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Corollary 6.2.5. Let \( m \) be a bounded \( {\mathcal{C}}^{n} \) function defined away from the coordinate axes on \( {\mathbf{R}}^{n} \) . Assume that for all \( k \in \{ 1,\ldots, n\} \), all distinct \( {j}_{1},\ldots ,{j}_{k} \in \{ 1,2,\ldots, n\} \), and all \( {\xi }_{r} \in \mathbf{R} \smallsetminus \{ 0\} \) ... | Proof. Simply observe that condition (6.2.9) implies (6.2.5). | No |
The following are examples of functions that satisfy the hypotheses of Corollary 6.2.5: | The functions \( {m}_{1} \) and \( {m}_{2} \) are defined on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) and \( {m}_{3} \) on \( {\mathbf{R}}^{3} \smallsetminus \{ 0\} \). The previous examples and many other examples that satisfy the hypothesis (6.2.9) of Corollary 6.2.5 are invariant under a set of dilations in the ... | Yes |
Theorem 6.2.7. Let \( m\left( \xi \right) \) be a complex-valued bounded function on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) that satisfies for some \( A < \infty \n\[ \n{\left( {\int }_{R < \left| \xi \right| < {2R}}{\left| {\partial }_{\xi }^{\alpha }m\left( \xi \right) \right| }^{2}d\xi \right) }^{\frac{1}{2}} ... | Proof. Since \( m \) is a bounded function, the operator given by convolution with \( W = \) \( {m}^{ \vee } \) is bounded on \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) . To prove that this operator maps \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{1,\infty }\left( {\mathbf{R}}^{n}\right) \), it suffices to ... | Yes |
Let \( m \) be a smooth function away from the origin that is homogeneous of imaginary order, i.e., for some fixed \( \tau \) real and all \( \lambda > 0 \) we have\n\n\[ m\left( {\lambda \xi }\right) = {\lambda }^{i\tau }m\left( \xi \right) \] | Indeed, differentiating both sides of (6.2.21) with respect to \( {\partial }_{\xi }^{\alpha } \) we obtain\n\n\[ {\lambda }^{\left| \alpha \right| }{\partial }_{\xi }^{\alpha }m\left( {\lambda \xi }\right) = {\lambda }^{i\tau }{\partial }_{\xi }^{\alpha }m\left( \xi \right) \]\n\nand taking \( \lambda = {\left| \xi \r... | Yes |
Theorem 6.3.1. Let \( m \) be a bounded function on \( {\mathbf{R}}^{n} \) that is \( {\mathcal{C}}^{1} \) in a neighborhood of the origin and satisfies \( m\left( 0\right) = 1 \) and \( \left| {m\left( \xi \right) }\right| \leq C{\left| \xi \right| }^{-\varepsilon } \) for some \( C,\varepsilon > 0 \) and all \( \xi \... | Proof. Select a Schwartz function \( \varphi \) such that \( \widehat{\varphi }\left( 0\right) = 1 \) . Then there are positive constants \( {C}_{1} \) and \( {C}_{2} \) such that \( \left| {m\left( \xi \right) - \widehat{\varphi }\left( \xi \right) }\right| \leq {C}_{1}{\left| \xi \right| }^{-\varepsilon } \) for \( \... | Yes |
Theorem 6.3.2. Let \( 1 < p < \infty \) and let \( U \) be the \( n \) -fold product of open intervals that contain zero. For each \( k \in \mathbf{Z} \) define \( {T}_{k}\left( f\right) \left( x\right) = {\left( \widehat{f}\left( \xi \right) {\chi }_{U}\left( {2}^{-k}\xi \right) \right) }^{ \vee }\left( x\right) \) . ... | Proof. Let us fix an open annulus \( A \) whose interior contains the boundary of \( U \) and take a smooth function with compact support \( \widehat{\psi } \) that vanishes in a neighborhood of zero and a neighborhood of infinity and is equal to 1 on the annulus \( A \) . Then the function \( \widehat{\varphi } = \lef... | Yes |
Corollary 6.3.3. (a) Let \( f \) be in \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) and let \( \Omega \) be an open set that contains the origin in \( {\mathbf{R}}^{n} \) . Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{\int }_{{2}^{k}\Omega }\widehat{f}\left( \xi \right) {e}^{{2\pi ix} \cdot \xi }{d\xi } = ... | Proof. Both limits exist everywhere for functions \( f \) in the Schwartz class. To obtain almost everywhere convergence for general \( f \) in \( {L}^{p} \) we appeal to Theorem 2.1.14. The required control of the corresponding maximal operator is a consequence of Theorem 6.3.1 with \( m = {\chi }_{\Omega } \) in case... | No |
Corollary 6.3.5. Suppose that \( \mu \) is a finite Borel measure on \( {\mathbf{R}}^{n} \) with compact support that satisfies \( \left| {\widehat{\mu }\left( \xi \right) }\right| \leq B\min \left( {{\left| \xi \right| }^{-b},{\left| \xi \right| }^{b}}\right) \) for some \( b > 0 \) and all \( \xi \neq 0 \) . Define m... | Proof. To obtain the boundedness of the square function in (6.3.11) we use the Rademacher functions \( {r}_{j}\left( t\right) \), introduced in Appendix C.1, reindexed so that their index set is the set of all integers (not the set of nonnegative integers). For each \( t \) we introduce the operators\n\n\[ {T}_{\mu }^{... | Yes |
Proposition 6.4.5. For every locally integrable function \( f \) on \( \mathbf{R} \) and for all \( k \in \mathbf{Z} \) we have the identity\n\n\[ \n{D}_{k}\left( f\right) = \mathop{\sum }\limits_{{I \in {\mathcal{D}}_{k - 1}}}\left\langle {f,{h}_{I}}\right\rangle {h}_{I} \n\]\n\n(6.4.2)\n\n\n\nand also\n\[ \n{\begin{V... | Proof. We observe that every interval \( J \) in \( {\mathcal{D}}_{k} \) is either an \( {I}_{L} \) or an \( {I}_{R} \) for some unique \( I \in {\mathcal{D}}_{k - 1} \) . Thus we can write\n\n\[ \n{E}_{k}\left( f\right) = \mathop{\sum }\limits_{{J \in {\mathcal{D}}_{k}}}\left( {\operatorname{Avg}f}\right) {\chi }_{J} ... | Yes |
Theorem 6.5.1. Let \( n \geq 3 \) . For each \( \frac{n}{n - 1} < p \leq \infty \), there is a constant \( {C}_{p} \) such that\n\n\[ \n\parallel \mathcal{M}\left( f\right) {\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {C}_{p}\parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }\n\]\n\nholds for... | The proof of this theorem is given in the rest of this section. Before we present the proof we explain the validity of (6.5.7). Clearly this assertion is valid for functions \( f \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . Using inequality (6.5.6) and Theorem 2.1.14 we obtain that (6.5.7) holds for all functions... | Yes |
Lemma 6.5.3. There exists a constant \( C = C\\left( n\\right) < \\infty \) such that for all \( j \\geq 1 \) and for all \( f \) in \( {L}^{1}\\left( {\\mathbf{R}}^{n}\\right) \) we have\n\n\[ \n{\\begin{Vmatrix}{\\mathcal{M}}_{j}\\left( f\\right) \\end{Vmatrix}}_{{L}^{1,\\infty }} \\leq C{2}^{j}\\parallel f{\\paralle... | Proof. Let \( {K}^{\\left( j\\right) } = {\\left( {\\varphi }_{j}\\right) }^{ \\vee } * {d\\sigma } = {\\Phi }_{{2}^{-j}} * {d\\sigma } \), where \( \\Phi \) is a Schwartz function. Setting\n\n\[ \n{\\left( {K}^{\\left( j\\right) }\\right) }_{t}\\left( x\\right) = {t}^{-n}{K}^{\\left( j\\right) }\\left( {{t}^{-1}x}\\ri... | Yes |
The single function \( \varphi \) in (6.6.2) therefore generates the Haar basis by taking translations and dilations. Moreover, we observed in Section 6.4 that the family \( {\left\{ {h}_{I}\right\} }_{I} \) is orthonormal. | Moreover, in Theorem 6.4.6 we obtained the representation \[ f = \mathop{\sum }\limits_{{I \in \mathcal{D}}}\left\langle {f,{h}_{I}}\right\rangle {h}_{I}\;\text{ in }{L}^{2}, \] which proves the completeness of the system \( {\left\{ {h}_{I}\right\} }_{I \in \mathcal{D}} \) in \( {L}^{2}\left( \mathbf{R}\right) \) . | Yes |
Proposition 6.6.3. Let \( g \in {L}^{1}\left( {\mathbf{R}}^{n}\right) \) . Then\n\n\[ \widehat{g}\left( m\right) = 0\;\text{ for all }m \in {\mathbf{Z}}^{n} \smallsetminus \{ 0\} \]\n\nif and only if\n\n\[ \mathop{\sum }\limits_{{k \in {\mathbf{Z}}^{n}}}g\left( {x + k}\right) = {\int }_{{\mathbf{R}}^{n}}g\left( t\right... | Proof. We define the periodic function\n\n\[ G\left( x\right) = \mathop{\sum }\limits_{{k \in {\mathbf{Z}}^{n}}}g\left( {x + k}\right) \]\n\nwhich is easily shown to be in \( {L}^{1}\left( {\mathbf{T}}^{n}\right) \) . Moreover, we have\n\n\[ \widehat{G}\left( m\right) = \widehat{g}\left( m\right) \]\n\nfor all \( m \in... | Yes |
