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Corollary 11.108. Let \( X \) be a thick Euclidean building of rank at least 3. Assume that \( {X}_{\infty } \) is Moufang and that there exists a twin building \( \left( {{X}_{ + },{X}_{ - }}\right) \) with \( X = {X}_{ + } \) . Then \( {\operatorname{lk}}_{X}\left( v\right) \) is Moufang for any special vertex \( v \... | Proof. By Proposition 11.107, \( {L}_{v} \mathrel{\text{:=}} {\operatorname{lk}}_{X}\left( v\right) \) can be embedded in \( {X}_{\infty } \) as a subbuilding for any special vertex \( v \) of \( X \) . So the corollary follows immediately from Proposition 7.37. | No |
Proposition 11.110. Suppose that the Euclidean building \( X \) is Moufang of rank at least 3 with root group system \( \left( {U}_{\alpha }\right) \), where \( \alpha \) ranges over the roots of E. Suppose further that the spherical building \( {X}_{\infty } \) is Moufang. Then for any root \( \alpha \) of \( E,{U}_{\... | Proof of Proposition 11.110. As we mentioned earlier, the assumption that \( X \) is Moufang implies that \( X \) is part of a Moufang twin building \( \left( {X,{X}_{ - }}\right) \) . Given \( \alpha \), we first choose a panel \( P \in \partial \alpha \) that contains a special vertex \( v \) . It is easy to see that... | No |
Lemma 12.9. Given \( x \in X \) of type \( \tau \left( x\right) = z \), the star of \( x \) contains the open ball \( \{ y \in X \mid d\left( {x, y}\right) < \epsilon \left( z\right) \} \) of radius \( \epsilon \left( z\right) \) centered at \( x \) . | Proof. We must show that \( y \notin \operatorname{st}x \Rightarrow d\left( {x, y}\right) \geq \epsilon \left( z\right) \) . Suppose \( y \notin \operatorname{st}x \) , and consider a chain from \( x \) to \( y \) represented as in (12.2). Assume, as we may, that \( {x}_{0},{x}_{1},{x}_{2} \) do not lie in a single \( ... | Yes |
Corollary 12.11. Suppose \( \Delta \) consists of a single apartment \( \sum = \sum \left( {W, S}\right) \) as in Example 12.2(b). Then \( W \) acts on \( X = Z\left( {W, S}\right) \) by isometries, and the identification of the fundamental domain with \( Z \) is compatible with the original metric on \( Z \) . | Proof. The first assertion is immediate from the definitions, and the second follows from part (2) of the proposition. | No |
Proposition 12.13. Under the hypotheses (1)-(4), the map \( \phi : Z\left( {W, S}\right) \rightarrow X \) is an isometry. | Proof. We claim that every geodesic segment \( \left\lbrack {x, y}\right\rbrack \) in \( X \) can be subdivided so that each piece is contained in a closed chamber. Accepting this for the moment, we obtain a chain from \( x \) to \( y \) of length \( d\left( {x, y}\right) \), whence \( {d}^{\prime }\left( {x, y}\right)... | Yes |
Proposition 12.16. Given a type-preserving chamber map \( \phi : \Delta \rightarrow {\Delta }^{\prime } \), the induced map \( \phi : X \rightarrow {X}^{\prime } \) is distance-decreasing, i.e., | \[ d\left( {\phi \left( x\right) ,\phi \left( y\right) }\right) \leq d\left( {x, y}\right) \] for all \( x, y \in X \) . | Yes |
Corollary 12.17. Let \( \Delta \) be a building and \( \sum \) an apartment in \( \Delta \), viewed as a building in its own right. Then the inclusion \( \sum \hookrightarrow \Delta \) induces an isometric embedding of \( Z\left( \sum \right) \) into \( Z\left( \Delta \right) \) . | Proof. If \( \iota : \sum \hookrightarrow \Delta \) is the inclusion and \( \rho : \Delta \rightarrow \sum \) is a retraction, then the induced maps on \( Z \) -realizations are distance-decreasing. Since \( {\rho \iota } = {\operatorname{id}}_{\sum } \), it follows that no distances can be strictly decreased by \( \io... | Yes |
Proposition 12.18. Let \( \rho = {\rho }_{\sum, C} \), where \( \sum \) is an apartment and \( C \) is a chamber of \( \sum \) . Then\n\n\[ d\left( {\rho \left( x\right) ,\rho \left( y\right) }\right) \leq d\left( {x, y}\right) \]\n\nfor all \( x, y \in X \), with equality if \( x \in Z\left( C\right) \) . | Proof. We need only prove the assertion about equality. Choose a chamber \( D \) with \( y \in Z\left( D\right) \), and let \( {\sum }^{\prime } \) be an apartment containing \( C \) and \( D \) . Recall that the restriction of \( \rho \) to \( {\sum }^{\prime } \) is given by a type-preserving isomorphism \( \phi : {\... | Yes |
Proposition 12.20. Two points of \( X \) with the same carrier and the same type are equal. | Proof. Let \( z \) be the common type of the two points, and write them as \( \left\lbrack {C, z}\right\rbrack \) and \( \left\lbrack {D, z}\right\rbrack \) . By hypothesis, the chambers \( C \) and \( D \) have the same face of cotype \( {S}_{z} \) and hence are in the same \( {S}_{z} \) -residue. But then \( \left\lb... | Yes |
Proposition 12.21. If \( x \in X \) has carrier \( A \in \Delta \), then \( \phi \left( x\right) \) has carrier \( \phi \left( A\right) \) . | Proof. Write \( x = \left\lbrack {C, z}\right\rbrack \), so that \( A \) is the face of \( C \) of cotype \( {S}_{z} \) . Then \( \phi \left( x\right) = \) \( \left\lbrack {\phi \left( C\right), z}\right\rbrack \), so the carrier of \( \phi \left( x\right) \) is the face of \( \phi \left( C\right) \) of cotype \( {S}_{... | Yes |
Corollary 12.22. Let \( {\phi }_{1},{\phi }_{2} : \Delta \rightarrow {\Delta }^{\prime } \) be type-preserving chamber maps between buildings of type \( \left( {W, S}\right) \) . Let \( x \in X \) have carrier \( A \in \Delta \) . Then \( {\phi }_{1}\left( x\right) = {\phi }_{2}\left( x\right) \) if and only if \( {\ph... | Proof. The second assertion follows from the first, applied with one map equal to the identity. The first assertion is an immediate consequence of Propositions 12.20 and 12.21. | No |
Corollary 12.26. Given \( x, y \in X \), there is a finite collection \( \mathcal{G} \) of galleries such that\n\n\[ d\left( {x, y}\right) = \mathop{\inf }\limits_{\gamma }l\left( \gamma \right) \]\n\nwhere \( \gamma \) ranges over the chains from \( x \) to \( y \) that can be represented as in (12.6) with the associa... | Proof. We may assume that \( \Delta = \sum \left( {W, S}\right) \) . Let \( A \) and \( B \) be the carriers of \( x \) and \( y \) as in the proposition. Then we can compute \( d\left( {x, y}\right) \) using only chains whose galleries are minimal from \( A \) to \( B \) . Recall now that there are only finitely many ... | Yes |
Theorem 12.27. Suppose that \( {Z}_{s} \) is a proper metric space for each \( s \in S \) . Then for any \( x, y \in X \), there is a chain from \( x \) to \( y \) of length \( d\left( {x, y}\right) \) . | Proof. In view of Corollary 12.26, it suffices to show that among the chains \( \gamma \) representable as in (12.6) with a given associated gallery \( \Gamma \), there is one of minimal length. Now the length of \( \gamma \) is given by\n\n\[ l\left( \gamma \right) = \mathop{\sum }\limits_{{i = 1}}^{m}{d}_{Z}\left( {{... | Yes |
