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Proposition 2.3.23. Given \( u \in {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \), there exists a sequence of \( {\mathcal{C}}_{0}^{\infty } \) functions \( {f}_{k} \) such that \( {f}_{k} \rightarrow u \) in the sense of tempered distributions; in particular, \( {\mathcal{C}}_{0}^{\infty }\left( {\mathbf{R}}... | Proof. Fix a function in \( {\mathcal{C}}_{0}^{\infty }\left( {\mathbf{R}}^{n}\right) \) with \( \varphi \left( x\right) = 1 \) in a neighborhood of the origin. Let \( {\varphi }_{k}\left( x\right) = {\delta }^{1/k}\left( \varphi \right) \left( x\right) = \varphi \left( {x/k}\right) \) . It follows from Exercise 2.3.5 ... | No |
Corollary 2.4.2. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \). If \( \widehat{u} \) is supported in the singleton \( \left\{ {\xi }_{0}\right\} \), then \( u \) is a finite linear combination of functions \( {\left( -2\pi i\xi \right) }^{\alpha }{e}^{{2\pi i\xi } \cdot {\xi }_{0}} \), where \( ... | Proof. Proposition 2.4.1 gives that \( \widehat{u} \) is a linear combination of derivatives of Dirac masses at \( {\xi }_{0} \). Then Proposition 2.3.22 (8) yields the required conclusion. | No |
Corollary 2.4.3. Let \( u \in {\mathcal{S}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) satisfy \( \Delta \left( u\right) = 0 \) . Then \( u \) is a polynomial. | Proof. Taking Fourier transforms, we obtain that \( \widehat{\Delta \left( u\right) } = 0 \) . Therefore,\n\n\[ \n- 4{\pi }^{2}{\left| \xi \right| }^{2}\widehat{u} = 0\;\text{ in }{\mathcal{S}}^{\prime }. \n\] \n\nThis implies that \( \widehat{u} \) is supported at the origin, and by Corollary 2.4.2 it follows that \( ... | Yes |
Proposition 2.4.4. For \( n \geq 3 \) we have\n\n\[ \Delta \left( {\left| x\right| }^{2 - n}\right) = - \left( {n - 2}\right) \frac{2{\pi }^{n/2}}{\Gamma \left( {n/2}\right) }{\delta }_{0}, \] | Proof. We use Green's identity\n\n\[ {\int }_{\Omega }\left( {{v\Delta }\left( u\right) - {u\Delta }\left( v\right) }\right) {dx} = {\int }_{\partial \Omega }\left( {v\frac{\partial u}{\partial v} - u\frac{\partial v}{\partial v}}\right) {ds} \]\n\nwhere \( \Omega \) is an open set in \( {\mathbf{R}}^{n} \) with smooth... | Yes |
Proposition 2.4.7. Suppose that \( m \) is a \( {\mathcal{C}}^{\infty } \) function on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) that is homogeneous of degree zero. Then there exist a scalar b and a \( {\mathcal{C}}^{\infty } \) function \( \Omega \) on \( {\mathbf{S}}^{n - 1} \) with integral zero such that\n\n\[ \... | To prove this result we need the following proposition, whose proof we postpone until the end of this section. | No |
Proposition 2.4.8. Suppose that \( u \) is a \( {\mathcal{C}}^{\infty } \) function on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) that is homogeneous of degree \( z \in \mathbf{C} \) . Then \( \widehat{u} \) is a \( {\mathcal{C}}^{\infty } \) function on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) . | Proof. Let \( u \in {\mathcal{S}}^{\prime } \) be homogeneous of degree \( z \) and \( {\mathcal{C}}^{\infty } \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) . We need to show that \( \widehat{u} \) is \( {\mathcal{C}}^{\infty } \) away from the origin. We prove that \( \widehat{u} \) is \( {\mathcal{C}}^{M} \) for... | Yes |
Let \( \eta \) be a smooth radial function on \( {\mathbf{R}}^{n} \) that is equal to 1 on the set \( \left| x\right| \geq 1/2 \) and vanishes on the set \( \left| x\right| \leq 1/4 \) . Fix \( z \in \mathbf{C} \) satisfy \( 0 < \operatorname{Re}z < n \) . Let \( g = {\left( \eta \left( x\right) {\left| x\right| }^{-z}... | To show that \( g \) is a function we write it as \( g = {\left( {\left| x\right| }^{-z}\right) }^{ \land } + {\left( \left( \eta \left( x\right) - 1\right) {\left| x\right| }^{-z}\right) }^{ \land } \) and we observe that the first term is a function, since \( 0 < \operatorname{Re}z < n \) . Using Theorem 2.4.6 we wri... | Yes |