Proposition 6.6.4. Let \( \varphi \in {L}^{2}\left( {\mathbf{R}}^{n}\right) \) . Then the sequence\n\n\[ \n\\{ \varphi \left( {x - k}\right) \\} _{k \in {\mathbf{Z}}^{n}}\n\]\n\nforms an orthonormal set in \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) if and only if\n\n\[ \n\mathop{\\sum }\\limits_{{k \in {\mathbf{Z}}^{n... | Proof. Observe that either (6.6.4) or the hypothesis that the sequence in (6.6.3) is orthonormal implies that \( \\parallel \varphi {\\parallel }_{{L}^{2}} = 1 \) . Also the orthonormality condition\n\n\[ \n{\\int }_{{\\mathbf{R}}^{n}}\varphi \left( {x - j}\\right) \\overline{\\varphi \left( {x - k}\\right) }{dx} = \\l... | Yes |
Let \( \varphi \in {L}^{2}\left( {\mathbf{R}}^{n}\right) \) and suppose that the sequence\n\n\[ \{ \varphi \left( {x - k}\right) {\} }_{k \in {\mathbf{Z}}^{n}} \]\n\nforms an orthonormal set in \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) . Then the measure of the support of \( \widehat{\varphi } \) is at least 1, that ... | Proof. It follows from (6.6.4) that \( \left| \widehat{\varphi }\right| \leq 1 \) for almost all \( \xi \in {\mathbf{R}}^{n} \) and thus\n\n\[ \left| {\operatorname{supp}\widehat{\varphi }}\right| \geq {\int }_{{\mathbf{R}}^{n}}{\left| \widehat{\varphi }\left( \xi \right) \right| }^{2}{d\xi } = {\int }_{\lbrack 0,1{)}^... | Yes |
Let \( A = \left\lbrack {-1, - \frac{1}{2}}\right) \bigcup \left\lbrack {\frac{1}{2},1}\right) \) and define a function \( \varphi \) on \( \mathbf{R} \) by setting\n\n\[ \widehat{\varphi } = {\chi }_{A} \]\n\nThen we assert that the family of functions\n\n\[ {\left\{ {\varphi }_{v, k}\left( x\right) \right\} }_{k \in ... | To verify this assertion, first note that \( {\left\{ {\varphi }_{0, k}\right\} }_{k \in \mathbf{Z}} \) is an orthonormal set, since (6.6.4) is easily seen to hold. Dilating by \( {2}^{v} \), it follows that \( {\left\{ {\varphi }_{v, k}\right\} }_{k \in \mathbf{Z}} \) is also an orthonormal set for every fixed \( v \i... | No |
Theorem 6.6.9. (a) Let \( f \) in \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) be band limited on the cube \( {\left\lbrack -B, B\right\rbrack }^{n} \) . Then \( f \) can be sampled by its values at the points \( x = k/{2B} \), where \( k \in {\mathbf{Z}}^{n} \) . In particular, we have\n\n\[ f\left( {{x}_{1},\ldots ,{x... | Proof. Since the function \( \widehat{f} \) is supported in \( {\left\lbrack -B, B\right\rbrack }^{n} \), we use Exercise 6.6.2 to obtain\n\n\[ \widehat{f}\left( \xi \right) = \frac{1}{{\left( 2B\right) }^{n}}\mathop{\sum }\limits_{{k \in {\mathbf{Z}}^{n}}}\widehat{\widehat{f}}\left( \frac{k}{2B}\right) {e}^{{2\pi i}\f... | Yes |
A positive measure \( {d\mu } \) is called doubling if for some \( C < \infty \) , \[ \mu \left( {2B}\right) \leq {C\mu }\left( B\right) \] for all balls \( B \) . We show that the measures \( {\left| x\right| }^{a}{dx} \) are doubling when \( a > - n \) . | We divide all balls \( B\left( {{x}_{0}, R}\right) \) in \( {\mathbf{R}}^{n} \) into two categories: balls of type I that satisfy \( \left| {x}_{0}\right| \geq \) \( {3R} \) and type II that satisfy \( \left| {x}_{0}\right| < {3R} \) . For balls of type I we observe that \[ {\int }_{B\left( {{x}_{0},{2R}}\right) }{\lef... | Yes |
We investigate for which real numbers \( a \) the power function \( {\left| x\right| }^{a} \) is an \( {A}_{p} \) weight on \( {\mathbf{R}}^{n} \) . For \( 1 < p < \infty \), we examine for which \( a \) the following expression is finite: | As in the previous example we split the balls in \( {\mathbf{R}}^{n} \) into those of type I and those of type II. If \( B = B\left( {{x}_{0}, R}\right) \) is of type I, then for \( x \) satisfying \( \left| {x - {x}_{0}}\right| \leq R \) we must have\n\n\[ \frac{2}{3}\left| {x}_{0}\right| \leq \left| {x}_{0}\right| - ... | Yes |
On \( {\mathbf{R}}^{n} \) the function\n\n\[ u\left( x\right) = \left\{ \begin{array}{ll} \log \frac{1}{\left| x\right| } & \text{ when }\left| x\right| < \frac{1}{e}, \\ 1 & \text{ otherwise } \end{array}\right. \]\n\nis an \( {A}_{1} \) weight. | Indeed, to check condition (7.1.19) it suffices to consider balls of type I and type II as defined in Example 7.1.6. In either case the required estimate follows easily. | No |
Theorem 7.1.9. (a) Let \( w \in {A}_{1} \) . Then we have\n\n\[{\begin{Vmatrix}{\mathcal{M}}_{c}\end{Vmatrix}}_{{L}^{1}\left( w\right) \rightarrow {L}^{1,\infty }\left( w\right) } \leq {3}^{n}{\left\lbrack w\right\rbrack }_{{A}_{1}}.\] | Proof. (a) Since \( {d\mu } = {wdx} \) is a doubling measure and \( {d\mu }\left( {3Q}\right) \leq {3}^{n}{\left\lbrack w\right\rbrack }_{{A}_{1}}\mu \left( Q\right) \) , using Proposition 7.1.5 (9) and Exercise 2.1.1 we obtain that \( {M}_{c}^{w} \) maps \( {L}^{1}\left( w\right) \) to \( {L}^{1,\infty }\left( w\right... | No |
Lemma 7.2.1. Let \( w \in {A}_{p} \) for some \( 1 \leq p < \infty \) and let \( 0 < \alpha < 1 \) . Then there exists \( \beta < 1 \) such that whenever \( S \) is a measurable subset of a cube \( Q \) that satisfies \( \left| S\right| \leq \) \( \alpha \left| Q\right| \), we have \( w\left( S\right) \leq {\beta w}\le... | Proof. Taking \( f = {\chi }_{A} \) in property (8) of Proposition 7.1.5, we obtain\n\n\[ \n{\left( \frac{\left| A\right| }{\left| Q\right| }\right) }^{p} \leq {\left\lbrack w\right\rbrack }_{{A}_{p}}\frac{w\left( A\right) }{w\left( Q\right) }\n\]\n\n(7.2.1)\n\nWe write \( S = Q \smallsetminus A \) to get\n\n\[ \n{\lef... | Yes |
Corollary 7.2.4. Let \( w \) be a weight and let \( \mu \) be a measure on \( {\mathbf{R}}^{n} \) satisfying (7.2.9). Suppose that there exist \( 0 < \alpha ,\beta < 1 \), such that\n\n\[ \mu \left( S\right) \leq {\alpha \mu }\left( Q\right) \Rightarrow {\int }_{S}w\left( t\right) {d\mu }\left( t\right) \leq \beta {\in... | Proof. The proof of the corollary can be obtained almost verbatim from that of Theorem 7.2.2 by replacing Lebesgue measure with the doubling measure \( {d\mu } \) and the constant \( {2}^{n} \) by \( {C}_{n} \) .\n\nPrecisely, we define \( {\alpha }_{k} = {\left( {C}_{n}{\alpha }^{-1}\right) }^{k}{\alpha }_{0} \), wher... | Yes |
Theorem 7.2.5. If \( w \in {A}_{p} \) for some \( 1 \leq p < \infty \), then there exists a number \( \gamma > 0 \) (that depends on \( n, p \), and \( {\left\lbrack w\right\rbrack }_{{A}_{p}} \) ) such that \( {w}^{1 + \gamma } \in {A}_{p} \) . | Proof. Let \( C \) be the constant in the proof of Theorem 7.2.2. When \( p = 1 \), we apply the reverse Hölder inequality of Theorem 7.2.2 to the weight \( w \) to obtain\n\n\[ \frac{1}{\left| Q\right| }{\int }_{Q}w{\left( t\right) }^{1 + \gamma }{dt} \leq {\left( \frac{C}{\left| Q\right| }{\int }_{Q}w\left( t\right) ... | Yes |
For any \( 1 < p < \infty \) and for every \( w \in {A}_{p} \) there is a \( q = q\left( {n, p,{\left\lbrack w\right\rbrack }_{{A}_{p}}}\right) \) with \( q < p \) such that \( w \in {A}_{q} \) . In other words, we have \[ {A}_{p} = \mathop{\bigcup }\limits_{{q \in \left( {1, p}\right) }}{A}_{q} \] | Proof. Given \( w \in {A}_{p} \), let \( \gamma ,{C}_{1},{C}_{2} \) be as in the proof of Theorem 7.2.5. In view of the result in Exercise 7.1.3 with \( \delta = 1/\left( {1 + \gamma }\right) \), if \( {w}^{1 + \gamma } \in {A}_{p} \) and \[ q = p\frac{1}{1 + \gamma } + 1 - \frac{1}{1 + \gamma } = \frac{p + \gamma }{1 ... | Yes |