Corollary 12.28. If \( Z \) is a geodesic metric space and each \( {Z}_{s} \) is a proper metric space, then \( X = Z\left( \Delta \right) \) is a geodesic metric space. | Proof. Given \( x, y \in X \), choose a chain \( x = {x}_{0},{x}_{1},\ldots ,{x}_{m} = y \) of length \( d\left( {x, y}\right) \) . Then there are geodesics \( \left\lbrack {{x}_{i - 1},{x}_{i}}\right\rbrack \) for each \( i = 1,\ldots, m \), and we can concatenate these to obtain a geodesic \( \left\lbrack {x, y}\righ... | No |
Proposition 12.29. Suppose that \( Z\left( {W, S}\right) \) is a \( \operatorname{CAT}\left( \kappa \right) \) space for some real number \( \kappa \) . Then the \( Z \) -realization of any building \( \Delta \) of type \( \left( {W, S}\right) \) is a \( \operatorname{CAT}\left( \kappa \right) \) space. | The proof is essentially the same as the proof that Euclidean buildings are CAT(0) spaces (Theorem 11.16), once we establish some basic properties of the \( \operatorname{CAT}\left( \kappa \right) \) property for \( \kappa \neq 0 \) . Note that we can scale the metric to reduce to the cases \( \kappa = \pm 1 \), so we ... | Yes |
Lemma 12.30. Let \( X \) be the 2-sphere \( {S}^{2} \) or the hyperbolic plane \( {\mathbb{H}}^{2} \), and let \( x, y, z \) be three points in \( X \) . If \( X = {S}^{2} \), assume that \( x \) and \( y \) are not antipodal. Set \( c \mathrel{\text{:=}} d\left( {x, y}\right) \), and for \( 0 \leq t \leq 1 \), let \( ... | Proof. Assume first that \( X = {S}^{2} \), and recall (from [48, Chapter I.2], for instance) that the distance function is given by \( \cos d\left( {u, v}\right) = \langle u, v\rangle \), where the right side denotes the inner product of unit vectors in \( {\mathbb{R}}^{3} \) . Our task, then, is to compute \( \langle... | Yes |
For every spherical building \( \Delta \), the ordinary geometric realization \( \left| \Delta \right| \) admits a canonical \( \operatorname{CAT}\left( 1\right) \) metric, obtained as follows. Let \( \Delta \) have type \( \left( {W, S}\right) \) ; then \( W \) is finite. Let \( Z \) be a simplex with vertex set \( S ... | This gives us a metric on \( X \) such that every apartment is isometric to the standard unit sphere of dimension \( \left| S\right| - 1 \) (see Example 12.14(a)). In view of Proposition 12.29, \( X \) is a (complete) \( \operatorname{CAT}\left( 1\right) \) space. | Yes |
For every Euclidean building \( \Delta \), the \( \operatorname{CAT}\left( 0\right) \) metric on the geometric realization \( \left| \Delta \right| \) that we discussed in detail in Chapter 11 is a special case of the construction in the present chapter. | As in the previous example, take \( Z \) to be a simplex with vertex set \( S \) . It has a canonical Euclidean metric. To show that the resulting metric on the \( Z \) -realization is the same as the metric constructed in Chapter 11, it suffices to consider the case of an apartment, since any two points are contained ... | No |
Let \( \Delta \) be a building of type \( \left( {W, S}\right) \), where \( W \) is hyperbolic in the sense of Gromov. | Moussong has characterized such Coxeter groups \( W \) ; see Section 12.3.9 below. Moussong has also shown that the Davis realization of an apartment admits a different metric in this case, obtained by using the same set \( Z \) but giving it a piecewise hyperbolic metric instead of a piecewise-Euclidean metric. The ne... | No |
Let \( W \) be the Coxeter group of type \( {\widetilde{\mathrm{A}}}_{2} \). Its Coxeter complex \( \sum \) is the plane tiled by equilateral triangles. If we draw the chamber graph (or Cayley graph) on top of a picture of \( \sum \) as in Figure 12.9, we see the honeycomb tiling of the plane by hexagons. This yields a... | We can describe the cells of \( {\sum }_{d} \) in the following way, which will be familiar to readers who have seen dual cell decompositions of triangulated manifolds as in [129, p. 232; 179, Section 64]: For each vertex \( v \in \sum \), its link in \( \sum \) is a hexagon. Inside the barycentric subdivision of \( \s... | Yes |
Let \( \sum \) be an arbitrary Coxeter complex. The link of every simplex is again a Coxeter complex; hence if it is finite, it triangulates a sphere. Intuitively, the construction that follows consists in coning off smaller copies of these spheres in the barycentric subdivision of \( \sum \) to get cells, as in Exampl... | Let \( {\sum }_{f} \) be the subposet of \( \sum \) consisting of the spherical simplices in \( \sum \) , i.e., the simplices \( A \in \sum \) whose link \( {\operatorname{lk}}_{\sum }A \) is finite. If \( \sum \) is given to us as \( \sum \left( {W, S}\right) \), then \( {\sum }_{f} \) is the set of simplices whose st... | Yes |
Proposition 12.52. The underlying space \( X = \left| {\sum }_{d}\right| \) is always contractible. | Sketch of proof. As in Section 4.12, our proof will be complete except for (routine) homotopy-theoretic details. The proposition is trivial if \( \sum \) is finite, so we may assume that it is infinite and hence contractible (Theorem 4.127). Let \( {\sum }^{\prime } \) be the set of nonempty simplices. Then the flag co... | No |
Theorem 12.58. For any Coxeter system \( \left( {W, S}\right) \), the space \( X = \left| {{\sum }_{d}\left( {W, S}\right) }\right| \) with its piecewise Euclidean metric is a \( \operatorname{CAT}\left( 0\right) \) space. | The proof is long and we will not give it here. Moussong's original proof can be found in [170, Theorem 14.1]. See [89, Section 12.3; 90, Corollary 6.7.5; 149, Appendix B] for other proofs. | No |
Corollary 12.59. If \( \left( {W, S}\right) \) is an arbitrary Coxeter system, then every finite subgroup of \( W \) is conjugate to a subgroup of a finite standard subgroup. | Proof. The point stabilizers \( {W}_{x} \) for the \( W \) -action on \( X \) are the conjugates of the finite standard subgroups. The corollary now follows from the Bruhat-Tits fixed-point theorem (Theorem 11.23). | Yes |
Theorem 12.60. The following conditions on a Coxeter system \( \left( {W, S}\right) \) are equivalent:\n\n(i) \( W \) is hyperbolic in the sense of Gromov.\n\n(ii) \( W \) does not contain a free abelian subgroup of rank 2.\n\n(iii) For all \( J \subseteq S \), the standard parabolic subgroup \( {W}_{J} \) has at most ... | Moussong's original proof can be found in [170, proof of Theorem 17.1]. See also [89, Section 12.6; 90, Theorem 11.1]. | No |
Proposition 12.63. Let \( \left( {W, S}\right) \) be an arbitrary Coxeter system. There is a cubical complex \( {\sum }_{c}\left( {W, S}\right) \) that subdivides \( {\sum }_{d}\left( {W, S}\right) \) . Its poset of cells can be identified with the set of closed intervals \( \left\lbrack {w{W}_{J}, w{W}_{K}}\right\rbra... | Sketch of proof. It is is easy to verify that the poset of finite standard cosets has a cubical realization, as defined in Section A.3. The existence of \( {\sum }_{c} \) now follows easily from Proposition A.38. (See also Remark A.39.) The remaining assertions are equally easy and are left to the interested reader. No... | No |
