\[ \n{\int }_{E}{\int }_{E}u\left( x\right) v\left( y\right) {dxdy} + {\int }_{E}{\int }_{... | Yes |
Theorem 18.1 ([67,68,148]) Let \( f \) and \( g \) be nonnegative measurable functions in \( {\mathbb{R}}^{N} \) and let \( p, q > 1 \) and \( \sigma \in \left( {0, N}\right) \) be linked by\n\n\[ \frac{1}{p} + \frac{1}{q} + \frac{\sigma }{N} = 2 \]\n\nThere exists a constant \( C\left( {p, q,\sigma, N}\right) \) depen... | ## 18.1 Proof of Theorem 18.1\n\nThe arithmetic mean of \( N \) positive numbers is more than the corresponding geometric mean (§ 14.1c of Chap. 5). Therefore\n\n\[ \left| {x - y}\right| = {\left( \mathop{\sum }\limits_{{i = 1}}^{N}{\left( {x}_{i} - {y}_{i}\right) }^{2}\right) }^{\frac{1}{2}} \geq \sqrt{N}\mathop{\prod... | Yes |
Theorem 19.1 There exists a constant \( {C}_{N} \) depending only upon \( \sigma, N \) and \( p \), such that \[ \parallel h{\parallel }_{{p}^{ * }} \leq {C}_{N}\parallel f{\parallel }_{p}\;\text{ where }\;\frac{1}{{p}^{ * }} = \frac{1}{p} + \frac{\sigma }{N} - 1. \] | Proof Since \( 1 < {p}^{ * } < \infty \) the norm \( \parallel h{\parallel }_{{p}^{ * }} \) is characterized by (§ 3.1 of Chap. 6) \[ \parallel h{\parallel }_{{p}^{ * }} = \mathop{\sup }\limits_{\substack{{g \in {L}^{q}\left( {\mathbb{R}}^{N}\right) } \\ {\parallel g{\parallel }_{q} = 1} }}{\int }_{{\mathbb{R}}^{N}}{hg... | Yes |
Proposition 21.1 Let \( E \) be of finite measure. For every \( r \in \left\lbrack {1,\frac{N}{N - 1}}\right) \) , \[ \mathop{\sup }\limits_{{x \in E}}{\int }_{E}\frac{dy}{{\left| x - y\right| }^{\left( {N - 1}\right) r}} \leq \frac{{\kappa }_{N}^{\frac{N - 1}{N}r}}{1 - \frac{N - 1}{N}r}\mu {\left( E\right) }^{1 - \fra... | Proof Fix \( x \in E \) . The symmetric rearrangement \( {\left( E - x\right) }^{ * } \) of \( \left( {E - x\right) \) is a ball about the origin of radius \( \rho > 0 \) such that \( \mu \left( E\right) = \mu \left( {{B}_{\rho }\left( x\right) }\right) \) . Then by Proposition 12.1, \[ {\int }_{E}\frac{dy}{{\left| x -... | Yes |
Proposition 21.2 Let \( E \) be of finite measure, and let \( f \in {L}^{1}\left( E\right) \) . Then \( {V}_{f} \in {L}^{q}\left( E\right) \) for all \( q \in \left\lbrack {1,\frac{N}{N - 1}}\right) \), and \[ {\begin{Vmatrix}{V}_{f}\end{Vmatrix}}_{q} \leq \frac{{\kappa }_{N}^{\frac{N - 1}{N}}}{{\left( 1 - \frac{N - 1}... | Proof Fix \( q \in \left( {1,\frac{N}{N - 1}}\right) \) and write \[ \frac{\left| f\left( y\right) \right| }{{\left| x - y\right| }^{N - 1}} = {\left| f\left( y\right) \right| }^{1 - \frac{1}{q}}\frac{{\left| f\left( y\right) \right| }^{\frac{1}{q}}}{{\left| x - y\right| }^{N - 1}}. \] Then by Hölder's inequality, \[ {... | Yes |
Proposition 21.3 Let \( E \) be of finite measure, and let \( f \in {L}^{p}\left( E\right) \) for some \( p > N \) . Then \( {V}_{f} \in {L}^{\infty }\left( E\right) \) and\n\n\[{\begin{Vmatrix}{V}_{f}\end{Vmatrix}}_{\infty } \leq C\left( {N, p}\right) \mu {\left( E\right) }^{\frac{p - N}{Np}}\parallel f{\parallel }_{p... | Proof By Hölder’s inequality, and the definition of \( {V}_{f} \) ,\n\n\[{\begin{Vmatrix}{V}_{f}\end{Vmatrix}}_{\infty } \leq \parallel f{\parallel }_{p}{\left( \mathop{\sup }\limits_{{x \in E}}{\int }_{E}\frac{dy}{{\left| x - y\right| }^{\left( {N - 1}\right) \frac{p}{p - 1}}}\right) }^{\frac{p - 1}{p}}.\]\n\nThe esti... | Yes |