Proposition 7.2.8. Let \( 1 \leq p < \infty \) and \( w \in {A}_{p} \) . Then there exist \( \delta \in \left( {0,1}\right) \) and \( C > 0 \) depending only on \( n, p \), and \( {\left\lbrack w\right\rbrack }_{{A}_{p}} \) such that for any cube \( Q \) and any measurable subset \( S \) of \( Q \) we have\n\n\[ \frac{... | Proof. Let \( C \) and \( \gamma \) be as in Theorem 7.2.2. We use Hölder’s inequality to write\n\n\[ \frac{w\left( S\right) }{w\left( Q\right) } = \frac{1}{w\left( Q\right) }{\int }_{Q}w\left( x\right) {\chi }_{S}\left( x\right) {dx} \]\n\n\[ \leq \frac{1}{w\left( Q\right) }{\left( {\int }_{Q}w{\left( x\right) }^{1 + ... | Yes |
Proposition 7.3.2. Let \( w \in {A}_{\infty } \) . Then\n\n(1) \( {\left\lbrack {\delta }^{\lambda }\left( w\right) \right\rbrack }_{{A}_{\infty }} = {\left\lbrack w\right\rbrack }_{{A}_{\infty }} \), where \( {\delta }^{\lambda }\left( w\right) \left( x\right) = w\left( {\lambda {x}_{1},\ldots ,\lambda {x}_{n}}\right)... | Proof. Properties (1)-(3) are elementary, while property (4) is a consequence of Exercise 1.1.3(b). To show (5), first observe that by taking \( f = {w}^{-1} \), the expression on the right in (5) is at least as big as \( {\left\lbrack w\right\rbrack }_{{A}_{\infty }} \) . Conversely,(7.3.1) gives\n\n\[ \exp \left( {\f... | No |
Theorem 7.4.3. Let \( 1 \leq p \leq \infty, w \in {A}_{p} \), and \( T \) in \( \operatorname{CZO}\left( {\delta, A, B}\right) \) . Then there exist positive constants \( {}^{1}{C}_{0} = {C}_{0}\left( {n, p,{\left\lbrack w\right\rbrack }_{{A}_{p}}}\right) ,{\varepsilon }_{0} = {\varepsilon }_{0}\left( {n, p,{\left\lbra... | Proof. We write the open set\n\n\[ \Omega = \left\{ {{T}^{\left( *\right) }\left( f\right) > \lambda }\right\} = \mathop{\bigcup }\limits_{j}{Q}_{j} \]\n\nwhere \( {Q}_{j} \) are the Whitney cubes (see Appendix J). We set\n\n\[ {Q}_{j}^{ * } = {10}\sqrt{n}{Q}_{j} \]\n\n\[ {Q}_{j}^{* * } = {10}\sqrt{n}{Q}_{j}^{ * } \]\n... | Yes |
Lemma 7.4.5. Let \( 1 \leq p < \infty ,\varepsilon > 0, w \in {A}_{p}, x \in {\mathbf{R}}^{n} \), and \( f \in {L}^{p}\left( w\right) \) . Then we have\n\n\[ \n{\int }_{\left| {x - y}\right| \geq \varepsilon }\frac{\left| f\left( y\right) \right| }{{\left| x - y\right| }^{n}}{dy} \leq {C}_{00}\left( {w, n, p, x,\vareps... | Proof. For each \( \varepsilon > 0 \) and \( x \) pick a cube \( {Q}_{0} = {Q}_{0}\left( {x,\varepsilon }\right) \) of side length \( {c}_{n}\varepsilon \) (for some constant \( \left. {c}_{n}\right) \) such that \( {Q}_{0} \subseteqq B\left( {x,\varepsilon }\right) \) . Set \( {Q}_{j} = {2}^{j}{Q}_{0} \) for \( j \geq... | Yes |
Theorem 7.5.3. Suppose that \( T \) is defined on \( \mathop{\bigcup }\limits_{{1 \leq q < \infty }}\mathop{\bigcup }\limits_{{w \in {A}_{q}}}{L}^{q}\left( w\right) \) and takes values in the space of measurable complex-valued functions. Let \( 1 \leq {p}_{0} < \infty \) and suppose that there exists a positive increas... | Proof. Let \( 1 < p < \infty \) and \( w \in {A}_{p} \) . We define an operator\n\n\[ \n{M}^{\prime }\left( f\right) = \frac{M\left( {fw}\right) }{w}\n\]\n\nwhere \( M \) is the Hardy-Littlewood maximal operator. We observe that since \( {w}^{1 - {p}^{\prime }} \) is in \( {A}_{{p}^{\prime }} \), the operator \( {M}^{\... | Yes |
Theorem 7.5.5. Suppose that \( T \) is a well defined operator on \( \mathop{\bigcup }\limits_{{1 < q < \infty }}\mathop{\bigcup }\limits_{{w \in {A}_{q}}}{L}^{q}\left( w\right) \) that takes values in the space of measurable complex-valued functions. Fix \( 1 \leq \) \( {p}_{0} < \infty \) and suppose that there is an... | Proof. For every fixed \( \lambda > 0 \) we define\n\n\[ {T}_{\lambda }\left( f\right) = \lambda {\chi }_{\left| {T\left( f\right) }\right| > \lambda }\](7.5.22)\n\nThe operator \( {T}_{\lambda } \) is not linear but is well defined on \( \mathop{\bigcup }\limits_{{1 < q < \infty }}\mathop{\bigcup }\limits_{{w \in {A}_... | Yes |
Corollary 7.5.6. Suppose that \( T \) is a sublinear operator on \( \mathop{\bigcup }\limits_{{1 < q < \infty }}\mathop{\bigcup }\limits_{{w \in {A}_{q}}}{L}^{q}\left( w\right) \) that takes values in the space of measurable complex-valued functions. Fix \( 1 \leq \) \( {p}_{0} < \infty \) and suppose that there is an ... | Proof. The proof follows from Theorem 7.5.5 and the Marcinkiewicz interpolation theorem. | No |
Corollary 7.5.7. Suppose that \( T \) is defined on \( \mathop{\bigcup }\limits_{{1 \leq q < \infty }}\mathop{\bigcup }\limits_{{w \in {A}_{q}}}{L}^{q}\left( w\right) \) and takes values in the space of all measurable complex-valued functions. Fix \( 1 \leq {p}_{0} < \infty \) and suppose that there is an increasing fu... | Proof. To derive the claimed vector-valued inequality follow the proof of Theorem\n\n7.5.3 replacing the function \( f \) by \( {\left( \mathop{\sum }\limits_{j}{\left| {f}_{j}\right| }^{{p}_{0}}\right) }^{\frac{1}{{p}_{0}}} \) and \( T\left( f\right) \) by \( {\left( \mathop{\sum }\limits_{j}{\left| T\left( {f}_{j}\ri... | Yes |
Theorem 7.5.8. (a) Let \( 0 < p < q, r < \infty \) . Let \( {\left\{ {T}_{j}\right\} }_{j} \) be a sequence of sublinear operators that map \( {L}^{q}\left( \mu \right) \) to \( {L}^{r}\left( \nu \right) \), where \( \mu \) and \( \nu \) are arbitrary measures. Then the vector-valued inequality\n\n\[ \n{\begin{Vmatrix}... | Proof. We begin with part (a). Given \( {f}_{j} \in {L}^{q}\left( {{\mathbf{R}}^{n},\mu }\right) \), we use (7.5.24) to obtain\n\n\[ \n{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{j}{\left| {T}_{j}\left( {f}_{j}\right) \right| }^{p}\right) }^{\frac{1}{p}}\end{Vmatrix}}_{{L}^{r}\left( v\right) } = {\begin{Vmatrix}\mat... | Yes |
Example 7.5.9. We use the previous theorem to obtain another proof of the vector-valued Hardy-Littlewood maximal inequality in Corollary 5.6.5. We take \( {T}_{j} = M \) for all \( j \) . For given \( 1 < p < q < \infty \) and \( u \) in \( {L}^{\frac{q}{q - p}} \) we set \( s = \frac{q}{q - p} \) and \( U = \parallel ... | \[ \parallel U{\parallel }_{{L}^{s}} \leq \parallel u{\parallel }_{{L}^{s}}\;\text{ and }\;{\int }_{{\mathbf{R}}^{n}}M{\left( f\right) }^{p}{udx} \leq {C}^{p}{\int }_{{\mathbf{R}}^{n}}{\left| f\right| }^{p}{Udx}. \] Using Theorem 7.5.8, we obtain \[ {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{j}{\left| M\left( {f}_{... | Yes |
Let \( \eta \left( \xi \right) \) be a compactly supported smooth function. The inverse Fourier transform of the function \( {e}^{-1/{\left| \xi \right| }^{2}}\eta \left( \xi \right) \) lies in \( {\mathcal{S}}_{0}\left( {\mathbf{R}}^{n}\right) \) . | Indeed, every derivative of \( {e}^{-1/{\left| \xi \right| }^{2}}\eta \left( \xi \right) \) at the origin is equal to a finite linear combination of expressions of the form \[ \mathop{\lim }\limits_{{\xi \rightarrow 0}}{\partial }_{\xi }^{\beta }\left( {\left| \xi \right| }^{-2}\right) {e}^{-1/{\left| \xi \right| }^{2}... | Yes |
Proposition 1.1.3. The dual space of \( {\mathcal{S}}_{0}\left( {\mathbf{R}}^{n}\right) \) under the topology inherited from \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) is\n\n\[ \n{\mathcal{S}}_{0}^{\prime }\left( {\mathbf{R}}^{n}\right) = {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) /\mathcal{P}\left( {\m... | Proof. To identify the dual of \( {\mathcal{S}}_{0}\left( {\mathbf{R}}^{n}\right) \) we argue as follows. For each \( u \) in \( {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) , let \( J\left( u\right) \) be the restriction of \( u \) on the subspace \( {\mathcal{S}}_{0}\left( {\mathbf{R}}^{n}\right) \) of \(... | Yes |