Theorem 12.66. For any building \( \Delta \), its Davis realization \( X = Z\left( \Delta \right) \) is a complete \( \operatorname{CAT}\left( 0\right) \) space. | Proof. \( X \) is a geodesic metric space by Corollary 12.28, and it is complete by Proposition 12.10. The hard part is that \( X \) is a \( \operatorname{CAT}\left( 0\right) \) space. In view of Proposition 12.29, we need only check this when \( \Delta \) consists of a single apartment, and for this we have Moussong's... | Yes |
Let \( H \) be a group of type-preserving automorphisms of a building \( \Delta \) . If \( H \) stabilizes a bounded set of chambers, then \( H \) fixes a spherical simplex. Equivalently, \( H \) stabilizes a spherical residue in \( \mathcal{C}\left( \Delta \right) \) . | Proof. Since \( Z \) is compact, a bound on the combinatorial distance between two chambers \( C, D \) yields a bound on the distance in \( X \) between points of \( Z\left( C\right) \) and points of \( Z\left( D\right) \) . So \( H \) stabilizes a bounded subset of \( X \) and therefore has a fixed point \( x \) . In ... | Yes |
Theorem 13.8. Let \( G \) be a simply connected and absolutely almost simple linear algebraic group defined over \( \mathbb{Q} \). Then, with the notation above, any torsion-free \( S \) -arithmetic subgroup of \( G\left( \mathbb{Q}\right) \) is finitely presented and of type FL and is a duality group of dimension \( d... | We now sketch the method actually used by Borel and Serre [42] to prove Theorem 13.8. Instead of letting the torsion-free \( S \) -arithmetic group \( \Gamma \) act on the various \( {X}_{p} \) one at a time, they let it act on them simultaneously. More precisely, let \( {L}_{p} \mathrel{\text{:=}} G\left( {\mathbb{Q}}... | Yes |
Theorem 13.14. Let \( F \) be an algebraic extension of \( \mathbb{Q} \) and let \( \Gamma \) be an arbitrary finitely generated subgroup of \( {\mathrm{{GL}}}_{n}\left( F\right) \) . Then \( \operatorname{vcd}\Gamma < \infty \) . | To prove this, we may assume that \( F \) is finite over \( \mathbb{Q} \), and then we can easily reduce to the case \( F = \mathbb{Q} \) . [An \( n \) -dimensional vector space over \( F \) is a finite-dimensional vector space over \( \mathbb{Q} \) .] Then \( \Gamma \leq {\mathrm{{GL}}}_{n}\left( A\right) \) for some ... | Yes |
Theorem 13.15. Let \( \Gamma \) be a finitely generated subgroup of \( {\mathrm{{GL}}}_{n}\left( F\right) \), where \( F \) is a field of characteristic 0 . Then \( \operatorname{vcd}\Gamma < \infty \) if and only if there is an upper bound on the Hirsch ranks of the unipotent subgroups of \( \Gamma \) . | Recall that any unipotent subgroup \( U \) of \( {\mathrm{{GL}}}_{n}\left( F\right) \) is torsion-free and nilpotent by Kolchin’s theorem (Section C.7). So the Hirsch rank of \( U \) is indeed defined and differs from \( \operatorname{cd}U \) by at most 1 (see Section 13.1.5). Thus \ | No |
Theorem 13.18. For any \( n \) there is an integer \( N \) such that \( {\mathrm{{SL}}}_{n}\left( {{\mathbb{F}}_{q}\left\lbrack t\right\rbrack }\right) \) has finiteness length \( n - 2 \) if \( q \geq N \) . | If \( n \leq 5 \), Abramenko has shown that one can take \( N = 2 \), i.e., there is no restriction on \( q \) . If \( n \geq 6 \), however, the best known value of \( N \) is \( N = \) \( \mathop{\max }\limits_{{1 \leq i \leq n - 2}}\left( \begin{matrix} n - 2 \\ i \end{matrix}\right) \), again due to Abramenko, but t... | Yes |
Proposition 14.1. Suppose a group \( G \) acts on a simply connected simplicial complex \( \Delta \), and suppose \( F \subseteq \Delta \) is a simplicial fundamental domain in the sense of Definition 3.74. Then \( G \) is the sum of the vertex stabilizers \( {G}_{v} \) amalgamated along the edge stabilizers \( {G}_{e}... | This means, by definition, that \( G \) is the direct limit of the system consisting of the groups \( {G}_{v} \) and \( {G}_{e} \), together with the inclusions\n\n\n\nwhenever \( e \) is an edge of \( F \) with vert... | Yes |
For pedagogical reasons we compute the distribution function \( {d}_{f} \) of a nonnegative simple function\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{N}{a}_{j}{\chi }_{{E}_{j}}\left( x\right) \]\n\nwhere the sets \( {E}_{j} \) are pairwise disjoint and \( {a}_{1} > \cdots > {a}_{N} > 0 \) . If \( \alpha... | Setting\n\n\[ {B}_{j} = \mathop{\sum }\limits_{{k = 1}}^{j}\mu \left( {E}_{k}\right) \]\n\nfor \( j \in \{ 1,\ldots, N\} ,{B}_{0} = {a}_{N + 1} = 0 \), and \( {a}_{0} = \infty \), we have\n\n\[ {d}_{f}\left( \mathbf{\alpha }\right) = \mathop{\sum }\limits_{{j = 0}}^{N}{B}_{j}{\chi }_{\left\lbrack {a}_{j + 1},{a}_{j}\ri... | Yes |
Proposition 1.1.3. Let \( f \) and \( g \) be measurable functions on \( \left( {X,\mu }\right) \) . Then for all \( \alpha ,\beta > 0 \) we have\n\n(1) \( \left| g\right| \leq \left| f\right| \mu \) -a.e. implies that \( {d}_{g} \leq {d}_{f} \) ;\n\n(2) \( {d}_{cf}\left( \alpha \right) = {d}_{f}\left( {\alpha /\left| ... | Proof. The simple proofs are left to the reader. | No |
Proposition 1.1.4. Let \( \left( {X,\mu }\right) \) be a \( \sigma \) -finite measure space. Then for \( f \) in \( {L}^{p}\left( {X,\mu }\right) \) , \( 0 < p < \infty \), we have\n\n\[ \parallel f{\parallel }_{{L}^{p}}^{p} = p{\int }_{0}^{\infty }{\alpha }^{p - 1}{d}_{f}\left( \alpha \right) {d\alpha }.\ ]\n\n(1.1.6)... | Proof. Indeed, we have\n\n\[ p{\int }_{0}^{\infty }{\alpha }^{p - 1}{d}_{f}\left( \alpha \right) {d\alpha } = p{\int }_{0}^{\infty }{\alpha }^{p - 1}{\int }_{X}{\chi }_{\{ x : \left| {f\left( x\right) }\right| > \alpha \} }{d\mu }\left( x\right) {d\alpha }\ ]\n\n\[ = {\int }_{X}{\int }_{0}^{\left| f\left( x\right) \rig... | Yes |
Proposition 1.1.6. For any \( 0 < p < \infty \) and any \( f \) in \( {L}^{p}\left( {X,\mu }\right) \) we have\n\n\[ \parallel f{\parallel }_{{L}^{p,\infty }} \leq \parallel f{\parallel }_{{L}^{p}} \]\n\nHence the embedding \( {L}^{p}\left( {X,\mu }\right) \subseteqq {L}^{p,\infty }\left( {X,\mu }\right) \) holds. | Proof. This is just a trivial consequence of Chebyshev's inequality:\n\n\[ {\alpha }^{p}{d}_{f}\left( \alpha \right) \leq {\int }_{\{ x : \left| {f\left( x\right) }\right| > \alpha \} }{\left| f\left( x\right) \right| }^{p}{d\mu }\left( x\right) \leq \parallel f{\parallel }_{{L}^{p}}^{p}. \]\n\nUsing (1.1.9) we obtain ... | Yes |
Proposition 1.1.9. Let \( 0 < p \leq \infty \) and \( {f}_{n}, f \) be in \( {L}^{p,\infty }\left( {X,\mu }\right) \) . (1) If \( {f}_{n}, f \) are in \( {L}^{p} \) and \( {f}_{n} \rightarrow f \) in \( {L}^{p} \), then \( {f}_{n} \rightarrow f \) in \( {L}^{p,\infty } \) . | Proof. Fix \( 0 < p < \infty \) . Proposition 1.1.6 gives that for all \( \varepsilon > 0 \) we have \[ \mu \left( \left\{ {x \in X : \left| {{f}_{n}\left( x\right) - f\left( x\right) }\right| > \varepsilon }\right\} \right) \leq \frac{1}{{\varepsilon }^{p}}{\int }_{X}{\left| {f}_{n} - f\right| }^{p}{d\mu }. \] This sh... | Yes |