Lemma 23.2 Let \( E \) be Lebesgue measurable. Then \( {E}_{\mathbf{u}}^{ * } \) is Lebesgue measurable and \( \mu \left( {E}_{\mathbf{u}}^{ * }\right) = \mu \left( E\right) \) | Proof By Proposition 5.2 of Chap. 3, the 1-dimensional Hausdorff outer measure coincides with the 1-dimensional Lebesgue outer measure and the latter coincides with the 1-dimensional Lebesgue measure \( {\mu }_{1} \) on measurable sets. Therefore\n\n\[ \n{\mathcal{H}}_{1}\left( {E \cap {\ell }_{P;\mathbf{u}}}\right) = ... | Yes |
Proposition 24.1 For every \( E \subset {\mathbb{R}}^{N} \)\n\n\[{\mu }_{e}\left( E\right) \leq {\kappa }_{N}{\left( \frac{1}{2}\operatorname{diam}E\right) }^{N}\]\n\nwhere \( {\mu }_{e} \) is the Lebesgue outer measure in \( {\mathbb{R}}^{N} \) and \( {\kappa }_{N} \) is the measure of the unit ball in \( {\mathbb{R}}... | Proof Assuming diam \( E < \infty \), let \( {E}_{N}^{ * } \) be constructed in (24.2). Since \( {E}_{N}^{ * } \) is symmetric about the origin, if \( x \in {E}_{N}^{ * } \) then also \( - x \in {E}_{N}^{ * } \). Therefore \( 2\left| x\right| \leq \operatorname{diam}{E}_{N}^{ * } \) and hence \( {E}_{N}^{ * } \) is con... | Yes |
Proposition 25.1 There exists a measurable set \( {F}_{ * } \subset D \), and a subsequence \( \left\{ {F}_{{n}^{\prime }}\right\} \subset \) \( \left\{ {F}_{n}\right\} \), such that\n\n\[ \lim {\chi }_{{F}_{{n}^{\prime }}} = {\chi }_{{F}_{ * }}\;\text{ a.e. in }{\mathbb{R}}^{2}. \]\n\n(25.1)\n\nMoreover\n\n\[ {S}_{y}{... | Proof Consider the portion of \( {F}_{n} \) in the right, upper, quarter plane, i.e.,\n\n\[ {F}_{n}^{ + } = {F}_{n} \cap \left\lbrack {x > 0}\right\rbrack \cap \left\lbrack {y > 0}\right\rbrack . \]\n\nThe least upper bound of \( {F}_{n}^{ + } \) in \( \left\lbrack {x > 0}\right\rbrack \cap \left\lbrack {y > 0}\right\r... | Yes |
Lemma 3.2 \( \parallel u{\parallel }_{\frac{Np}{N - p}} \leq \frac{p\left( {N - 1}\right) }{N - p}\parallel {Du}{\parallel }_{p} \) . | Proof Write\n\n\[ \parallel u{\parallel }_{\frac{Np}{N - p}} = {\left( {\int }_{E}{\left( {\left| u\right| }^{\frac{p\left( {N - 1}\right) }{N - p}}\right) }^{\frac{N}{N - 1}}dx\right) }^{\frac{N - 1}{N}\frac{N - p}{p\left( {N - 1}\right) }} \]\n\nand apply (3.1) to the function \( w = {\left| u\right| }^{p\left( {N - ... | Yes |
Theorem 7.1 Let \( u \in {W}_{o}^{1, N}\left( E\right) \) . There exist constants \( {C}_{1} \) and \( {C}_{2} \) depending only upon \( N \), such that\n\n\[{\int }_{E}\exp {\left\{ \frac{\left| u\right| }{{C}_{1}\parallel {Du}{\parallel }_{N}}\right\} }^{\frac{N}{N - 1}}{dx} \leq {C}_{1}\mu \left( E\right) . | Proof We may assume \( u \in {C}_{o}^{\infty }\left( E\right) \) . From the representation formula (5.1)\n\n\[ \left| {u\left( x\right) }\right| \leq \frac{1}{{\omega }_{N}}{\int }_{E}\frac{\left| Du\right| }{{\left| x - y\right| }^{N - 1}}{dy}.\n\nThe embedding now follows from the limiting potential estimates. | No |
Theorem 8.1 (Sobolev-Nikol'skii [149]) Let \( E \) satisfy the cone condition for a fixed circular cone \( {\mathcal{C}}_{o} \) of solid angle \( \omega \), height \( h \) and vertex at the origin, and let \( u \in {W}^{1, p}\left( E\right) \) . If \( 1 < p < N \) then \( u \in {L}^{{p}^{ * }}\left( E\right) \) where \... | Proof of (8.1) and (8.2): It suffices to prove the various assertions for \( u \in {C}^{\infty }\left( E\right) \) . Fix \( x \in E \) and let \( {\mathcal{C}}_{x} \subset \bar{E} \) be a cone congruent to \( {\mathcal{C}}_{o} \) and claimed by the cone property. Th | No |