Proposition 1.1.5. Let \( N \in {\mathbf{Z}}^{ + } \) . Suppose that \( {\left\{ {g}_{i}\right\} }_{i \in \mathbf{Z}} \) are functions in \( {\mathcal{C}}^{\left| \alpha \right| }\left( {\mathbf{R}}^{n}\right) \) for all multi-indices \( \alpha \) with \( \left| \alpha \right| \leq N \) and that \( \mathop{\sum }\limit... | Proof. Let \( {e}_{j} \) be the vector in \( {\mathbf{R}}^{n} \) with 1 in the \( j \) th coordinate and zero in the remaining ones. For \( h \in \mathbf{R} \smallsetminus \{ 0\} \) we have\n\n\[ \n\frac{g\left( {x + h{e}_{j}}\right) - g\left( x\right) }{h} = \mathop{\sum }\limits_{{i \in \mathbf{Z}}}\frac{{g}_{i}\left... | Yes |
Proposition 1.1.6. (a) Let \( \\widehat{\\Phi } \) be a \( {\\mathcal{C}}_{0}^{\\infty } \) function that is equal to 1 on \( \\overline{B\\left( {0,1}\\right) } \) . Then for all \( \\varphi \\in \\mathcal{S}\\left( {\\mathbf{R}}^{n}\\right) \) we have\n\n\[ \n{S}_{N}^{\\Phi }\\left( \\varphi \\right) \\rightarrow \\v... | Proof. (a) Let \( \\widetilde{\\Phi }\\left( x\\right) = \\Phi \\left( {-x}\\right) \) . We observe that for any \( f \\in {\\mathcal{S}}^{\\prime }\\left( {\\mathbf{R}}^{n}\\right) \) and \( \\varphi \\in \\mathcal{S}\\left( {\\mathbf{R}}^{n}\\right) \) we have\n\n\[ \n\\left\\langle {{S}_{N}^{\\Phi }\\left( f\\right)... | Yes |
Corollary 1.1.7. (Calderón reproducing formula) Let \( \Psi ,\Omega \) be Schwartz functions whose Fourier transforms are supported in annuli that do not contain the origin and satisfy \[ \mathop{\sum }\limits_{{j \in \mathbf{Z}}}\widehat{\Psi }\left( {{2}^{-j}\xi }\right) \widehat{\Omega }\left( {{2}^{-j}\xi }\right) ... | Proof. The assertion is contained in the conclusion of Proposition 1.1.6(c) with \( \Psi * \Omega \) in place of \( \Psi \) . | No |
Corollary 1.2.6. (a) For all \( 0 < s < \infty \), the operator \( {\mathcal{J}}_{s} \) maps \( {L}^{r}\left( {\mathbf{R}}^{n}\right) \) to itself with norm 1 for all \( 1 \leq r \leq \infty \) . | Proof. (a) Since \( \widehat{{G}_{s}}\left( 0\right) = 1 \) and \( {G}_{s} > 0 \), it follows that \( {G}_{s} \) has \( {L}^{1} \) norm 1 . The operator \( {\mathcal{J}}_{s} \) is given by convolution with the positive function \( {G}_{s} \), which has \( {L}^{1} \) norm 1 ; thus, it maps \( {L}^{r}\left( {\mathbf{R}}^... | No |
Theorem 1.3.5. (Sobolev embedding theorem) (a) Let \( 0 < s < \frac{n}{p} \) and \( 1 < p < \infty \) . Then the Sobolev space \( {L}_{s}^{p}\left( {\mathbf{R}}^{n}\right) \) continuously embeds in \( {L}^{q}\left( {\mathbf{R}}^{n}\right) \) when\n\n\[ \n\frac{1}{p} - \frac{1}{q} = \frac{s}{n} \n\] | Proof. (a) If \( f \in {L}_{s}^{p} \), then \( {f}_{s}\left( x\right) = {\left( {\left( 1 + {\left| \xi \right| }^{2}\right) }^{\frac{s}{2}}\widehat{f}\right) }^{ \vee }\left( x\right) \) is in \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) . Thus,\n\n\[ \nf\left( x\right) = {\left( {\left( 1 + {\left| \xi \right| }^{2}\r... | Yes |
Theorem 1.3.6. Let \( \Psi \) satisfy (1.3.6), \( \Phi \) be as in (1.3.8), and let \( {\Delta }_{j}^{\Psi },{S}_{0}^{\Phi } \) be as in (1.3.7) and (1.3.10), respectively. Fix \( s \in \mathbf{R} \) and \( 1 < p < \infty \) . Then there exists a constant \( {C}_{1} \) that depends only on \( n, s, p,\Phi \), and \( \P... | Proof. We denote by \( C \) a generic constant that depends on the parameters \( n, s, p,\Phi \) , and \( \Psi \) and that may vary in different occurrences. For a given tempered distribution \( f \) we define another tempered distribution \( {f}_{s} \) by setting\n\n\[ \n{f}_{s} = {\left( {\left( 1 + {\left| \cdot \ri... | Yes |
Theorem 1.3.8. Let \( \Psi \) satisfy (1.3.6), and let \( {\Delta }_{j}^{\Psi } \) be the Littlewood-Paley operator associated with \( \Psi \). Let \( s \in \mathbf{R} \) and \( 1 < p < \infty \). Then there exists a constant \( {C}_{1} \) that depends only on \( n, s, p \), and \( \Psi \) such that for all \( f \in {\... | Proof. The proof of the theorem is similar to but a bit simpler than that of Theorem 1.3.6. To obtain (1.3.23), we start with \( f \in {\dot{L}}_{s}^{p} \) and note that\n\n\[{2}^{js}{\Delta }_{j}\left( f\right) = {2}^{js}{\left( {\left| \xi \right| }^{s}{\left| \xi \right| }^{-s}\widehat{\Psi }\left( {2}^{-j}\xi \righ... | Yes |
Proposition 1.4.5. Let \( f \) be a \( {\mathcal{C}}^{m} \) function on \( {\mathbf{R}}^{n} \) for some \( m \in {\mathbf{Z}}^{ + } \) . Then for all \( h = \left( {{h}_{1},\ldots ,{h}_{n}}\right) \) and \( x \in {\mathbf{R}}^{n} \) the following identity holds:\n\n\[ \n{D}_{h}\left( f\right) \left( x\right) = {\int }_... | Proof. Identity (1.4.4) is a consequence of the fundamental theorem of calculus applied to the function \( t \mapsto f\left( {\left( {1 - t}\right) x + t\left( {x + h}\right) }\right) \) on \( \left\lbrack {0,1}\right\rbrack \), whereas identity (1.4.5) follows from (1.4.4) by induction. | No |
Theorem 1.4.6. Let \( \Psi ,{\Delta }_{j}^{\Psi } \) be as above and \( \gamma > 0 \) . Then there is a constant \( C = \) \( C\left( {n,\gamma ,\Psi }\right) \) such that for all \( f \) in \( {\dot{\Lambda }}_{\gamma } \) we have the estimate\n\n\[ \mathop{\sup }\limits_{{j \in \mathbf{Z}}}{2}^{j\gamma }{\begin{Vmatr... | Proof. We begin with the proof of (1.4.7). We first consider the case \( 0 < \gamma < 1 \) , which is very simple. Since each \( {\Delta }_{j}^{\Psi } \) is given by convolution with a function with mean value zero, for a function \( f \in {\dot{\Lambda }}_{\gamma } \) and every \( x \in {\mathbf{R}}^{n} \) we write\n\... | Yes |
Lemma 1.4.7. Let \( {h}_{k}, k = 1,2,\ldots \) be \( {\mathcal{C}}^{N} \) functions on \( {\mathbf{R}}^{n} \) such that \( {h}_{k} \rightarrow h \) uniformly on compact subsets of \( {\mathbf{R}}^{n} \) as \( k \rightarrow \infty \) . Suppose that there exist finite constants \( {C}_{\alpha },{M}_{\alpha } \) such that... | Proof. It follows from the hypothesis that \( h \) has at most polynomial growth at infinity, and thus it can be thought of as an element on \( {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) . Then \( {\partial }^{\alpha }h \) exist as elements of \( {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) fo... | Yes |
Corollary 1.4.8. Any function \( f \) in \( {\dot{\Lambda }}_{\gamma } \) lies in \( {\mathcal{C}}^{\left| \beta \right| } \) for any \( \left| \beta \right| < \gamma \), and its derivatives \( {\partial }^{\beta }f \) lie in \( {\dot{\Lambda }}_{\gamma - \left| \beta \right| } \) and satisfy\n\n\[{\begin{Vmatrix}{\par... | Proof. We proved in Theorem 1.4.6 that if \( f \) lies in \( {\dot{\Lambda }}_{\gamma } \), then (1.4.7) holds, and that (1.4.7) implies that there exists a polynomial \( Q \) such that \( f - Q \) lies in \( {\mathcal{C}}^{\left\lbrack \gamma \right\rbrack } \) and in \( {\dot{\Lambda }}_{\gamma } \) . It follows that... | Yes |
Theorem 1.4.9. Let \( \Psi ,\Phi ,{\Delta }_{j}^{\Psi } \), and \( {S}_{0}^{\Phi } \) be as above, and let \( \gamma > 0 \) . Then there is a constant \( C = C\left( {n,\gamma }\right) \) such that for every function \( f \) in \( {\Lambda }_{\gamma } \) the following estimate holds:\n\n\[ \n{\begin{Vmatrix}{S}_{0}^{\P... | Proof. The proof of (1.4.23) is immediate since we trivially have\n\n\[ \n{\begin{Vmatrix}{S}_{0}^{\Phi }\left( f\right) \end{Vmatrix}}_{{L}^{\infty }} = \parallel f * \Phi {\parallel }_{{L}^{\infty }} \leq \parallel \Phi {\parallel }_{{L}^{1}}\parallel f{\parallel }_{{L}^{\infty }} \leq C\parallel f{\parallel }_{{\Lam... | Yes |