Theorem 1.1.13. Let \( \left( {X,\mu }\right) \) be a measure space and let \( \left\{ {f}_{n}\right\} \) be a complex-valued sequence on \( X \) that is Cauchy in measure. Then some subsequence of \( {f}_{n} \) converges \( \mu \) -a.e. | Proof. The proof is very similar to that of Theorem 1.1.11. For all \( k = 1,2,\ldots \) choose \( {n}_{k} \) inductively such that\n\n\[ \mu \left( \left\{ {x \in X : \left| {{f}_{{n}_{k}}\left( x\right) - {f}_{{n}_{k + 1}}\left( x\right) }\right| > {2}^{-k}}\right\} \right) < {2}^{-k} \]\n\n(1.1.20)\n\nand such that ... | Yes |
Proposition 1.1.14. Let \( 0 < p < q \leq \infty \) and let \( f \) in \( {L}^{p,\infty }\left( {X,\mu }\right) \cap {L}^{q,\infty }\left( {X,\mu }\right) \), where \( X \) is a \( \sigma \) -finite measure space. Then \( f \) is in \( {L}^{r}\left( {X,\mu }\right) \) for all \( p < r < q \) and\n\n\[ \parallel f{\para... | Proof. Let us take first \( q < \infty \) . We know that\n\n\[ {d}_{f}\left( \mathbf{\alpha }\right) \leq \min \left( {\frac{\parallel f{\parallel }_{{L}^{p,\infty }}^{p}}{{\mathbf{\alpha }}^{p}},\frac{\parallel f{\parallel }_{{L}^{q,\infty }}^{q}}{{\mathbf{\alpha }}^{q}}}\right) . \]\n\nSet\n\n\[ B = {\left( \frac{\pa... | Yes |
Let \( G = {\mathbf{R}}^{ * } = \mathbf{R} \smallsetminus \{ 0\} \) with group law the usual multiplication. It is easy to verify that the measure \( \lambda = {dx}/\left| x\right| \) is invariant under multiplicative translations, that is, | \[ {\int }_{-\infty }^{\infty }f\left( {tx}\right) \frac{dx}{\left| x\right| } = {\int }_{-\infty }^{\infty }f\left( x\right) \frac{dx}{\left| x\right| } \] for all \( f \) in \( {L}^{1}\left( {G,\mu }\right) \) and all \( t \in {\mathbf{R}}^{ * } \) . Therefore, \( {dx}/\left| x\right| \) is a Haar measure. [Taking \(... | Yes |
Example 1.2.5. The Heisenberg group \( {\mathbf{H}}^{n} \) is the set \( {\mathbf{C}}^{n} \times \mathbf{R} \) with the group operation\n\n\[ \n\left( {{z}_{1},\ldots ,{z}_{n}, t}\right) \left( {{w}_{1},\ldots ,{w}_{n}, s}\right) = \left( {{z}_{1} + {w}_{1},\ldots ,{z}_{n} + {w}_{n}, t + s + 2\operatorname{Im}\mathop{\... | It can easily be seen that the identity element \( e \) of this group is \( 0 \in {\mathbf{C}}^{n} \times \mathbf{R} \) and \( {\left( {z}_{1},\ldots ,{z}_{n}, t\right) }^{-1} = \left( {-{z}_{1},\ldots , - {z}_{n}, - t}\right) \) . Topologically the Heisenberg group is identified with \( {\mathbf{C}}^{n} \times \mathbf... | Yes |
On R let \( f\left( x\right) = 1 \) when \( - 1 \leq x \leq 1 \) and zero otherwise. We see that \( \left( {f * f}\right) \left( x\right) \) is equal to the length of the intersection of the intervals \( \left\lbrack {-1,1}\right\rbrack \) and \( \left\lbrack {x - 1, x + 1}\right\rbrack \) . | It follows that \( \left( {f * f}\right) \left( x\right) = 2 - \left| x\right| \) for \( \left| x\right| \leq 2 \) and zero otherwise. Observe that \( f * f \) is a smoother function than \( f \) . Similarly, we obtain that \( f * f * f \) is a smoother function than \( f * f \) . | Yes |
Proposition 1.2.9. For all \( f, g, h \) in \( {L}^{1}\left( G\right) \), the following properties are valid:\n\n(1) \( f * \left( {g * h}\right) = \left( {f * g}\right) * h \) (associativity)\n\n\( \left( 2\right) \;f * \left( {g + h}\right) = f * g + f * h\; \) and \( \;\left( {f + g}\right) * h = f * h + g * h\; \) ... | Proof. The easy proofs are omitted. | No |
Theorem 1.2.10. (Minkowski’s inequality) Let \( 1 \leq p \leq \infty \) . For \( f \) in \( {L}^{p}\left( G\right) \) and \( g \) in \( {L}^{1}\left( G\right) \) we have that \( g * f \) exists \( \lambda \) -a.e. and satisfies\n\n\[ \parallel g * f{\parallel }_{{L}^{p}\left( G\right) } \leq \parallel g{\parallel }_{{L... | Proof. Estimate (1.2.7) follows directly from Exercise 1.1.6. Here we give a direct proof. We may assume that \( 1 < p < \infty \), since the cases \( p = 1 \) and \( p = \infty \) are simple. We first show that the convolution \( \left| g\right| * \left| f\right| \) exists \( \lambda \) -a.e. Indeed,\n\n\[ \left( {\le... | Yes |
Theorem 1.2.13. (Young’s inequality for weak type spaces) Let \( G \) be a locally compact group with left Haar measure \( \lambda \) that satisfies (1.2.12). Let \( 1 \leq p < \infty \) and \( 1 < q, r < \infty \) satisfy \[ \frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r} \] Then there exists a constant \( {C}_{p, q, r} >... | Proof. As in the proofs of Theorems 1.2.10 and 1.2.12, we first obtain (1.2.16) for the convolution of the absolute values of the functions. This implies that \( \left| f\right| * \left| g\right| < \infty \) \( \lambda \) -a.e., and thus \( f * g \) exists \( \lambda \) -a.e. and satisfies \( \left| {f * g}\right| \leq... | No |
Theorem 1.2.13 may fail at some endpoints: | (1) \( r = 1 \) and \( 1 \leq p = q \leq \infty \) . On \( \mathbf{R} \) take \( g\left( x\right) = 1/\left| x\right| \) and \( f = {\chi }_{\left\lbrack 0,1\right\rbrack } \) . Clearly, \( g \) is in \( {L}^{1,\infty } \) and \( f \) in \( {L}^{p} \) for all \( 1 \leq p \leq \infty \), but the convolution of \( f \) a... | Yes |
On R let \( P\left( x\right) = {\left( \pi \left( {x}^{2} + 1\right) \right) }^{-1} \) and \( {P}_{\varepsilon }\left( x\right) = {\varepsilon }^{-1}P\left( {{\varepsilon }^{-1}x}\right) \) for \( \varepsilon > 0 \) . Since \( {P}_{\varepsilon } \) and \( P \) have the same \( {L}^{1} \) norm and | \[ {\int }_{-\infty }^{+\infty }\frac{1}{{x}^{2} + 1}{dx} = \mathop{\lim }\limits_{{x \rightarrow + \infty }}\left\lbrack {\arctan \left( x\right) - \arctan \left( {-x}\right) }\right\rbrack = \left( {\pi /2}\right) - \left( {-\pi /2}\right) = \pi ,\] property (ii) is satisfied. Property (iii) follows from the fact tha... | Yes |
On \( {\mathbf{R}}^{n} \) let \( k\left( x\right) \) be an integrable function with integral one. Let \( {k}_{\varepsilon }\left( x\right) = {\varepsilon }^{-n}k\left( {{\varepsilon }^{-1}x}\right) \) . It is straightforward to see that \( {k}_{\varepsilon }\left( x\right) \) is an approximate identity. Property (iii) ... | \[ {\int }_{\left| x\right| \geq \delta /\varepsilon }\left| {k\left( x\right) }\right| {dx} \rightarrow 0 \] as \( \varepsilon \rightarrow 0 \) for \( \delta \) fixed. | Yes |
On the circle group \( {\mathbf{T}}^{1} \) let\n\n\[ \n{F}_{N}\left( t\right) = \mathop{\sum }\limits_{{j = - N}}^{N}\left( {1 - \frac{\left| j\right| }{N + 1}}\right) {e}^{2\pi ijt} = \frac{1}{N + 1}{\left( \frac{\sin \left( {\pi \left( {N + 1}\right) t}\right) }{\sin \left( {\pi t}\right) }\right) }^{2}. \n\]\n\n(1.2... | To check the previous equality we use that\n\n\[ \n{\sin }^{2}\left( x\right) = \left( {2 - {e}^{2ix} - {e}^{-{2ix}}}\right) /4 \n\]\n\nand we carry out the calculation. | No |
Theorem 1.2.19. Let \( {k}_{\varepsilon } \) be an approximate identity on a locally compact group \( G \) with left Haar measure \( \lambda \) . (1) If \( f \) lies in \( {L}^{p}\left( G\right) \) for \( 1 \leq p < \infty \), then \( {\begin{Vmatrix}{k}_{\varepsilon } * f - f\end{Vmatrix}}_{{L}^{p}\left( G\right) } \r... | Proof. We start with the case \( 1 \leq p < \infty \) . We recall that continuous functions with compact support are dense in \( {L}^{p} \) of locally compact Hausdorff spaces equipped with measures arising from nonnegative linear functionals; see [152, Theorem 12.10]. For a continuous function \( g \) supported in a c... | Yes |