Lemma 9.1 For every pair \( x, y \in \bar{E} \) such that \( \left| {x - y}\right| = \rho \leq h \) | Proof Fix \( x, y \in E \) such that \( \left| {x - y}\right| = \rho \leq h \) . Since \( E \) is convex, for all \( \xi \in {\mathcal{C}}_{x,\rho } \)\n\n\[ \left| {u\left( y\right) - u\left( \xi \right) }\right| = \left| {{\int }_{0}^{1}\frac{\partial }{\partial t}u\left( {y + t\left( {\xi - y}\right) }\right) {dt}}\... | Yes |
Theorem 10.1 Let \( E \) be bounded and convex and let \( u \in {W}^{1, p}\left( E\right) \) for some \( 1 < \) \( p < N \) . There exists a constant \( C \) depending only upon \( N \) and \( p \), such that\n\n\[ \n{\begin{Vmatrix}u - {u}_{E}\end{Vmatrix}}_{{p}^{ * }} \leq C\frac{{\left( \operatorname{diam}E\right) }... | Proof Having fixed \( x, y \in E \), denote by \( R\left( {x, y}\right) \) the distance from \( x \) to \( \partial E \), along the direction of \( \left( {y - x}\right) \) and write\n\n\[ \n\left| {u\left( x\right) - u\left( y\right) }\right| \leq {\int }_{0}^{R\left( {x, y}\right) }\left| {\frac{\partial }{\partial \... | Yes |
Proposition 10.1 Let \( E \) be a bounded and convex subset of \( {\mathbb{R}}^{N} \), and let \( u \in {W}^{1, p}\left( E\right) \) for some \( 1 < p < N \). There exists a constant \( C \) depending only upon \( N \) and \( p \), such that \[ {\begin{Vmatrix}u - {u}_{E}\end{Vmatrix}}_{q} \leq {C}^{\theta }{\left\lbra... | Proof The case \( \theta = 0 \) corresponds to \( q = r \) and (10.7) is trivial. The case \( \theta = 1 \) corresponds to \( q = \frac{Np}{N - p} \) and coincides with (10.1). Let \( r \in \left( {1,\infty }\right) \) and \( p \in \left( {1, N}\right) \) be fixed and choose \( \theta \in \left( {0,1}\right) \) and \( ... | Yes |
Proposition 10.2 Let \( E \) be a bounded, convex domain in \( {\mathbb{R}}^{N} \) with boundary \( \partial E \) of class \( {C}^{1} \), and let \( u \in {W}^{1, p}\left( E\right) \) for some \( 1 \leq p < \infty \) . Then \( \left( {u - {u}_{E}}\right) \) admits an extension \( w \in {W}_{o}^{1, p}\left( {\mathbb{R}}... | Proof Write down the estimate (18.1), of Chap. 8, for \( \left( {u - {u}_{E}}\right) \) and apply the Poincaré inequality to estimate the last term. | No |
Corollary 10.1 Let \( {B}_{R} \) be the ball of radius \( R \) centered at the origin of \( {\mathbb{R}}^{N} \), let \( u \in \) \( {W}^{1, p}\left( {B}_{R}\right) \) for some \( 1 \leq p < \infty \) . Then \( \left( {u - {u}_{{B}_{R}}}\right) \) admits an extension \( w \in {W}_{o}^{1, p}\left( {\mathbb{R}}^{N}\right)... | Proof It follows from (10.8) and Remark 18.2 of Chap. 8. | No |
Corollary 10.2 Let \( {B}_{R} \) be the ball of radius \( R \) centered at the origin of \( {\mathbb{R}}^{N} \), let \( u \in \) \( {W}^{1, p}\left( {B}_{R}\right) \) for some \( 1 \leq p < \infty \) . Then \( {\left( u - {u}_{{B}_{R}}\right) }_{ \pm } \) admit extensions \( {w}_{ \pm } \in {W}_{o}^{1, p}\left( {\mathb... | Proof Extend \( {w}_{ \pm } \) as in Proposition 19.1 of Chap. 8, and following Remark 18.2, write down (10.4) of Chap. 8. In the last of these majorize\n\n\[ \n{\begin{Vmatrix}{\left( u - {u}_{{B}_{R}}\right) }_{ \pm }\end{Vmatrix}}_{p;{B}_{R}} \leq {\begin{Vmatrix}u - {u}_{{B}_{R}}\end{Vmatrix}}_{p;{B}_{R}} \]\n\nand... | No |