Corollary 1.4.10. For \( 0 < \gamma < \delta < \infty \) there is a constant \( {C}_{n,\gamma ,\delta } < \infty \) such that for all \( f \in {\Lambda }_{\delta }\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \parallel f{\parallel }_{{\Lambda }_{\gamma }} \leq {C}_{n,\gamma ,\delta }\parallel f{\parallel }_{{\Lambda ... | Proof. If \( 0 < \gamma < \delta \) and \( j \geq 1 \), then we must have \( {2}^{j\gamma } < {2}^{j\delta } \), and thus\n\n\[ \mathop{\sup }\limits_{{j \geq 1}}{2}^{j\gamma }{\begin{Vmatrix}{\Delta }_{j}^{\Psi }\left( f\right) \end{Vmatrix}}_{{L}^{\infty }} \leq \mathop{\sup }\limits_{{j \geq 1}}{2}^{j\delta }{\begin... | Yes |
Corollary 1.4.11. Let \( \gamma > 0 \), and let \( \alpha \) be a multi-index with \( \left| \alpha \right| < \gamma \) . Then any function \( f \) in \( {\Lambda }_{\gamma } \) lies in \( {\mathcal{C}}^{\alpha },{\partial }^{\alpha }f \) lies in \( {\Lambda }_{\gamma - \left| \alpha \right| } \), and the estimate \[ {... | Proof. Let \( \alpha \) be a multi-index with \( \left| \alpha \right| < \gamma \) . We denote by \( {\Delta }_{j}^{{\partial }^{\alpha }\Psi } \) the Littlewood-Paley operator associated with the bump \( {\left( {\partial }^{\alpha }\Psi \right) }_{{2}^{-j}} \) . Let \( f \in {\Lambda }_{\gamma } \) . Theorem 1.4.9 im... | Yes |
Theorem 2.1.2. (a) Let \( 1 < p < \infty \) . Then every bounded tempered distribution \( f \) in \( {H}^{p} \) is an element of \( {L}^{p} \) . Moreover, there is a constant \( {C}_{n, p} \) such that for all such \( f \) we have\n\n\[ \parallel f{\parallel }_{{L}^{p}} \leq \parallel f{\parallel }_{{H}^{p}} \leq {C}_{... | Proof. (a) Let \( f \in {H}^{p}\left( {\mathbf{R}}^{n}\right) \) for some \( 1 < p < \infty \) . The set \( \left\{ {{P}_{t} * f : t > 0}\right\} \) lies in a multiple of the unit ball of \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \), which is the dual space of the separable Banach space \( {L}^{{p}^{\prime }}\left( {\ma... | Yes |
Corollary 2.1.8. For any two Schwartz functions \( \Phi \) and \( \Theta \) with nonvanishing integral we have\n\n\[{\begin{Vmatrix}\mathop{\sup }\limits_{{t > 0}}\left| {\Theta }_{t} * f\right| \end{Vmatrix}}_{{L}^{p}} \approx {\begin{Vmatrix}\mathop{\sup }\limits_{{t > 0}}\left| {\Phi }_{t} * f\right| \end{Vmatrix}}_... | Proof. See the discussion after Theorem 2.1.4. | No |
Corollary 2.1.9. (a) For any \( 0 < p \leq 1 \), every \( f \in {H}^{p}\left( {\mathbf{R}}^{n}\right) \), and any \( \varphi \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) , we have\n\n\[ \left| {\langle f,\varphi \rangle }\right| \leq {\mathfrak{N}}_{N}\left( \varphi \right) \mathop{\inf }\limits_{{\left| z\right| \... | Proof. (a) We use that \( \langle f,\varphi \rangle = \left( {\widetilde{\varphi } * f}\right) \left( 0\right) \), where \( \widetilde{\varphi }\left( x\right) = \varphi \left( {-x}\right) \) and we observe that \( {\mathfrak{N}}_{N}\left( \varphi \right) = {\mathfrak{N}}_{N}\left( \widetilde{\varphi }\right) \) . Then... | Yes |
Proposition 2.1.10. Let \( 0 < p \leq 1 \) . Then the following statements are valid:\n\n(a) Convergence in \( {H}^{p} \) implies convergence in \( {\mathcal{S}}^{\prime } \) . | Proof. (a) Let \( {f}_{j}, f \) in \( {H}^{p}\left( {\mathbf{R}}^{n}\right) \) and suppose that \( {f}_{j} \rightarrow f \) in \( {H}^{p}\left( {\mathbf{R}}^{n}\right) \) . Applying (2.1.50) we obtain that for any \( \varphi \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) we have \( \left\langle {{f}_{j} - f,\varphi }... | Yes |
Theorem 2.1.13. Let \( 0 < p < \infty, L \in {\mathbf{Z}}^{ + } \) . Then the following statements are valid: (a) There exists a Schwartz function \( {\Phi }^{o} \) with \( {\int }_{{\mathbf{R}}^{n}}{\Phi }^{o}\left( x\right) {dx} = 1 \) and a constant \( {C}_{1} \) \( \left( {{C}_{1} = {500}}\right. \) works) such tha... | Proof. The proof of this theorem is obtained via a step-by-step repetition of the proof of Theorem 2.1.4 in which the scalar absolute values of complex numbers are replaced by \( {\ell }_{L}^{2} \) norms. The verification of the details of this extension is omitted. The crucial observation in the adaptation of the proo... | No |
Theorem 2.1.14. Suppose that a finite sequence of kernels \( {\left\{ {K}_{j}\right\} }_{j = 1}^{L} \) satisfies (2.1.62) and (2.1.63) with \( N = \left\lbrack \frac{n}{p}\right\rbrack + 1 \), for some \( 0 < p \leq 1 \) . Then there exists a constant \( {C}_{n, p} \) that depends only on the dimension \( n \) and on \... | Proof. We fix a smooth positive function \( \Phi \) supported in the unit ball \( B\left( {0,1}\right) \) with \( {\int }_{{\mathbf{R}}^{n}}\Phi \left( x\right) {dx} = 1 \) and we consider the maximal function\n\n\[ M\left( {\mathop{\sum }\limits_{{j = 1}}^{L}{K}_{j} * {f}_{j};\Phi }\right) = \mathop{\sup }\limits_{{\v... | Yes |
Lemma 2.2.3. Let \( 0 < r < \infty \) . Then there exist constants \( {C}_{1} \) and \( {C}_{2} \) such that for all \( t > 0 \) and for all \( {\mathcal{C}}^{1} \) functions \( u \) on \( {\mathbf{R}}^{n} \) whose distributional Fourier transform is supported in the ball \( \left| \xi \right| \leq t \) we have\n\n\[ \... | Proof. Select a Schwartz function \( \Phi \) whose Fourier transform is supported in the ball \( \left| \xi \right| \leq 2 \) and is equal to 1 on the unit ball \( \left| \xi \right| \leq 1 \) . Then \( \widehat{\Phi }\left( \frac{\xi }{t}\right) \) is equal to 1 on the support of \( \widehat{u} \) and we can write\n\n... | Yes |
Corollary 2.2.5. Let \( \Phi ,\Omega ,\Psi \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \). Suppose that the Fourier transforms of \( \Omega ,\Psi \) are supported in the annulus \( 1 - \frac{1}{7} \leq \left| \xi \right| \leq 2 \). Let \( 0 < r < \infty \). Then for all \( f \) in \( {\mathcal{S}}^{\prime }\left( {\m... | Proof. Given \( r \) pick \( N = \frac{n}{r} + n + 1 \). Then we have\n\n\[ \left| {\left( {{\Phi }_{t} * {\Delta }_{j}^{\Psi }\left( f\right) }\right) \left( x\right) }\right| \leq {C}_{\Phi, N}{\int }_{{\mathbf{R}}^{n}}\frac{\left| {\Delta }_{j}^{\Psi }\left( f\right) \left( x - z\right) \right| }{{\left( 1 + {t}^{-1... | Yes |
Theorem 2.2.9. Let \( \Psi \) be a Schwartz function on \( {\mathbf{R}}^{n} \) whose Fourier transform is nonnegative, supported in \( \frac{6}{7} \leq \left| \xi \right| \leq 2 \), equal to 1 on \( 1 \leq \left| \xi \right| \leq \frac{12}{7} \), and satisfies for all \( \xi \neq 0 \n\n\[ \mathop{\sum }\limits_{{j \in ... | Proof. We fix \( \Phi \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) with integral equal to 1 and we take \( f \in {H}^{p} \cap {L}^{1} \) and \( M \) in \( {\mathbf{Z}}^{ + } \). Let \( {r}_{j} \) be the Rademacher functions, defined in Appendix C. 1 in [156], reindexed so that their index set is the set of all inte... | Yes |
Corollary 2.2.10. Fix \( \Psi \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) with Fourier transform supported in \( \frac{6}{7} \leq \left| \xi \right| \leq 2 \) , equal 1 on the annulus \( 1 \leq \left| \xi \right| \leq \frac{12}{7} \), and satisfying \( \mathop{\sum }\limits_{{j \in \mathbf{Z}}}\widehat{\Psi }... | Proof. Inequality (2.2.31) is a direct consequence of (2.2.25) since \( {\Delta }_{k}^{\Omega } \) can be written as a finite sum of \( {\Delta }_{j}^{\Omega } \) ’s. Conversely, we introduce a Schwartz function \( \eta \) whose Fourier transform \( \widehat{\eta } \) is supported in an annulus of the form \( 0 < {c}_{... | Yes |