Theorem 1.3.2. Let \( \left( {X,\mu }\right) \) be a \( \sigma \) -finite measure space, let \( \left( {Y,\nu }\right) \) be another measure space, and let \( 0 < {p}_{0} < {p}_{1} \leq \infty \) . Let \( T \) be a sublinear operator defined on \( {L}^{{p}_{0}}\left( X\right) + {L}^{{p}_{1}}\left( X\right) = \left\{ {{... | Proof. Assume first that \( {p}_{1} < \infty \) . Fix \( f \) a function in \( {L}^{p}\left( X\right) \) and \( \alpha > 0 \) . We split \( f = {f}_{0}^{\alpha } + {f}_{1}^{\alpha } \), where \( {f}_{0}^{\alpha } \) is in \( {L}^{{p}_{0}} \) and \( {f}_{1}^{\alpha } \) is in \( {L}^{{p}_{1}} \) . The splitting is obtai... | Yes |
Theorem 1.3.4. Let \( \\left( {X,\\mu }\\right) \) and \( \\left( {Y, v}\\right) \) be two \( \\sigma \) -finite measure spaces. Let \( T \) be a linear operator defined on the set of all finitely simple functions on \( X \) and taking values in the set of measurable functions on \( Y \) . Let \( 1 \\leq {p}_{0},{p}_{1... | Proof. Let\n\n\[ \nf = \\mathop{\\sum }\\limits_{{k = 1}}^{m}{a}_{k}{e}^{i{\\alpha }_{k}}{\\chi }_{{A}_{k}} \n\]\n\nbe a finitely simple function on \( X \), where \( {a}_{k} > 0,{\\alpha }_{k} \) are real, and \( {A}_{k} \) are pairwise disjoint subsets of \( X \) with finite measure.\n\nWe need to control\n\n\[ \n\\p... | Yes |
Lemma 1.3.5. Let \( F \) be analytic in the open strip \( S = \{ z \in \mathbf{C} : 0 < \operatorname{Re}z < 1\} \) , continuous and bounded on its closure, such that \( \left| {F\left( z\right) }\right| \leq {B}_{0} \) when \( \operatorname{Re}z = 0 \) and \( \left| {F\left( z\right) }\right| \leq {B}_{1} \) when \( \... | To prove the lemma we define analytic functions\n\n\[ G\left( z\right) = F\left( z\right) {\left( {B}_{0}^{1 - z}{B}_{1}^{z}\right) }^{-1}\;\text{ and }\;{G}_{n}\left( z\right) = G\left( z\right) {e}^{\left( {{z}^{2} - 1}\right) /n} \]\n\nfor \( z \) in the unit strip \( S \), for \( n = 1,2,\ldots \) Since \( F \) is ... | Yes |
One may prove Young's inequality (Theorem 1.2.12) using the Riesz-Thorin interpolation theorem (Theorem 1.3.4). Fix a function \( g \) in \( {L}^{r} \) and let \( T\left( f\right) = f * g \) . Since \( T : {L}^{1} \rightarrow {L}^{r} \) with norm at most \( \parallel g{\parallel }_{{L}^{r}} \) and \( T : {L}^{{r}^{\pri... | \[ \frac{1}{p} = \frac{1 - \theta }{1} + \frac{\theta }{{r}^{\prime }}\;\text{ and }\;\frac{1}{q} = \frac{1 - \theta }{r} + \frac{\theta }{\infty }. \] (1.3.19) Finally, observe that equations (1.3.19) give (1.2.13). | Yes |
Theorem 1.3.7. Let \( {T}_{z} \) be an analytic family of linear operators of admissible growth defined on the space of finitely simple functions of a \( \sigma \) -finite measure space \( \left( {X,\mu }\right) \) and taking values in the set of measurable functions of another \( \sigma \) -finite measure space \( \le... | Note that in view of (1.3.24), the integral defining \( M\left( t\right) \) converges absolutely. The proof of the previous theorem is based on an extension of Lemma 1.3.5. | No |
Consider the simple function of Example 1.1.2,\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{N}{a}_{j}{\chi }_{{E}_{j}}\left( x\right) \]\n\nwhere \( {E}_{j} \) are pairwise disjoint sets of finite measure and \( {a}_{1} > \cdots > {a}_{N} > 0 \) . We saw in Example 1.1.2 that\n\n\[ {d}_{f}\left( \mathbf{\a... | Observe that for \( {B}_{0} \leq t < {B}_{1} \), the smallest \( s > 0 \) with \( {d}_{f}\left( s\right) \leq t \) is \( {a}_{1} \) . Similarly, for \( {B}_{1} \leq t < {B}_{2} \), the smallest \( s > 0 \) with \( {d}_{f}\left( s\right) \leq t \) is \( {a}_{2} \) . Arguing this way, it is not difficult to see that\n\n\... | Yes |
On \( \left( {{\mathbf{R}}^{n},{dx}}\right) \) let\n\n\[ f\left( x\right) = \frac{1}{1 + {\left| x\right| }^{p}},\;0 < p < \infty . \]\n | A computation shows that\n\n\[ {d}_{f}\left( \alpha \right) = \left\{ \begin{array}{ll} {v}_{n}{\left( \frac{1}{\alpha } - 1\right) }^{\frac{n}{p}} & \text{ if }\alpha < 1, \\ 0 & \text{ if }\alpha \geq 1, \end{array}\right. \]\n\nand therefore\n\n\[ {f}^{ * }\left( t\right) = \frac{1}{{\left( t/{v}_{n}\right) }^{p/n} ... | Yes |
Example 1.4.4. Again on \( \left( {{\mathbf{R}}^{n},{dx}}\right) \) let \( g\left( x\right) = 1 - {e}^{-{\left| x\right| }^{2}} \) . We can easily see that \( {d}_{g}\left( \alpha \right) = 0 \) if \( \alpha \geq 1 \) and \( {d}_{g}\left( \alpha \right) = \infty \) if \( \alpha < 1 \) . We conclude that \( {g}^{ * }\le... | This example indicates that although quantitative information is preserved, significant qualitative information is lost in passing from a function to its decreasing rearrangement. | No |
Proposition 1.4.5. For \( f, g,{f}_{n}\mu \) -measurable, \( k \in \mathbf{C} \), and \( 0 \leq t, s,{t}_{1},{t}_{2} < \infty \) we have\n\n(1) \( {f}^{ * }\left( {{d}_{f}\left( \alpha \right) }\right) \leq \alpha \) whenever \( \alpha > 0 \) . | Proof. Property (1): The set \( A = \left\{ {s > 0 : {d}_{f}\left( s\right) \leq {d}_{f}\left( \alpha \right) }\right\} \) contains \( \alpha \) and thus \( {f}^{ * }\left( {{d}_{f}\left( \alpha \right) }\right) = \inf A \leq \alpha . | Yes |
Using the notation of Example 1.4.2, when \( 0 < p, q < \infty \) we have\n\n\[ \parallel f{\parallel }_{{L}^{p, q}} = {\left( \frac{p}{q}\right) }^{\frac{1}{q}}{\left\lbrack {a}_{1}^{q}{B}_{1}^{\frac{q}{p}} + {a}_{2}^{q}\left( {B}_{2}^{\frac{q}{p}} - {B}_{1}^{\frac{q}{p}}\right) + \cdots + {a}_{N}^{q}\left( {B}_{N}^{\... | and also\n\n\[ \parallel f{\parallel }_{{L}^{p,\infty }} = \mathop{\sup }\limits_{{1 \leq j \leq N}}{a}_{j}{B}_{j}^{\frac{1}{p}}. \] | Yes |
Proposition 1.4.9. For \( 0 < p < \infty \) and \( 0 < q \leq \infty \), we have the identity\n\n\[ \n\parallel f{\parallel }_{{L}^{p, q}} = \left\{ \begin{array}{ll} {p}^{\frac{1}{q}}{\left( {\int }_{0}^{\infty }{\left\lbrack {d}_{f}{\left( s\right) }^{\frac{1}{p}}s\right\rbrack }^{q}\frac{ds}{s}\right) }^{\frac{1}{q}... | Proof. The case \( q = \infty \) is statement (16) in Proposition 1.4.5, and we may therefore concentrate on the case \( q < \infty \) . If \( f \) is the simple function of Example 1.1.2, then\n\n\[ \n{d}_{f}\left( s\right) = \mathop{\sum }\limits_{{j = 1}}^{N}{B}_{j}{\chi }_{\left\lbrack {a}_{j + 1},{a}_{j}\right) }\... | No |
Proposition 1.4.10. Suppose \( 0 < p \leq \infty \) and \( 0 < q < r \leq \infty \) . Then there exists a constant \( {c}_{p, q, r} \) (which depends on \( p, q \), and \( r \) ) such that\n\n\[ \parallel f{\parallel }_{{L}^{p, r}} \leq {c}_{p, q, r}\parallel f{\parallel }_{{L}^{p, q}} \]\n\n(1.4.6)\n\nIn other words, ... | Proof. We may assume \( p < \infty \), since the case \( p = \infty \) is trivial. We have\n\n\[ {t}^{1/p}{f}^{ * }\left( t\right) = {\left\{ \frac{q}{p}{\int }_{0}^{t}{\left\lbrack {s}^{1/p}{f}^{ * }\left( t\right) \right\rbrack }^{q}\frac{ds}{s}\right\} }^{1/q} \]\n\n\[ \leq {\left\{ \frac{q}{p}{\int }_{0}^{t}{\left\... | Yes |