Theorem 11.1 (DeGiorgi [32]) Let \( E \) be a bounded convex open set in \( {\mathbb{R}}^{N} \) and let \( u \in {W}^{1,1}\left( E\right) \) . Assume that the set where \( u \) vanishes has positive measure. Then\n\n\[ \parallel u{\parallel }_{1} \leq {\kappa }_{N}^{\frac{N - 1}{N}}\frac{{\left( \operatorname{diam}E\ri... | Proof Let \( \mathbf{n} \) denote the unit vector ranging over the unit sphere in \( {\mathbb{R}}^{N} \) . For almost all \( x \in E \) and almost all \( y \in \left\lbrack {u = 0}\right\rbrack \)\n\n\[ \left| {u\left( x\right) }\right| = \left| {{\int }_{0}^{\left| y - x\right| }\frac{\partial }{\partial \rho }u\left(... | Yes |
Proposition 12.1 Let \( f \in {M}^{p}\left( E\right) \) for some \( p > \frac{N}{\alpha } \) . Then\n\n\[ \n{\begin{Vmatrix}{V}_{\alpha, f}\end{Vmatrix}}_{\infty } \leq \frac{N\left( {p - 1}\right) }{{\alpha p} - N}{\left( \operatorname{diam}E\right) }^{\frac{{\alpha p} - N}{p}}\parallel f{\parallel }_{{M}^{p}}.\n\] | Proof For \( x \) and \( y \) in \( {\mathbb{R}}^{N} \) let \( \mathbf{n} = \frac{y - x}{\left| x - y\right| } \) . Then by making use of polar coordinates, compute\n\n\[ \n\frac{d}{d\rho }{\int }_{{B}_{\rho }\left( x\right) }\left| {f\left( y\right) }\right| {dy} = \frac{d}{d\rho }{\int }_{\left| \mathbf{n}\right| = 1... | Yes |
Theorem 12.1 Let \( u \in {W}^{1,1}\left( E\right) \) and assume that \( \left| {Du}\right| \in {M}^{p}\left( E\right) \) for some \( p > N \) . Then \( u \in {C}^{\eta }\left( E\right) \) with \( \eta = \frac{p - N}{p} \) . Moreover for every ball \( {B}_{\rho }\left( x\right) \subset E \)\n\n\[ \underset{{B}_{\rho }\... | Proof Consider the potential inequality (10.3) written for the ball \( {B}_{\rho }\left( x\right) \) replacing \( E \) . It gives\n\n\[ \left| {u\left( x\right) - {\left( u\right) }_{x,\rho }}\right| \leq \frac{{2}^{N}}{{\kappa }_{N}}{\int }_{{B}_{\rho }\left( x\right) }\frac{\left| Du\left( y\right) \right| }{{\left| ... | Yes |
Theorem 13.1 (John-Nirenberg ([77]) Let \( u \in {W}^{1,1}\left( E\right) \) and let (13.1) hold. There exist constants \( {C}_{1} \) and \( {C}_{2} \) depending only upon \( N \), such that\n\n\[ \n{\int }_{{B}_{\rho }\left( x\right) }\exp \left( \frac{\left| u - {\left( u\right) }_{x,\rho }\right| }{{C}_{1}\parallel ... | Proof It follows from Proposition 13.1, starting from the potential inequality (10.3). | No |
Theorem 14.1 (Rellich-Kondrachov ([123,85]) Let \( E \) be a bounded domain in \( {\mathbb{R}}^{N} \) with the cone property, and let \( 1 \leq p < N \) . Then the embedding of \( {W}^{1, p}\left( E\right) \) into \( {L}^{q}\left( E\right) \) is compact for all \( 1 \leq q < {p}^{ * } \) . | Proof The proof consists of verifying that a bounded subset of \( {W}^{1, p}\left( E\right) \) satisfies the conditions for a subset of \( {L}^{q}\left( E\right) \) to be compact, given in \( §{19} \) of Chap. 6. For \( \delta > 0 \) let\n\n\[ \n{E}_{\delta } = \{ x \in E \mid \operatorname{dist}\left( {x,\partial E}\r... | Yes |