Proposition 2.3.1. Let \( 0 < p, q < \infty \), and \( \alpha \in \mathbf{R} \). The homogeneous Triebel–Lizorkin space \( {\dot{F}}_{p}^{\alpha, q}\left( {\mathbf{R}}^{n}\right) \) is continuously embedded in the Besov space \( {\dot{B}}_{p}^{\alpha ,\infty }\left( {\mathbf{R}}^{n}\right) \) which is in turn continuou... | Proof. Given \( f \in {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) /\mathcal{P}\left( {\mathbf{R}}^{n}\right) \) we have the sequence of inequalities\n\n\[ \mathop{\sup }\limits_{{j \in \mathbf{Z}}}{2}^{j\alpha }{\begin{Vmatrix}{\Delta }_{j}^{\Psi }\left( f\right) \end{Vmatrix}}_{{L}^{p}} \leq {\begin{Vmatrix}... | Yes |
Theorem 2.3.4. Let \( 0 < p, q < \infty ,\alpha \in \mathbf{R} \), and let\n\n\[ L = \left\lbrack {\max \left( {n\max \left( {1,\frac{1}{p},\frac{1}{q}}\right) - n - \alpha ,\alpha }\right) }\right\rbrack .\n\]\n\nThen there is a constant \( {C}_{n, p, q,\alpha } \) such that for every sequence of smooth L-atoms \( {\l... | Proof. We prove the first assertion of the theorem. We let \( {\Delta }_{j}^{\Psi } \) be the Littlewood-Paley operator associated with a Schwartz function \( \Psi \) whose Fourier transform is compactly supported away from the origin in \( {\mathbf{R}}^{n} \) . Let \( {a}_{Q} \) be a smooth \( L \) -atom supported in ... | Yes |
Corollary 2.3.9. Let \( \alpha \in \mathbf{R},0 < p \leq 1, L \geq \left\lbrack {\max \left( {\frac{n}{p} - n - \alpha ,\alpha }\right) }\right\rbrack \), and let \( q \) satisfy \( p \leq q < \infty \) . Then for a given \( f \in {\dot{F}}_{p}^{\alpha, q} \) we have the following equivalence:\n\n\[ \parallel f{\parall... | Proof. Let \( {\lambda }_{j},{A}_{j} \) be as above such that\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}{\begin{Vmatrix}f - \mathop{\sum }\limits_{{j = 1}}^{N}{\lambda }_{j}{A}_{j}\end{Vmatrix}}_{{\dot{F}}_{p}^{\alpha, q}} = 0. \]\n\nIn view of the subadditivity of the expression \( h \mapsto \parallel f{\pa... | No |
Theorem 2.3.11. Let \( 0 < p \leq 1 \) . There is a constant \( {C}_{n, p} < \infty \) such that every \( {L}^{2} \) - atom A for \( {H}^{p}\left( {\mathbf{R}}^{n}\right) \) satisfies\n\n\[ \parallel A{\parallel }_{{H}^{p}} \leq {C}_{n, p} \] | Proof. We could prove this theorem either by showing that the smooth maximal function \( M\left( {A;\Phi }\right) \) is in \( {L}^{p} \) or by showing that the square function \( {\left( \mathop{\sum }\limits_{{j \in \mathbf{Z}}}{\left| {\Delta }_{j}^{\Psi }\left( A\right) \right| }^{2}\right) }^{1/2} \) is in \( {L}^{... | Yes |
Theorem 2.3.12. Let \( 0 < p \leq 1 \) . Given a distribution \( f \in {H}^{p}\left( {\mathbf{R}}^{n}\right) \), there exists a sequence of \( {L}^{2} \) -atoms for \( {H}^{p},{\left\{ {A}_{j}\right\} }_{j = 1}^{\infty } \), and a sequence of scalars \( {\left\{ {\lambda }_{j}\right\} }_{j = 1}^{\infty } \) such that\n... | Proof. Fix \( f \in {H}^{p}\left( {\mathbf{R}}^{n}\right) \) . Let \( {A}_{j} \) be \( {L}^{2} \) -atoms for \( {H}^{p} \) and \( \mathop{\sum }\limits_{{j = 1}}^{\infty }{\left| {\lambda }_{j}\right| }^{p} < \infty \) such that (2.3.27) holds. It follows from Theorem 2.3.11 and the sublinearity of the expression \( g ... | Yes |
Corollary 2.4.2. Let \( K \) satisfy (2.4.1),(2.4.2), and (2.4.3), and let \( T \) be defined as in (2.4.4). Let \( 1 \leq p \leq \infty ,0 < q \leq \infty \), and \( \alpha \in \mathbf{R} \). Then there is a constant \( {C}_{n, p, q,\alpha } \) such that for all \( f \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) ... | Proof. Let \( \Psi \) be a Schwartz function whose Fourier transform is supported in the annulus \( 1 - \frac{1}{7} \leq \left| \xi \right| \leq 2 \) and that satisfies\n\n\[ \mathop{\sum }\limits_{{j \in \mathbf{Z}}}\widehat{\Psi }\left( {{2}^{-j}\xi }\right) = 1,\;\xi \neq 0. \]\n\nPick a Schwartz function \( \zeta \... | Yes |
Lemma 2.4.4. Let \( {\left\{ {\zeta }_{\varepsilon }\right\} }_{\varepsilon > 0} \) be a family of Schwartz functions and for each \( \varepsilon > 0 \) let \( {T}_{\varepsilon } \) be the operator given by convolution with \( {\zeta }_{\varepsilon } \) . Suppose that the \( {T}_{\varepsilon } \) ’s are uniformly (in \... | Proof. We begin the proof by observing that as a consequence of (2.4.23) we have\n\n\[ \n\parallel T\left( a\right) {\parallel }_{{L}^{p}} \leq {C}_{0}\n\]\n\nfor all \( a \) that are \( {L}^{2} \) -atoms for \( {H}^{p} \) . Indeed,(2.4.22) implies that for a given \( {L}^{2} \) atom \( a \) for \( {H}^{p} \), there is... | Yes |
Corollary 2.4.7. An integrable function on the line lies in the Hardy space \( {H}^{1}\left( \mathbf{R}\right) \) if and only if its Hilbert transform is integrable. For \( n \geq 2 \), an integrable function on \( {\mathbf{R}}^{n} \) lies in the Hardy space \( {H}^{1}\left( {\mathbf{R}}^{n}\right) \) if and only its R... | Proof. The corollary follows by combining Theorems 2.4.1 and 2.4.6. | No |
Corollary 2.4.8. Functions in \( {H}^{1}\left( {\mathbf{R}}^{n}\right), n \geq 1 \), have integral zero. | Proof. Indeed, if \( f \in {H}^{1}\left( {\mathbf{R}}^{n}\right) \), we must have \( {R}_{1}\left( f\right) \in {L}^{1}\left( {\mathbf{R}}^{n}\right) \) by Theorem 2.4.1; thus \( \widehat{{R}_{1}\left( f\right) } \) is uniformly continuous. But since\n\n\[ \widehat{{R}_{1}\left( f\right) }\left( \xi \right) = - i\wideh... | Yes |
Proposition 3.1.2. The following properties of the space \( \operatorname{BMO}\left( {\mathbf{R}}^{n}\right) \) are valid:\n\n(1) If \( \parallel f{\parallel }_{BMO} = 0 \), then \( f \) is a.e. equal to a constant. | Proof. To prove (1) note that \( f \) has to be a.e. equal to its average \( {c}_{N} \) over every cube \( {\left\lbrack -N, N\right\rbrack }^{n} \) . Since \( {\left\lbrack -N, N\right\rbrack }^{n} \) is contained in \( {\left\lbrack -N - 1, N + 1\right\rbrack }^{n} \), it follows that \( {c}_{N} = {c}_{N + 1} \) for ... | Yes |
We indicate why \( {L}^{\infty }\left( {\mathbf{R}}^{n}\right) \) is a proper subspace of \( {BMO}\left( {\mathbf{R}}^{n}\right) \) . We claim that the unbounded function \( \log \left| x\right| \) is in \( {BMO}\left( {\mathbf{R}}^{n}\right) \) . | To prove this, for every \( {x}_{0} \in {\mathbf{R}}^{n} \) and \( R > 0 \), we find a constant \( {C}_{{x}_{0}, R} \) such that the average of \( \left| \log \right| x\left| {-{C}_{{x}_{0}, R}}\right| \) over the ball \( \overline{B\left( {0, R}\right) } = \left\{ {x \in {\mathbf{R}}^{n} : \left| {x - {x}_{0}}\right| ... | Yes |
The function \( h\left( x\right) = {\chi }_{x > 0}\log \frac{1}{x} \) is not in \( {BMO}\left( \mathbf{R}\right) \) . | Indeed, the problem is at the origin. Consider the intervals \( \left( {-\varepsilon ,\varepsilon }\right) \), where \( 0 < \varepsilon < \frac{1}{2} \) . We have that\n\n\[{\operatorname{Avg}}_{\left( -\varepsilon ,\varepsilon \right) }h = \frac{1}{2\varepsilon }{\int }_{-\varepsilon }^{+\varepsilon }h\left( x\right) ... | Yes |
Proposition 3.1.5. (i) Let \( f \) be in \( {BMO}\left( {\mathbf{R}}^{n}\right) \) . Given a ball \( B \) and a positive integer \( m \), we have\n\n\[ \left| {\underset{B}{\operatorname{Avg}f} - \underset{{2}^{m}}{\operatorname{Avg}f}}\right| \leq {2}^{n}m\parallel f{\parallel }_{BMO} \] | Proof. (i) We have\n\n\[ \left| {\mathop{\operatorname{Avg}}\limits_{B}f - \mathop{\operatorname{Avg}}\limits_{{2B}}f}\right| = \frac{1}{\left| B\right| }\left| {{\int }_{B}\left( {f\left( t\right) - \mathop{\operatorname{Avg}}\limits_{{2B}}f}\right) {dt}}\right| \]\n\n\[ \leq \frac{{2}^{n}}{\left| 2B\right| }{\int }_{... | Yes |
Proposition 3.1.2 (3) now implies that\n\n\[ \parallel f{\parallel }_{BMO} \leq {2A}/\left( {{v}_{n}{c}_{n}^{\prime }}\right) \] | This concludes the proof of the proposition. | No |