Theorem 1.4.11. Let \( \left( {X,\mu }\right) \) be a measure space. Then for all \( 0 < p, q \leq \infty \), the spaces \( {L}^{p, q}\left( {X,\mu }\right) \) are complete with respect to their quasi-norm and they are therefore quasi-Banach spaces. | Proof. We consider only the case \( p < \infty \) . First we note that convergence in \( {L}^{p, q} \) implies convergence in measure. When \( q = \infty \), this is proved in Proposition 1.1.9. When \( q < \infty \), in view of Proposition 1.4.5 (16) and (1.4.7), it follows that\n\n\[ \mathop{\sup }\limits_{{t > 0}}{t... | Yes |
Theorem 1.4.13. Finitely simple functions are dense in \( {L}^{p, q}\left( {X,\mu }\right) \) when \( 0 < q < \infty \) . | Proof. Let \( f \in {L}^{p, q}\left( {X,\mu }\right) \) . Assume without loss of generality that \( f \geq 0 \) . Since \( f \) lies in \( {L}^{p, q} \subseteq {L}^{p,\infty } \) we have \( \mu {\left( \{ f > \varepsilon \} \right) }^{1/p}\varepsilon \leq \parallel f{\parallel }_{{L}^{p, q}} < \infty \) for every \( \v... | Yes |
Lemma 1.4.20. Let \( 0 < p < \infty \) and \( 0 < q \leq \infty \) and let \( \left( {X,\mu }\right) ,\left( {Y,\nu }\right) \) be \( \sigma \) -finite measure spaces. Let \( T \) be a quasi-linear operator defined on \( S\left( X\right) \) and taking values in the set of measurable functions on \( Y \) . Suppose that ... | Proof. A function \( f \) in \( {S}_{0}\left( X\right) \) can be written as \( f = {h}_{1} - {h}_{2} + i\left( {{h}_{3} - {h}_{4}}\right) \), where \( {h}_{j} \) are in \( {S}_{0}^{ + }\left( X\right) \) . We write \( f = {f}_{1} - {f}_{2} + i\left( {{f}_{3} - {f}_{4}}\right) \), where \( {f}_{1} = \max \left( {{h}_{1}... | Yes |
Proposition 1.4.5(2) implies \( \mu \left( {A}_{k}\right) \leq {d}_{f}\left( {{f}^{ * }\left( {2}^{k + 1}\right) }\right) \leq {2}^{k + 1} \) | \[ \parallel T\left( f\right) {\parallel }_{{L}^{q,\infty }\left( Y\right) } \leq 4{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{k = - N}}^{N}{\left| T\left( f{\chi }_{{A}_{k}}\right) \right| }^{\alpha }\right) }^{\frac{1}{\alpha }}\end{Vmatrix}}_{{L}^{q,\infty }\left( Y\right) } = 4{\begin{Vmatrix}\mathop{\sum }\lim... | Yes |
Corollary 1.4.22. Let \( T \) be as in the statement of Theorem 1.4.19 and let \( 0 < {p}_{0} \neq \) \( {p}_{1} \leq \infty \) and \( 0 < {q}_{0} \neq {q}_{1} \leq \infty \) . If \( T \) is restricted weak type \( \left( {{p}_{0},{q}_{0}}\right) \) and \( \left( {{p}_{1},{q}_{1}}\right) \) with constants \( {M}_{0} \)... | Proof. Since \( \theta \in \left( {0,1}\right) \) we must have \( p, q < \infty \) . Take \( r = q \) in Theorem 1.4.19 and note that \( \parallel f{\parallel }_{{L}^{p, r}} \leq \parallel f{\parallel }_{{L}^{p}} \) since \( p \leq q = r \) ; see Proposition 1.4.10. The last assertion follows using Exercise 1.4.17. | No |
Example 1.4.23. Let \( X = Y = \mathbf{R} \) and \[ T\left( f\right) \left( x\right) = {\left| x\right| }^{-1/2}{\int }_{0}^{1}f\left( t\right) {dt} \] Then \( \alpha {\left| \left\{ x : \left| T\left( {\chi }_{A}\right) \left( x\right) \right| > \alpha \right\} \right| }^{1/2} = {2}^{1/2}\left| {A \cap \left\lbrack {0... | The dual operator \[ S\left( f\right) \left( x\right) = {\chi }_{\left\lbrack 0,1\right\rbrack }\left( x\right) {\int }_{-\infty }^{+\infty }f\left( t\right) {\left| t\right| }^{-1/2}{dt} \] satisfies \( \alpha {\left| \left\{ x : \left| S\left( {\chi }_{A}\right) \left( x\right) \right| > \alpha \right\} \right| }^{1/... | Yes |
Corollary 1.4.24. Let \( 1 \leq r < \infty ,1 \leq {p}_{0} \neq {p}_{1} < \infty \), and \( 0 < {q}_{0} \neq {q}_{1} \leq \infty \) and let \( \left( {X,\mu }\right) \) and \( \left( {Y,\nu }\right) \) be \( \sigma \) -finite measure spaces. Let \( T \) be a quasi-linear operator defined on \( {L}^{{p}_{0}}\left( X\rig... | Proof. Since \( {L}^{p}\left( X\right) \) is contained in the sum \( {L}^{{p}_{0}}\left( X\right) + {L}^{{p}_{1}}\left( X\right) \), the operator \( T \) is well defined on \( {L}^{p}\left( X\right) \) . Hypothesis (1.4.42) implies that (1.4.30) holds for all \( f \in {L}^{{p}_{j},1} \) . Repeat the proof of Theorem 1.... | Yes |
Theorem 1.4.25. (Young’s inequality for weak type spaces) Let \( G \) be a locally compact group with left Haar measure \( \lambda \) that satisfies (1.2.12) for all measurable subsets A of G. Let \( 1 < p, q, r < \infty \) satisfy\n\n\[ \n\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r} \n\]\n\n(1.4.45)\n\nThen there exist... | Proof. We fix \( 1 < p, q < \infty \) . Since \( p \) and \( q \) range in an open interval, we can find \( {p}_{0} < p < {p}_{1},{q}_{0} < q < {q}_{1} \), and \( 0 < \theta < 1 \) such that (1.4.23) and (1.4.45) hold. Let \( T\left( f\right) = f * g \), defined for all functions \( f \) on \( G \) . By Theorem 1.2.13,... | Yes |
On \( \mathbf{R} \), let \( f \) be the characteristic function of the interval \( \left\lbrack {a, b}\right\rbrack \) . For \( x \in \left( {a, b}\right) \), clearly \( \mathcal{M}\left( f\right) = 1 \) . For \( x \geq b \), a simple calculation shows that the largest average of \( f \) over all intervals \( \left( {x... | \[ \mathcal{M}\left( f\right) \left( x\right) = \left\{ \begin{array}{ll} \left( {b - a}\right) /2\left| {x - b}\right| & \text{ when }x \leq a, \\ 1 & \text{ when }x \in \left( {a, b}\right) , \\ \left( {b - a}\right) /2\left| {x - a}\right| & \text{ when }x \geq b. \end{array}\right. \] | Yes |
On \( \mathbf{R} \), let \( f \) be the characteristic function of the interval \( I = \left\lbrack {a, b}\right\rbrack \) . For \( x \in \left( {a, b}\right) \), clearly \( M\left( f\right) \left( x\right) = 1 \) . For \( x > b \), a calculation shows that the largest average of \( f \) over all intervals \( \left( {y... | \[ M\left( f\right) \left( x\right) = \left\{ \begin{array}{ll} \left( {b - a}\right) /\left| {x - b}\right| & \text{ when }x \leq a, \\ 1 & \text{ when }x \in \left( {a, b}\right) , \\ \left( {b - a}\right) /\left| {x - a}\right| & \text{ when }x \geq b. \end{array}\right. \] | Yes |
Lemma 2.1.5. Let \( \left\{ {{B}_{1},{B}_{2},\ldots ,{B}_{k}}\right\} \) be a finite collection of open balls in \( {\mathbf{R}}^{n} \) . Then there exists a finite subcollection \( \left\{ {{B}_{{j}_{1}},\ldots ,{B}_{{j}_{l}}}\right\} \) of pairwise disjoint balls such that\n\n\[ \mathop{\sum }\limits_{{r = 1}}^{l}\le... | Proof. Let us reindex the balls so that\n\n\[ \left| {B}_{1}\right| \geq \left| {B}_{2}\right| \geq \cdots \geq \left| {B}_{k}\right| \]\n\nLet \( {j}_{1} = 1 \) . Having chosen \( {j}_{1},{j}_{2},\ldots ,{j}_{i} \), let \( {j}_{i + 1} \) be the least index \( s > {j}_{i} \) such that \( \mathop{\bigcup }\limits_{{m = ... | Yes |