Proposition 15.1 \( {W}^{1, p}\left( {\mathbb{R}}^{N}\right) \) is continuously embedded into \( {W}^{s, p}\left( {\mathbb{R}}^{N}\right) \) for all \( s \in \) \( \left( {0,1}\right) \) . Moreover\n\n\[ \parallel u{\parallel }_{s, p} \leq {\left( \frac{2{\omega }_{N}}{{ps}\left( {1 - s}\right) }\right) }^{\frac{1}{p}}... | Proof A change of variable in the definition of the semi-norm in (15.1) gives\n\n\[ \parallel u{\parallel }_{s, p}^{p} = {\int }_{{\mathbb{R}}^{N}}{\left| \xi \right| }^{-\left( {N + {sp}}\right) }{d\xi }{\int }_{{\mathbb{R}}^{N}}{\left| u\left( x + \xi \right) - u\left( x\right) \right| }^{p}{dx} \]\n\n\[ = {\int }_{\... | Yes |
Proposition 16.1 Let \( u \in {W}^{1, p}\left( {\mathbb{R}}_{ + }^{N}\right) \) be continuous in \( {\overline{\mathbb{R}}}_{ + }^{N} \) . Then\n\n\[ \parallel u\left( {\cdot ,0}\right) {\parallel }_{r,{\mathbb{R}}^{N - 1}}^{r} \leq r\parallel {Du}{\parallel }_{p,{\mathbb{R}}_{ + }^{N}}\parallel u{\parallel }_{q,{\math... | Proof May assume that \( u \in {C}_{o}^{\infty }\left( {\mathbb{R}}^{N}\right) \) . For \( \bar{x} \in {\mathbb{R}}^{N - 1} \) and \( r \geq 1 \)\n\n\[ {\left| u\left( \bar{x},0\right) \right| }^{r} \leq r{\int }_{0}^{\infty }{\left| u\left( \bar{x},{x}_{N}\right) \right| }^{r - 1}\left| {\frac{\partial }{\partial {x}_... | Yes |
Proposition 16.2 Let \( u \in {W}^{1, p}\left( {\mathbb{R}}_{ + }^{N}\right) \) for some \( p \geq 1 \) . Then\n\n\[ \parallel u\left( {\cdot ,0}\right) {\parallel }_{p,{\mathbb{R}}^{N - 1}} \leq {p}^{\frac{1}{p}}\parallel u{\parallel }_{p,{\mathbb{R}}_{ + }^{N}}^{1 - \frac{1}{p}}\parallel {Du}{\parallel }_{p,{\mathbb{... | Proof Inequality (16.2) follows from (16.1) with \( r = p \) . The domain \( {\mathbb{R}}_{ + }^{N} \) satisfies the cone condition with cone \( {\mathcal{C}}_{o} \) of solid angle \( \frac{1}{2}{\omega }_{N} \) and height \( h \in \left( {0,\infty }\right) \) . Then (16.5) follows from (8.4) of Theorem 8.1, whereas (1... | Yes |
Proposition 17.1 Let \( u \in {W}^{1, p}\left( {\mathbb{R}}_{ + }^{N + 1}\right) \) for some \( p > 1 \) . Then the trace of \( u \) on the hyperplane \( t = 0 \) belongs to the fractional Sobolev space \( {W}^{1 - \frac{1}{p}, p}\left( {\mathbb{R}}^{N}\right) \) . Moreover\n\n\[ \parallel u\left( {\cdot ,0}\right) {\p... | Proof For every pair \( x, y \in {\mathbb{R}}^{N} \) set \( {2\xi } = x - y \) and consider the point \( z \in {\mathbb{R}}_{ + }^{N + 1} \) of coordinates \( z = \left( {\frac{1}{2}\left( {x + y}\right) ,\lambda \left| \xi \right| }\right) \), where \( \lambda \) is a positive parameter to be chosen. Then\n\n\[ \left|... | Yes |
Proposition 21.1 There exists a map \( \mathcal{F} : \bar{E} \rightarrow \bar{Q} \) and positive, absolute constants \( C > c > 0 \), independent of the geometry of \( \partial E \) and \( \Gamma \), such that \( Q = {\mathcal{F}}^{-1}\left( E\right) \) , \( {\Gamma }_{o} \subset {\mathcal{F}}^{-1}\left( \Gamma \right)... | Proof Let \( \mathbf{n} \) be the unit vector in \( {\mathbb{R}}^{N} \) ranging over the unit sphere \( {S}_{1} \), and consider the map \( {\phi }_{{x}_{o}, E} : {S}_{1} \rightarrow \partial E \) defined by\n\n\[ \n{\phi }_{{x}_{o}, E}\left( \mathbf{n}\right) = {x}_{o} + t\mathbf{n}\n\]\n\n(21.3)\n\nwhere \( t \) is t... | No |