Corollary 3.1.7. Every BMO function is exponentially integrable over any cube. Precisely, for any \( \gamma < 1/\left( {{2}^{n}e}\right) \), for all \( f \in {BMO}\left( {\mathbf{R}}^{n}\right) \), and all cubes \( Q \) we have\n\n\[ \frac{1}{\left| Q\right| }{\int }_{Q}{e}^{\gamma \left| {f\left( x\right) - {\operator... | Proof. Using identity\n\n\[ {\int }_{X}{e}^{\left| f\right| } - {1d\mu } = {\int }_{0}^{\infty }{e}^{\alpha }\mu \left( {\{ x \in X : \left| {f\left( x\right) }\right| > \alpha \} }\right) {d\alpha }, \]\n\nproved in Proposition 1.1.4 in [156] for a \( \sigma \) -finite measure space \( \left( {X,\mu }\right) \), we wr... | Yes |
Corollary 3.1.8. For all \( 0 < p < \infty \), there exists a finite constant \( {B}_{p, n} \) such that for all \( f \in {BMO} \) we have\n\n\[ \mathop{\sup }\limits_{Q}{\left( \frac{1}{\left| Q\right| }{\int }_{Q}{\left| f\left( x\right) - {\left. {\operatorname{Avg}}_{Q}f\right| }^{p}dx\right| }^{\frac{1}{p}} \leq {... | Proof. This result can be obtained from the one in the preceding corollary or directly in the following way:\n\n\[ \frac{1}{\left| Q\right| }{\int }_{Q}{\left| f\left( x\right) - \mathop{\operatorname{Avg}}\limits_{Q}f\right| }^{p}{dx} = \frac{p}{\left| Q\right| }{\int }_{0}^{\infty }{\alpha }^{p - 1}\left| {\{ x \in Q... | Yes |
Corollary 3.1.9. For all \( 1 < p < \infty \) and \( f \) in \( {L}_{\text{loc }}^{1}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \mathop{\sup }\limits_{Q}{\left( \frac{1}{\left| Q\right| }{\int }_{Q}{\left| f\left( x\right) - \operatorname{Avg}f\right| }^{p}dx\right) }^{\frac{1}{p}} \approx \parallel f{\parallel }_... | Proof. One direction follows from Corollary 3.1.8. Conversely, the supremum in (3.1.16) is bigger than or equal to the corresponding supremum with \( p = 1 \), which is equal to the \( {BMO} \) norm of \( f \), by definition. | Yes |
Proposition 3.3.4. (Whitney decomposition) Let \( \Omega \) be an open nonempty proper subset of \( {\mathbf{R}}^{n} \) . Then there exists a family of closed cubes \( {\left\{ {Q}_{j}\right\} }_{j} \) such that\n\n(a) \( \mathop{\bigcup }\limits_{j}{Q}_{j} = \Omega \) and the \( {Q}_{j} \) ’s have disjoint interiors;\... | The proof of Proposition 3.3.4 is given in Appendix J in [156]. | Yes |
For any Carleson measure \( \mu \) and every \( \mu \) -measurable function \( F \) on \( {\mathbf{R}}_{ + }^{n + 1} \) we have\n\n\[ \n{\int }_{{\mathbf{R}}_{ + }^{n + 1}}{\left| F\left( x, t\right) \right| }^{p}{d\mu }\left( {x, t}\right) \leq {C}_{n}\parallel \mu {\parallel }_{\mathcal{C}}{\int }_{{\mathbf{R}}^{n}}{... | Start with (3.3.5), multiply by \( p{\alpha }^{p - 1} \) and integrate in \( \alpha \) from zero to infinity. We obtain\n\n\[ \n{\int }_{0}^{\infty }p{\alpha }^{p - 1}\mu \left( {\{ \left| F\right| > \alpha \} }\right) {d\alpha } \leq {C}_{n}\parallel \mu {\parallel }_{\mathcal{C}}{\int }_{0}^{\infty }p{\alpha }^{p - 1... | No |
Let \( \Phi \) be a function on \( {\mathbf{R}}^{n} \) that satisfies for some \( 0 < C,\delta < \infty \), \[ \left| {\Phi \left( x\right) }\right| \leq \frac{C}{{\left( 1 + \left| x\right| \right) }^{n + \delta }}. \] Let \( \mu \) be a Carleson measure on \( {\mathbf{R}}_{ + }^{n + 1} \). Then for every \( 1 < p < \... | If \( \mu \) is a Carleson measure, we may obtain (3.3.11) as a consequence of Corollary 3.3.6. Indeed, for \( F\left( {x, t}\right) = \left( {{\Phi }_{t} * f}\right) \left( x\right) \), we have \[ {F}^{ * }\left( x\right) = \mathop{\sup }\limits_{{t > 0}}\mathop{\sup }\limits_{\substack{{y \in {\mathbf{R}}^{n}} \\ {\l... | No |
Theorem 3.3.8. Let \( b \) be a BMO function on \( {\mathbf{R}}^{n} \) and let \( \Psi \) be an integrable function with mean value zero on \( {\mathbf{R}}^{n} \) that satisfies\n\n\[ \n\left| {\Psi \left( x\right) }\right| \leq A{\left( 1 + \left| x\right| \right) }^{-n - \delta } \n\]\n\n(3.3.12)\n\nfor some \( 0 < A... | Proof. We prove (a). The measure \( \mu \) is defined so that for every \( \mu \) -integrable function \( F \) on \( {\mathbf{R}}_{ + }^{n + 1} \) we have\n\n\[ \n{\int }_{{\mathbf{R}}_{ + }^{n + 1}}F\left( {x, t}\right) {d\mu }\left( {x, t}\right) = \mathop{\sum }\limits_{{j \in \mathbf{Z}}}{\int }_{{\mathbf{R}}^{n}}{... | Yes |
Proposition 3.4.2. Let \( f, g \) be a locally integrable functions on \( {\mathbf{R}}^{n} \) . Then\n\n(1) \( {M}^{\# }\left( f\right) \leq 2{M}_{c}\left( f\right) \), where \( {M}_{c} \) is the Hardy-Littlewood maximal operator with respect to cubes in \( {\mathbf{R}}^{n} \) . | Proof. The proof of (1) is trivial. | No |
Theorem 3.4.4. For all \( \gamma > 0 \), all \( \lambda > 0 \), and all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \), we have the estimate\n\n\[ \left| \left\{ {x \in {\mathbf{R}}^{n} : {M}_{d}\left( f\right) \left( x\right) > {2\lambda },{M}^{\# }\left( f\right) \left( x\right) \leq {\gamma \lambda }... | Proof. We may suppose that the set \( {\Omega }_{\lambda } = \left\{ {x \in {\mathbf{R}}^{n} : {M}_{d}\left( f\right) \left( x\right) > \lambda }\right\} \) has finite measure; otherwise, there is nothing to prove. Then for each \( x \in {\Omega }_{\lambda } \) there is a maximal dyadic cube \( {Q}^{x} \) that contains... | Yes |
Theorem 3.4.5. Let \( 0 < {p}_{0} \leq p < \infty \) . Then there is a constant \( {C}_{n}\left( p\right) \) such that for all functions \( f \) in \( {L}_{\mathrm{{loc}}}^{1}\left( {\mathbf{R}}^{n}\right) \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \n{\begin{Vmatr... | Proof. Fix \( p \geq {p}_{0} \) with \( p < \infty \) . For a positive real number \( N \) we set\n\n\[ \n{I}_{N} = {\int }_{0}^{N}p{\lambda }^{p - 1}\left| \left\{ {x \in {\mathbf{R}}^{n} : {M}_{d}\left( f\right) \left( x\right) > \lambda }\right\} \right| {d\lambda }.\n\]\n\nWe note that \( {I}_{N} \) is finite, sinc... | Yes |
Corollary 3.4.6. Let \( 0 < {p}_{0} < \infty \) . Then for any \( p \) with \( {p}_{0} \leq p < \infty \) and for all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \parallel f{\parallel }_{{L}^{p}\left( {\ma... | Proof. Since for every point in \( {\mathbf{R}}^{n} \) there is a sequence of dyadic cubes shrinking to it, the Lebesgue differentiation theorem yields that for almost every point \( x \) in \( {\mathbf{R}}^{n} \) the averages of the locally integrable function \( f \) over the dyadic cubes containing \( x \) converge ... | Yes |
Theorem 3.4.7. Let \( 1 \leq {p}_{0} < \infty \) . Let \( T \) be a linear operator that maps \( {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) with bound \( {A}_{0} \), and \( {L}^{\infty }\left( {\mathbf{R}}^{n}\right) \) to \( \operatorname{BMO}\left( {\mathbf{R}}... | Proof. We consider the operator\n\n\[ S\left( f\right) = {M}^{\# }\left( {T\left( f\right) }\right) \]\n\ndefined for \( f \in {L}^{{p}_{0}} + {L}^{\infty } \) . It is easy to see that \( S \) is a sublinear operator. We prove that \( S \) maps \( {L}^{\infty } \) to itself and \( {L}^{{p}_{0}} \) to itself if \( {p}_{... | Yes |
Corollary 3.4.10. Let \( T \) be given by convolution with a distribution \( W \) that coincides with a function \( K \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) that satisfies (3.4.14). Assume that \( T \) has an extension that is \( {L}^{2} \) bounded with norm \( B \) . Then there is a constant \( {C}_{n} \) ... | Proof. We take \( s = 2 \) in Theorem 3.4.9 and we observe that\n\n\[ \parallel T\left( f\right) {\parallel }_{BMO} = {\begin{Vmatrix}{M}^{\# }\left( T\left( f\right) \right) \end{Vmatrix}}_{{L}^{\infty }} \leq {C}_{n}\left( {{A}_{2} + B}\right) {\begin{Vmatrix}M{\left( {\left| f\right| }^{2}\right) }^{\frac{1}{2}}\end... | Yes |