Theorem 2.1.6. The uncentered and centered Hardy-Littlewood maximal operators \( M \) and \( \mathcal{M} \) map \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{1,\infty }\left( {\mathbf{R}}^{n}\right) \) with constant at most \( {3}^{n} \) and also \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{p}\left( ... | Proof. We claim that the set \( {E}_{\alpha } = \left\{ {x \in {\mathbf{R}}^{n} : M\left( f\right) \left( x\right) > \alpha }\right\} \) is open. Indeed, for \( x \in {E}_{\alpha } \), there is an open ball \( {B}_{x} \) that contains \( x \) such that the average of \( \left| f\right| \) over \( {B}_{x} \) is strictly... | Yes |
Example 2.1.8. Let \( R > 0 \) . Then we have\n\n\[ \frac{{R}^{n}}{{\left( \left| x\right| + R\right) }^{n}} \leq M\left( {\chi }_{B\left( {0, R}\right) }\right) \left( x\right) \leq \frac{{6}^{n}{R}^{n}}{{\left( \left| x\right| + R\right) }^{n}}. \] | The lower estimate in (2.1.6), is an easy consequence of the fact that the ball \( B\left( {x,\left| x\right| + R}\right) \) contains the ball \( B\left( {0, R}\right) \) . For the upper estimate, we first consider the case where \( \left| x\right| \leq {2R} \), when clearly \( M\left( {\chi }_{B\left( {0, R}\right) }\... | Yes |
Example 2.1.13. Let\n\n\[ \nP\left( x\right) = \frac{{c}_{n}}{{\left( 1 + {\left| x\right| }^{2}\right) }^{\frac{n + 1}{2}}}, \]\n\nwhere \( {c}_{n} \) is a constant such that\n\n\[ \n{\int }_{{\mathbf{R}}^{n}}P\left( x\right) {dx} = 1 \]\n\nThe function \( P \) is called the Poisson kernel. We define \( {L}^{1} \) dil... | Let us now compute the value of the constant \( {c}_{n} \) . Denote by \( {\omega }_{n - 1} \) the surface area of \( {\mathbf{S}}^{n - 1} \) . Using polar coordinates, we obtain\n\n\[ \n\frac{1}{{c}_{n}} = {\int }_{{\mathbf{R}}^{n}}\frac{dx}{{\left( 1 + {\left| x\right| }^{2}\right) }^{\frac{n + 1}{2}}} \]\n\n\[ \n= {... | Yes |
Theorem 2.1.14. Let \( 0 < p < \infty ,0 < q < \infty \), and \( {T}_{\varepsilon } \) and \( {T}_{ * } \) as previously. Suppose that for some \( B > 0 \) and all \( f \in {L}^{p}\left( X\right) \) we have\n\n\[ \n{\begin{Vmatrix}{T}_{ * }\left( f\right) \end{Vmatrix}}_{{L}^{q,\infty }} \leq B\parallel f{\parallel }_{... | Proof. Given \( f \) in \( {L}^{p} \), we define the oscillation of \( f \) :\n\n\[ \n{O}_{f}\left( y\right) = \mathop{\limsup }\limits_{{\varepsilon \rightarrow 0}}\mathop{\limsup }\limits_{{\theta \rightarrow 0}}\left| {{T}_{\varepsilon }\left( f\right) \left( y\right) - {T}_{\theta }\left( f\right) \left( y\right) }... | Yes |
Fix \( 1 \leq p < \infty \) and \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \). Let\n\n\[ P\left( x\right) = \frac{\Gamma \left( \frac{n + 1}{2}\right) }{{\pi }^{\frac{n + 1}{2}}}\frac{1}{{\left( 1 + {\left| x\right| }^{2}\right) }^{\frac{n + 1}{2}}} \]\n\nbe the Poisson kernel on \( {\mathbf{R}}^{n} \) and let \( {... | Let \( D \) be the set of all continuous functions with compact support on \( {\mathbf{R}}^{n} \). Since the family \( {\left( {P}_{\varepsilon }\right) }_{\varepsilon > 0} \) is an approximate identity, Theorem 1.2.19 (2) implies that for \( f \) in \( D \) we have that \( f * {P}_{\varepsilon } \rightarrow f \) unifo... | Yes |
Corollary 2.1.16. (Lebesgue's differentiation theorem) For any locally integrable function \( f \) on \( {\mathbf{R}}^{n} \) we have\n\n\[ \mathop{\lim }\limits_{{r \rightarrow 0}}\frac{1}{\left| B\left( x, r\right) \right| }{\int }_{B\left( {x, r}\right) }f\left( y\right) {dy} = f\left( x\right) \]\n\nfor almost all \... | Proof. Since \( {\mathbf{R}}^{n} \) is the union of the balls \( B\left( {0, N}\right) \) for \( N = 1,2,3\ldots \), it suffices to prove the required conclusion for almost all \( x \) inside a fixed ball \( B\left( {0, N}\right) \) . Given a locally integrable function \( f \) on \( {\mathbf{R}}^{n} \), consider the f... | Yes |
Corollary 2.1.17. (Differentiation theorem for approximate identities) Let \( K \) be an \( {L}^{1} \) function on \( {\mathbf{R}}^{n} \) with integral 1 that has a continuous integrable radially decreasing majorant. Then \( f * {K}_{\varepsilon } \rightarrow f \) a.e. as \( \varepsilon \rightarrow 0 \) for all \( f \i... | Proof. It follows from Example 1.2.17 that \( {K}_{\varepsilon } \) is an approximate identity. Theorem 1.2.19 now implies that \( f * {K}_{\varepsilon } \rightarrow f \) uniformly on compact sets when \( f \) is continuous. Let \( D \) be the space of all continuous functions with compact support. Then \( f * {K}_{\va... | Yes |
Corollary 2.1.19. (Differentiation theorem for multiples of approximate identities) Let \( K \) be a function on \( {\mathbf{R}}^{n} \) that has an integrable radially decreasing majorant.\n\nLet \( a = {\int }_{{\mathbf{R}}^{n}}K\left( x\right) {dx} \) . Then for all \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) a... | Proof. Use Theorem 1.2.21 instead of Theorem 1.2.19 in the proof of Corollary 2.1.17. | No |
Proposition 2.1.20. Given a nonnegative integrable function \( f \) on \( {\mathbf{R}}^{n} \) and \( \alpha > 0 \) , there exists a collection of disjoint (possibly empty) open cubes \( {Q}_{j} \) such that for almost all \( x \in {\left( \mathop{\bigcup }\limits_{j}{Q}_{j}\right) }^{c} \) we have \( f\left( x\right) \... | Proof. The proof provides an excellent paradigm of a stopping-time argument. Start by decomposing \( {\mathbf{R}}^{n} \) as a union of cubes of equal size, whose interiors are disjoint, and whose diameter is so large that \( {\left| Q\right| }^{-1}{\int }_{Q}f\left( x\right) {dx} \leq \alpha \) for every \( Q \) in thi... | Yes |
Corollary 2.1.21. Let \( f \geq 0 \) be an integrable function over a cube \( Q \) in \( {\mathbf{R}}^{n} \) and let \( \alpha \geq \frac{1}{\left| Q\right| }{\int }_{Q}{fdx} \) . Then there exist disjoint (possibly empty) open subcubes \( {Q}_{j} \) of \( Q \) such that for almost all \( x \in Q \smallsetminus \mathop... | Proof. The proof easily follows by a simple modification of Proposition 2.1.20 in which \( {\mathbf{R}}^{n} \) is replaced by the fixed cube \( Q \) . To apply Corollary 2.1.16, we extend \( f \) to be zero outside the cube \( Q \) . | Yes |
Proposition 2.2.6. Let \( f,{f}_{k}, k = 1,2,3,\ldots \), be in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . If \( {f}_{k} \rightarrow f \) in \( \mathcal{S} \) then \( {f}_{k} \rightarrow f \) in \( {L}^{p} \) for all \( 0 < p \leq \infty \) . Moreover, there exists a \( {C}_{p, n} > 0 \) such that\n\n\[ \n{\begi... | Proof. Observe that when \( p < \infty \) we have\n\n\[ \n{\begin{Vmatrix}{\partial }^{\beta }f\end{Vmatrix}}_{{L}^{p}} \leq {\left\lbrack {\int }_{\left| x\right| \leq 1}{\left| {\partial }^{\beta }f\left( x\right) \right| }^{p}dx + {\int }_{\left| x\right| \geq 1}{\left| x\right| }^{n + 1}{\left| {\partial }^{\beta }... | Yes |
Proposition 2.2.7. Let \( f, g \) be in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . Then \( {fg} \) and \( f * g \) are in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . Moreover,\n\n\[ \n{\partial }^{\alpha }\left( {f * g}\right) = \left( {{\partial }^{\alpha }f}\right) * g = f * \left( {{\partial }^{\alpha }... | Proof. Fix \( f \) and \( g \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . Let \( {e}_{j} \) be the unit vector \( \left( {0,\ldots ,1,\ldots ,0}\right) \) with 1 in the \( j \) th entry and zeros in all the other entries. Since \n\n\[ \n\frac{f\left( {y + h{e}_{j}}\right) - f\left( y\right) }{h} - \left( {{\p... | Yes |