Lemma 21.1 There exists a constant \( {C}_{o} \), depending on \( {x}_{o} \) and \( E \), such that\n\n\[ \rho \left| {{\mathbf{n}}_{1} - {\mathbf{n}}_{2}}\right| \leq \left| {{\phi }_{{x}_{o}, E}\left( {\mathbf{n}}_{1}\right) - {\phi }_{{x}_{o}, E}\left( {\mathbf{n}}_{2}\right) }\right| \leq {4R}\frac{R}{\rho }\left| ... | Proof Up to a translation we may assume that \( {x}_{o} = 0 \) and set \( {\phi }_{{x}_{o}, E} = \phi \) . Having fixed \( {\mathbf{n}}_{1} \) and \( {\mathbf{n}}_{2} \) on \( {S}_{1} \), the bound below in (21.4) follows from the definition (21.3) since the sphere \( {S}_{\rho } \) centered at the origin is contained ... | Yes |
Lemma 22.1 For all \( x, y \in {B}_{1} \)\n\n\[ \frac{1}{4}\rho \frac{\rho }{R}\left| {x - y}\right| \leq \left| {{\varphi }_{{x}_{o}, E}\left( x\right) - {\varphi }_{{x}_{o}, E}\left( y\right) }\right| \leq {5R}\frac{R}{\rho }\left| {x - y}\right| . \] | Proof Assume \( {x}_{o} = 0 \), set \( {\varphi }_{{x}_{o}, E} = \varphi \), and fix any two nonzero vectors \( x, y \in {B}_{1} \) . By intersecting \( E \) with the hyperplane through the origin and containing \( x \) and \( y \) , it suffices to consider the case \( N = 2 \) . Assume for example that \( \left| y\rig... | Yes |
Proposition 24.1 Let \( u \in {W}_{{p}^{ * }}^{1, p}\left( E\right) \) . For all \( \varepsilon > 0 \) there exists \( {u}_{\varepsilon } \in {W}^{1, p}\left( E\right) \) such that \( {\begin{Vmatrix}u - {u}_{\varepsilon }\end{Vmatrix}}_{1, p;{p}^{ * }} < \varepsilon \) . Moreover \( u \) and \( {Du} \) satisfy (1.6). | Proof May assume \( E = {\mathbb{R}}^{N} \) . Let \( {\zeta }_{n} \) be a nonnegative, piecewise smooth cutoff function in \( {\mathbb{R}}^{N} \), such that\n\n\[ \n\begin{array}{l} {\zeta }_{n} = 1\text{ for }\left| x\right| < n \\ {\zeta }_{n} = 0\text{ for }\left| x\right| \geq {2n} \end{array}\;\text{ and }\;\left|... | Yes |
Lemma 1.1 \( {\mu }_{1}\left\lbrack {f\left( {D}_{N}\right) }\right\rbrack = 0 \) . | Proof Fix \( \epsilon \in \left( {0,1}\right) \) . For each \( x \in {D}_{N} \) and for all \( 0 < \varepsilon \leq \epsilon \) there exists an open ball \( {B}_{r}\left( x\right) \) centered at \( x \) and radius \( 0 < r \leq \varepsilon \) such that\n\n\[ \mathop{\sup }\limits_{{y \in {B}_{r}\left( x\right) }}\left|... | Yes |
Proposition 2.1 Let \( E \) be a bounded domain in \( {\mathbb{R}}^{N} \) and let \( u \in {C}^{\infty }\left( E\right) \) . Then for all nonnegative \( \varphi \in C\left( E\right) \)\n\n\[{\int }_{E}\varphi \left| {\nabla u}\right| {dx} = {\int }_{0}^{\infty }\left( {{\int }_{\partial \left\lbrack {\left| u\right| > ... | Proof Assume first \( \varphi \in {C}_{o}^{\infty }\left( E\right) \) and, for \( \varepsilon > 0 \) set\n\n\[ \mathbf{h} = \varphi \frac{\nabla u}{\sqrt{{\left| \nabla u\right| }^{2} + \varepsilon }} \in {C}_{o}^{\infty }\left( E\right) . \]\n\nThen, by integration by parts and formula (15.5) of Chap. 4\n\n\[ {\int }_... | Yes |
Proposition 3.1 Let \( E \) be a bounded, open set in \( {\mathbb{R}}^{N} \), with smooth boundary \( \partial E \). Then\n\n\[ \frac{\sigma {\left( \partial E\right) }^{\frac{N}{N - 1}}}{\mu \left( E\right) } \geq N{\omega }_{N}^{\frac{1}{N - 1}} \] | Proof Denote by \( E \ni x \rightarrow {\delta }_{E}\left( x\right) \) the distance from \( x \) to \( \partial E \) and for \( \delta > 0 \) let \( {E}_{\delta } \) be defined as\n\n\[ {E}_{\delta } = \left\{ {x \in E \mid {\delta }_{E}\left( x\right) > \delta }\right\} \]\n\nwhere \( \delta \) is so small that \( {E}... | Yes |