The function \( K\left( {x, y}\right) = {\left| x - y\right| }^{-n} \) defined away from the diagonal of \( {\mathbf{R}}^{n} \times {\mathbf{R}}^{n} \) is in \( {SK}\left( {1, n{4}^{n + 1}}\right) \) . | Indeed, for\n\n\[\n\left| {x - {x}^{\prime }}\right| \leq \frac{1}{2}\max \left( {\left| {x - y}\right| ,\left| {{x}^{\prime } - y}\right| }\right)\n\]\n\nthe mean value theorem gives\n\n\[\n\left| \right| x - y\left| {{}^{-n} - {\left| {x}^{\prime } - y}\right| }^{-n}}\right| \leq \frac{n\left| {x - {x}^{\prime }}\rig... | Yes |
Suppose that \( K\left( {x, y}\right) \) satisfies (4.1.1) and (4.1.2) and is antisymmetric, in the sense that\n\n\[ K\left( {x, y}\right) = - K\left( {y, x}\right) \]\n\nfor all \( x \neq y \) in \( {\mathbf{R}}^{n} \) . Then \( K \) also satisfies (4.1.3), and moreover, there is a distribution \( W \) on \( {\mathbf{... | Indeed, define\n\n\[ \langle W, F\rangle = \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{\iint }_{\begin{matrix} {x - y \mid > \varepsilon } \end{matrix}}K\left( {x, y}\right) F\left( {x, y}\right) {dydx} \]\n\n(4.1.10)\n\nfor all \( F \) in the Schwartz class of \( {\mathbf{R}}^{2n} \) . In view of antisymmetry,... | Yes |
Let \( A \) be a real-valued Lipschitz function on \( \mathbf{R} \). This means that it satisfies the estimate \( \left| {A\left( x\right) - A\left( y\right) }\right| \leq L\left| {x - y}\right| \) for some \( L < \infty \) and all \( x, y \in \mathbf{R} \). For \( x, y \in \mathbf{R}, x \neq y \), we let\n\n\[ \n{K}_{... | A simple calculation gives that when \( \left| {y - {y}^{\prime }}\right| \leq \frac{1}{2}\max \left( {\left| {x - y}\right| ,\left| {x - {y}^{\prime }}\right| }\right) \) then\n\n\[ \n\left| {{K}_{A}\left( {x, y}\right) - {K}_{A}\left( {x,{y}^{\prime }}\right) }\right| \leq \frac{\left| {y - {y}^{\prime }}\right| + \l... | Yes |
Proposition 4.1.11. Let \( K \) be a kernel in \( {SK}\left( {\delta, A}\right) \) and let \( T \) in \( {CZO}\left( {\delta, A, B}\right) \) be associated with \( K \) . For \( \varepsilon > 0 \), let \( {T}^{\left( \varepsilon \right) } \) be the truncated operators obtained from \( T \) . Assume that there exists a ... | Proof. Since \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) is separable, by the Banach-Alaoglu theorem the unit ball of its dual is weak* compact and metrizable for the weak* topology. Let \( {\left\{ {f}_{k}\right\} }_{k = 1}^{\infty } \) be a dense countable subset of \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) . By (4... | Yes |
Proposition 4.2.1. Let \( f \in {L}^{1}\left( {\mathbf{R}}^{n}\right) \) and \( \alpha > 0 \) . Then there exist functions \( g \) and \( b \) on \( {\mathbf{R}}^{n} \) such that\n\n(1) \( f = g + b \) .\n\n(2) \( \parallel g{\parallel }_{{L}^{1}} \leq \parallel f{\parallel }_{{L}^{1}} \) and \( \parallel g{\parallel }... | A proof of Proposition 4.2.1 can be given by considering the level set of the uncentered maximal function with respect to cubes at height \( \alpha \) ; see Exercise 4.2.6. Another proof is given in Proposition 5.3.1 in [156]. | No |
Proposition 4.2.3. Let \( T \) be an operator in \( {CZO}\left( {\delta, A, B}\right) \) associated with a kernel \( K \) . Then for \( g \in {L}^{p}\left( {\mathbf{R}}^{n}\right) ,1 \leq p < \infty \), the following absolutely convergent integral representation is valid:\n\n\[ T\left( g\right) \left( x\right) = {\int ... | Proof. Set \( {g}_{k}\left( x\right) = g\left( x\right) {\chi }_{\left| {g\left( x\right) }\right| \leq k}{\chi }_{\left| x\right| \leq k} \) . These are \( {L}^{p} \) functions with compact support contained in the support of \( g \) . Also, the \( {g}_{k} \) converge to \( g \) in \( {L}^{p} \) as \( k \rightarrow \i... | Yes |
Theorem 4.2.4. Let \( K \) be in \( {SK}\left( {\delta, A}\right) \) and \( T \) in \( {CZO}\left( {\delta, A, B}\right) \) be associated with \( K \) . Let \( r \in \left( {0,1}\right) \) . Then there is a constant \( C\left( {n, r}\right) \) such that Cotlar’s inequality\n\n\[ \left| {{T}^{\left( *\right) }\left( f\r... | Proof. We fix \( r \) so that \( 0 < r < 1 \) and \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) for some \( p \) satisfying \( 1 \leq p < \infty \) . To prove (4.2.2), we also fix \( \varepsilon > 0 \) and we set \( {f}_{0}^{\varepsilon, x} = f{\chi }_{B\left( {x,\varepsilon }\right) } \) and \( {f}_{\infty }^{\var... | Yes |
Lemma 4.3.5. Under assumptions (4.1.1),(4.1.2), and (4.1.3), there is a constant \( {C}_{n} \) such that for all normalized bumps \( \varphi \) we have\n\n\[ \mathop{\sup }\limits_{{{x}_{0} \in {\mathbf{R}}^{n}}}{\int }_{\left| {x - {x}_{0}}\right| \geq {20R}}{\left| {\int }_{{\mathbf{R}}^{n}}K\left( x, y\right) {\tau ... | Proof. Note that the interior integral in (4.3.4) is absolutely convergent, since \( {\tau }^{{x}_{0}}{\varphi }_{R} \) is supported in the ball \( B\left( {{x}_{0},{10R}}\right) \) and \( x \) lies in the complement of the double of this ball. To prove (4.3.4), simply observe that since \( \left| {K\left( {x, y}\right... | Yes |
Corollary 4.3.7. Let \( K \) be a standard kernel that is antisymmetric, i.e., it satisfies \( K\left( {x, y}\right) = - K\left( {y, x}\right) \) for all \( x \neq y \) . Then a linear continuous operator \( T \) associated with \( K \) is \( {L}^{2} \) bounded if and only if \( T\left( 1\right) \) is in BMO. | Proof. In view of Exercise 4.3.3, \( T \) automatically satisfies the weak boundedness property. Moreover, \( {T}^{t} = - T \) . Therefore, the three conditions of Theorem 4.3.3 (iv) reduce to the single condition \( T\left( 1\right) \in {BMO} \) . | Yes |
Let us recall the kernels \( {K}_{m} \) of Example 4.1.7. These arise in the expansion of the kernel in Example 4.1.6 in geometric series\n\n\[ \n\frac{1}{x - y + i\left( {A\left( x\right) - A\left( y\right) }\right) } = \frac{1}{x - y}\mathop{\sum }\limits_{{m = 0}}^{\infty }{\left( i\frac{A\left( x\right) - A\left( y... | We use the \( T\left( 1\right) \) theorem to show that the operators \( {\mathcal{C}}_{m} \) are \( {L}^{2} \) bounded.\n\nWe show that there exists a constant \( R > 0 \) such that for all \( m \geq 0 \) we have\n\n\[ \n{\begin{Vmatrix}{\mathcal{C}}_{m}\end{Vmatrix}}_{{L}^{2} \rightarrow {L}^{2}} \leq {R}^{m}{L}^{m} \... | Yes |
Lemma 4.5.1. Let \( {\left\{ {T}_{j}\right\} }_{j \in \mathbf{Z}} \) be a family of operators mapping a Hilbert space \( H \) to itself. Assume that there is a a function \( \gamma : \mathbf{Z} \rightarrow {\mathbf{R}}^{ + } \) such that\n\n\[ \n{\begin{Vmatrix}{T}_{j}^{ * }{T}_{k}\end{Vmatrix}}_{H \rightarrow H} + {\b... | Proof. As usual we denote by \( {S}^{ * } \) the adjoint of a linear operator \( S \) . It is a simple fact that any bounded linear operator \( S : H \rightarrow H \) satisfies\n\n\[ \n\parallel S{\parallel }_{H \rightarrow H}^{2} = {\begin{Vmatrix}S{S}^{ * }\end{Vmatrix}}_{H \rightarrow H} \n\]\n\n(4.5.2)\n\nSee Exerc... | No |
Theorem 4.5.4. Let \( K \) be in \( {SK}\left( {\delta, A}\right) \) and let \( T \) be a continuous linear operator from \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) to \( {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) associated with \( K \) . Assume that\n\n\[ \parallel T\left( 1\right) {\parallel }_{BM... | Proof. Consider the paraproduct operators \( {P}_{T\left( 1\right) } \) and \( {P}_{{T}^{t}\left( 1\right) } \) introduced in the previous section. Then, as we showed in Proposition 4.4.4, we have\n\n\[ {P}_{T\left( 1\right) }\left( 1\right) = T\left( 1\right) ,\;{\left( {P}_{T\left( 1\right) }\right) }^{t}\left( 1\rig... | Yes |
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