If \( f\left( x\right) = {e}^{-\pi {\left| x\right| }^{2}} \) defined on \( {\mathbf{R}}^{n} \), then \( \widehat{f}\left( \xi \right) = f\left( \xi \right) \). | To prove this, observe that the function\n\n\[ s \mapsto {\int }_{-\infty }^{+\infty }{e}^{-\pi {\left( t + is\right) }^{2}}{dt},\;s \in \mathbf{R}, \]\n\ndefined on the line is constant (and thus equal to \( {\int }_{-\infty }^{+\infty }{e}^{-\pi {t}^{2}}{dt} \) ), since its derivative is\n\n\[ {\int }_{-\infty }^{+\i... | Yes |
Proposition 2.2.11. Given \( f, g \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right), y \in {\mathbf{R}}^{n}, b \in \mathbf{C},\alpha \) a multi-index, and \( t > 0 \) , we have\n\n(1) \( \parallel \widehat{f}{\parallel }_{{L}^{\infty }} \leq \parallel f{\parallel }_{{L}^{1}} \) | Proof. Property (1) follows directly from Definition 2.2.8. | Yes |
Corollary 2.2.12. The Fourier transform of a radial function is radial. Products and convolutions of radial functions are radial. | Proof. Let \( {\xi }_{1},{\xi }_{2} \) in \( {\mathbf{R}}^{n} \) with \( \left| {\xi }_{1}\right| = \left| {\xi }_{2}\right| \) . Then for some orthogonal matrix \( A \) we have \( A{\xi }_{1} = {\xi }_{2} \) . Since \( f \) is radial, we have \( f = f \circ A \) . Then\n\n\[ \widehat{f}\left( {\xi }_{2}\right) = \wide... | Yes |
Theorem 2.2.14. Given \( f, g \), and \( h \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \), we have\n\n(1) \( {\int }_{{\mathbf{R}}^{n}}f\left( x\right) \widehat{g}\left( x\right) {dx} = {\int }_{{\mathbf{R}}^{n}}\widehat{f}\left( x\right) g\left( x\right) {dx} \) | Proof. (1) follows immediately from the definition of the Fourier transform and Fubini's theorem. | No |
Corollary 2.2.15. The Fourier transform is a homeomorphism from \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) onto itself. | Proof. The continuity of the Fourier transform (and its inverse) follows from Exercise 2.2.2, while Fourier inversion yields that this map is bijective. | No |
Proposition 2.2.16. (Hausdorff-Young inequality) For every function \( f \) in \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) we have the estimate\n\n\[ \parallel \widehat{f}{\parallel }_{{L}^{{p}^{\prime }}} \leq \parallel f{\parallel }_{{L}^{p}} \]\n\nwhenever \( 1 \leq p \leq 2 \) . | Proof. This follows easily from Theorem 1.3.4. Interpolate between the estimates \( \parallel \widehat{f}{\parallel }_{{L}^{\infty }} \leq \parallel f{\parallel }_{{L}^{1}} \) (Proposition 2.2.11 (1)) and \( \parallel \widehat{f}{\parallel }_{{L}^{2}} \leq \parallel f{\parallel }_{{L}^{2}} \) to obtain \( \parallel \wi... | Yes |
Proposition 2.2.17. (Riemann-Lebesgue lemma) For a function \( f \) in \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) we have that\n\n\[ \left| {\widehat{f}\left( \xi \right) }\right| \rightarrow 0\;\text{ as }\;\left| \xi \right| \rightarrow \infty . \]\n | Proof. Consider the function \( {\chi }_{\left\lbrack a, b\right\rbrack } \) on \( \mathbf{R} \) . A simple computation gives\n\n\[ \widehat{{\chi }_{\left\lbrack a, b\right\rbrack }}\left( \xi \right) = {\int }_{a}^{b}{e}^{-{2\pi ix\xi }}{dx} = \frac{{e}^{-{2\pi i\xi a}} - {e}^{-{2\pi i\xi b}}}{2\pi i\xi }, \]\n\nwhic... | Yes |
We would like to find a Schwartz function \( f\left( {{x}_{1},{x}_{2},{x}_{3}}\right) \) on \( {\mathbf{R}}^{3} \) that satisfies the partial differential equation\n\n\[ f\left( x\right) + {\partial }_{1}^{2}{\partial }_{2}^{2}{\partial }_{3}^{4}f\left( x\right) + {4i}{\partial }_{1}^{2}f\left( x\right) + {\partial }_{... | Taking the Fourier transform on both sides of this identity and using Proposition 2.2.11 (2), (9) and the result of Example 2.2.9, we obtain\n\n\[ \widehat{f}\left( \xi \right) \left\lbrack {1 + {\left( 2\pi i{\xi }_{1}\right) }^{2}{\left( 2\pi i{\xi }_{2}\right) }^{2}{\left( 2\pi i{\xi }_{3}\right) }^{4} + {4i}{\left(... | Yes |
Let \( \varphi \) be a nonzero \( {\mathcal{C}}_{0}^{\infty } \) function on \( \mathbf{R} \) . We call such functions smooth bumps. Define the sequence of smooth bumps \( {\varphi }_{k}\left( x\right) = \varphi \left( {x - k}\right) /k \) . Then \( {\varphi }_{k}\left( x\right) \) does not converge to zero in \( {\mat... | Clearly \( {\varphi }_{k} \rightarrow 0 \) in \( {\mathcal{C}}^{\infty }\left( \mathbf{R}\right) \) . | No |
Proposition 2.3.4. (a) A linear functional \( u \) on \( {\mathcal{C}}_{0}^{\infty }\left( {\mathbf{R}}^{n}\right) \) is a distribution if and only if for every compact \( K \subseteq {\mathbf{R}}^{n} \), there exist \( C > 0 \) and an integer \( m \) such that | Proof. We prove only (2.3.3), since the proofs of (2.3.2) and (2.3.4) are similar. It is clear that (2.3.3) implies continuity of \( u \) . Conversely, it was pointed out in Section 2.2 that the family of sets \( \left\{ {f \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) : {\rho }_{\alpha ,\beta }\left( f\right) < \delta... | No |
We observe that \( \widehat{{\delta }_{0}} = 1 \) . More generally, for any multi-index \( \alpha \) we have\n\n\[ \n{\left( {\partial }^{\alpha }{\delta }_{0}\right) }^{ \land } = {\left( 2\pi ix\right) }^{\alpha }\n\] | To see this, observe that for all \( f \in \mathcal{S} \) we have\n\n\[ \n\left\langle {{\left( {\partial }^{\alpha }{\delta }_{0}\right) }^{ \land }, f}\right\rangle = \left\langle {{\partial }^{\alpha }{\delta }_{0},\widehat{f}}\right\rangle \n\]\n\n\[ \n= {\left( -1\right) }^{\left| \alpha \right| }\left\langle {{\d... | Yes |
Recall that for \( {x}_{0} \in {\mathbf{R}}^{n},{\delta }_{{x}_{0}}\left( f\right) = \left\langle {{\delta }_{{x}_{0}}, f}\right\rangle = f\left( {x}_{0}\right) \) . Then | \n\left\langle {\widehat{{\delta }_{{x}_{0}}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widehat{h}}\right\rangle = \widehat{h}\left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}h\left( x\right) {e}^{-{2\pi ix} \cdot {x}_{0}}{dx},\;h \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) ,\n\nthat is, \( \widehat{{\de... | Yes |
Example 2.3.14. Let \( u = {\delta }_{{x}_{0}} \) and \( f \in \mathcal{S} \) . Then \( f * {\delta }_{{x}_{0}} \) is the function \( x \mapsto f\left( {x - {x}_{0}}\right) \) , for when \( h \in \mathcal{S} \), we have | \[ \left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widetilde{f} * h}\right\rangle = \left( {\widetilde{f} * h}\right) \left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x - {x}_{0}}\right) h\left( x\right) {dx}. \] | Yes |
Proposition 2.3.22. Given \( u, v \) in \( {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) ,{f}_{j}, f \in \mathcal{S}, y \in {\mathbf{R}}^{n}, b \) a complex scalar, \( \alpha \) a multi-index, and \( a > 0 \), we have\n\n(1) \( \widehat{u + v} = \widehat{u} + \widehat{v} \) ,\n\n(2) \( \overset{⏜}{bu} = b\overs... | Proof. All the statements can be proved easily using duality and the corresponding statements for Schwartz functions. | No |
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