Proposition 4.1 \( {c}_{p}\left( K\right) = {c}_{p}^{o}\left( K\right) = {c}_{p}^{1}\left( K\right) \) . | Proof By construction \( {c}_{p}\left( K\right) \leq {c}_{p}^{o}\left( K\right) \leq {c}_{p}^{1}\left( K\right) \) . To establish the converse inequalities, may assume that \( {c}_{p}\left( K\right) < \infty \) . For \( \varepsilon > 0 \) there exists \( {u}_{\varepsilon } \in {C}_{o}^{\infty }\left( {\mathbb{R}}^{N}\r... | Yes |
Proposition 4.2 \( {c}_{p}^{ * }\left( K\right) = {c}_{p}\left( K\right) \) . | Proof By construction \( {c}_{p}^{ * }\left( K\right) \leq {c}_{p}\left( K\right) \) . For \( \varepsilon > 0 \) there exists \( {u}_{\varepsilon } \in {\Gamma }_{ * }\left( K\right) \) such\n\nthat\n\[ \n{\int }_{{\mathbb{R}}^{N}}{\left| D{u}_{\varepsilon }\right| }^{p}{dx} - \varepsilon \leq {c}_{p}^{ * }\left( K\rig... | Yes |
Theorem 5.1 Let \( K \) be a compact subset of \( {\mathbb{R}}^{N} \) . Then for \( 1 < p < N \)\n\n\[ \n{c}_{p}\left( K\right) = \mathop{\inf }\limits_{{u \in \Gamma \left( K\right) }}{\left\lbrack {\int }_{0}^{1}{\left( {\int }_{\partial \left\lbrack {u > t}\right\rbrack }{\left| Du\right| }^{p - 1}d\sigma \right) }^... | Proof (The Case \( 1 < p < N \) . The Lower Estimate in (5.2)) Let \( u \) be a nonnegative function in \( {C}_{o}^{\infty }\left( {\mathbb{R}}^{N}\right) \) . By the co-area formula (2.1), for smooth functions\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{\left| Du\right| }^{p}{dx} \geq {\int }_{0}^{1}f\left( t\right) {dt} \n\]\... | Yes |
Corollary 6.1 Let \( {B}_{\rho } \) be an open ball of radius \( \rho \) in \( {\mathbb{R}}^{N} \) . Then,\n\n\[ \n{c}_{p}\left( {\bar{B}}_{\rho }\right) = {\omega }_{N}{\left( \frac{N - p}{p - 1}\right) }^{p - 1}{\rho }^{N - p}\;\text{ for }1 < p < N.\n\]\n\n(6.5)\n\nWhen \( p = 1 \) the power in round brackets is mea... | Proof If \( p = 1 \) the statement follows from (5.3) of Theorem 5.1. If \( 1 < p < N \) , from the previous lower estimate with \( K = {\bar{B}}_{\rho } \) one computes\n\n\[ \n{c}_{p}\left( {\bar{B}}_{\rho }\right) \geq {\left( \frac{N - p}{p - 1}\right) }^{p - 1}{\omega }_{N}{\rho }^{N - p}.\n\]\n\nOn the other hand... | Yes |
Lemma 7.1 \( {h}^{\prime } \in {L}^{p}\left( {0,{s}_{u}}\right) \) and, \[ {\int }_{0}^{{s}_{u}}{h}^{\prime p}\left( s\right) {ds} \leq {\int }_{{\mathbb{R}}^{N}}{\left| Du\right| }^{p}{dx} \] | Proof Let \[ 0 = {s}_{o} < {s}_{1} < \cdots < {s}_{n - 1} < {s}_{n} = {s}_{u}\;\text{ and } \] \[ 0 = {t}_{o} < {t}_{1} < \cdots < {t}_{n - 1} < {t}_{n} = {t}_{u} \] be a partition of \( \left( {0,{s}_{u}}\right) \) and the corresponding partition of \( \left( {0,{t}_{u}}\right) \) by which \( h\left( {s}_{j}\right) = ... | Yes |
Proposition 8.1 Let \( p \geq 1 \) and \( q > p \) . Then the Gagliardo type embedding inequality\n\n\[ \n\parallel u{\parallel }_{q} \leq \gamma \parallel {Du}{\parallel }_{p}\;\text{ for all }u \in {W}_{o}^{1, p}\left( {\mathbb{R}}^{N}\right)\n\]\n\n(8.1)\n\nfor a constant \( \gamma > 0 \) depending only on \( p, q \... | Proof Let (8.1) hold. Having fixed a compact set \( K \subset {\mathbb{R}}^{N} \), let \( u \) be in the class \( {\Gamma }_{o}\left( K\right) \) introduced in (4.1). Then from (8.1)\n\n\[ \n\mu {\left( K\right) }^{\frac{p}{q}} \leq {\gamma }^{p}{\int }_{{\mathbb{R}}^{N}}{\left| Du\right| }^{p}{d\mu }.\n\]\n\nminimizin... | Yes |
Theorem 10.1 Let \( K \) be a compact subset of \( {\mathbb{R}}^{N} \) and let \( 1 \leq p < N \) . Then\n\n\[ \n{c}_{p}\left( K\right) = 0\; \Rightarrow \;{\mathcal{H}}^{N - p + \varepsilon }\left( K\right) = 0\;\text{ for all }\varepsilon > 0.\n\] | Proof For every \( j \in \mathbb{N} \) there exists a nonnegative function \( {u}_{j} \in {C}_{o}^{\infty }\left( {\mathbb{R}}^{N}\right) \) such that \( {u}_{j} \geq 1 \) in a open neighborhood of \( K \), and\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}{\left| D{u}_{j}\right| }^{p}{dy} \leq \frac{1}{{2}^{j}}\n\]\n\nSet \( u = ... | Yes |
Proposition 11.1 Let \( E \) and \( F \) be subsets of \( {\mathbb{R}}^{N} \) and let \( p \geq 1 \) . Then\n\n\[{\bar{c}}_{p}\left( {E \cup F}\right) + {\bar{c}}_{p}\left( {E \cap F}\right) \leq {\bar{c}}_{p}\left( E\right) + {\bar{c}}_{p}\left( F\right) . | Proof (of Proposition 11.1) May assume that the right hand side of (11.4) is finite. Assume first that \( E \) and \( F \) are compact and let \( u \in {\Gamma }_{1}\left( E\right) \) and \( v \in {\Gamma }_{1}\left( F\right) \) . Then \( \left( {u \vee v}\right) \) and \( \left( {u \land v}\right) \) are Lipschitz con... | Yes |
Proposition 11.2 Let \( \\left\\{ {E}_{n}\\right\\} \) and \( \\left\\{ {F}_{n}\\right\\} \) be countable collections of sets in \( {\\mathbb{R}}^{N} \), each of finite outer p-capacity, and such that \( {E}_{n} \\subset {F}_{n} \). Then for all \( n \\in \\mathbb{N} \), \[ {\\bar{c}}_{p}\\left( {\\mathop{\\bigcup }\\l... | Proof It suffices to prove (11.5) for \( n = 2 \). Apply (11.4) to the pair of sets \[ E = {E}_{1} \\cup {F}_{1}\\;\\text{ and }\\;F = {E}_{1} \\cup {F}_{2} \] to get \[ {\\bar{c}}_{p}\\left( {{F}_{1} \\cup {F}_{2}}\\right) + {\\bar{c}}_{p}\\left( {{E}_{1} \\cup \\left( {{F}_{1} \\cap {F}_{2}}\\right) }\\right) \\leq {... | Yes |
Proposition 12.1 Let \( \\left\\{ {K}_{n}\\right\\} \) be a countable collection of compact sets such that \( {K}_{n + 1} \\subset {K}_{n} \) . Then\n\n\[ \n{c}_{p}\\left( {\\bigcap {K}_{n}}\\right) = \\lim {c}_{p}\\left( {K}_{n}\\right) .\n\] | Proof (of Proposition 12.1) We may assume that \( {c}_{p}\\left( {K}_{1}\\right) < \\infty \) . The set \( K = \\cap {K}_{n} \) is compact and hence \( p \) -capacitable. By the monotonicity of \( {c}_{p}\\left( \\cdot \\right) \)\n\n\[ \n{c}_{p}\\left( K\\right) \\leq \\lim {c}_{p}\\left( {K}_{n}\\right) \n\]\n\nTo es... | Yes |
Proposition 12.2 Let \( \left\{ {E}_{n}\right\} \) be a countable collection of sets in \( {\mathbb{R}}^{N} \) such that \( {E}_{n} \subset \) \( {E}_{n + 1} \), and with \( {c}_{p}\left( {\bigcup {E}_{n}}\right) < \infty \) . Then\n\n\[ \n{\bar{c}}_{p}\left( {\bigcup {E}_{n}}\right) = \lim {\bar{c}}_{p}\left( {E}_{n}\... | Proof Set \( E = \bigcup {E}_{n} \) . By monotonicity of \( {\bar{c}}_{p}\left( \cdot \right) \)\n\n\[ \n{\bar{c}}_{p}\left( E\right) \geq \lim {\bar{c}}_{p}\left( {E}_{n}\right) .\n\]\n\nTo establish the converse inequality, assume first that all \( {E}_{n} \) are open. Having fixed \( \varepsilon > 0 \), let \( K \) ... | Yes |
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