Lemma 2.5.3. Under the hypotheses of Theorem 2.5.2 and for \( f \in \mathcal{S}\left( {\mathbf{R}}^{n}\right) \), the distributional derivatives of \( T\left( f\right) \) are \( {L}^{q} \) functions that satisfy\n\n\[{\partial }^{\alpha }\left( {T\left( f\right) }\right) = T\left( {{\partial }^{\alpha }f}\right) ,\;\te... | (2.5.1) | No |
Theorem 2.5.6. \( {\mathcal{M}}^{p, q} = \{ 0\} \) whenever \( 1 \leq q < p < \infty \) . | Proof. Let \( f \) be a nonzero \( {\mathcal{C}}_{0}^{\infty } \) function and let \( h \in {\mathbf{R}}^{n} \) . We have\n\n\[ \n{\begin{Vmatrix}{\tau }^{h}\left( T\left( f\right) \right) + T\left( f\right) \end{Vmatrix}}_{{L}^{q}} = {\begin{Vmatrix}T\left( {\tau }^{h}\left( f\right) + f\right) \end{Vmatrix}}_{{L}^{q}... | No |
Theorem 2.5.7. Let \( 1 < p \leq q < \infty \) and \( T \in {\mathcal{M}}^{p, q}\left( {\mathbf{R}}^{n}\right) \) . Then \( T \) can be defined on \( {L}^{{q}^{\prime }}\left( {\mathbf{R}}^{n}\right) \), coinciding with its previous definition on the subspace \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \cap {L}^{{q}^{\pri... | Proof. We first observe that if \( T : {L}^{p} \rightarrow {L}^{q} \) is given by convolution with \( u \in {\mathcal{S}}^{\prime } \), then the adjoint operator \( {T}^{ * } : {L}^{{q}^{\prime }} \rightarrow {L}^{{p}^{\prime }} \) is given by convolution with \( \overline{\widetilde{u}} \in {\mathcal{S}}^{\prime } \) ... | Yes |
Let \( \left( {X,\parallel \cdot {\parallel }_{{L}^{\infty }}}\right) \) be the space of all complex-valued bounded functions on the real line such that\n\n\[ \Phi \left( f\right) = \mathop{\lim }\limits_{{R \rightarrow + \infty }}\frac{1}{R}{\int }_{0}^{R}f\left( t\right) {dt} \]\n\n exists. Then \( \Phi \) is a bound... | We note that \( \widetilde{\Phi } \) commutes with translations, since for all \( f \in {L}^{\infty }\left( \mathbf{R}\right) \) and \( x \in \mathbf{R} \) we have\n\n\[ \widetilde{\Phi }\left( {{\tau }^{\mathrm{x}}\left( f\right) }\right) - {\tau }^{\mathrm{x}}\left( {\widetilde{\Phi }\left( f\right) }\right) = \widet... | Yes |
The function \( m\left( \xi \right) = {e}^{{2\pi i\xi } \cdot b} \) is an \( {L}^{p} \) multiplier for all \( b \in {\mathbf{R}}^{n} \) | since the corresponding operator \( {T}_{m}\left( f\right) \left( x\right) = f\left( {x + b}\right) \) is bounded on \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) . Clearly \( \parallel m{\parallel }_{{\mathcal{M}}_{p}} = 1. \) | Yes |
Proposition 2.5.14. For all \( m \in {\mathcal{M}}_{p},1 \leq p < \infty, x \in {\mathbf{R}}^{n} \), and \( h > 0 \) we have\n\n\[{\begin{Vmatrix}{\tau }^{x}\left( m\right) \end{Vmatrix}}_{{\mathcal{M}}_{p}} = \parallel m{\parallel }_{{\mathcal{M}}_{p}}\]\n\n(2.5.18)\n\n\[{\begin{Vmatrix}{\delta }^{h}\left( m\right) \e... | Proof. See Exercise 2.5.2. | No |
We show that for \( - \infty < a < b < \infty \) we have \( {\begin{Vmatrix}{\chi }_{\left\lbrack a, b\right\rbrack }\end{Vmatrix}}_{{\mathcal{M}}_{p}} = {\begin{Vmatrix}{\chi }_{\left\lbrack 0,1\right\rbrack }\end{Vmatrix}}_{{\mathcal{M}}_{p}} \) . | Indeed, using (2.5.18) we obtain that \( {\begin{Vmatrix}{\chi }_{\left\lbrack a, b\right\rbrack }\end{Vmatrix}}_{{\mathcal{M}}_{p}} = {\begin{Vmatrix}{\chi }_{\left\lbrack 0, b - a\right\rbrack }\end{Vmatrix}}_{{\mathcal{M}}_{p}} \), and the latter is equal to \( {\begin{Vmatrix}{\chi }_{\left\lbrack 0,1\right\rbrack ... | Yes |
Example 2.5.17. (The cone multiplier) On \( {\mathbf{R}}^{n + 1} \) define the function\n\n\[ \n{m}_{\lambda }\left( {{\xi }_{1},\ldots ,{\xi }_{n + 1}}\right) = {\left( 1 - \frac{{\xi }_{1}^{2} + \cdots + {\xi }_{n}^{2}}{{\xi }_{n + 1}^{2}}\right) }_{ + }^{\lambda },\;\lambda > 0, \n\]\n\nwhere the plus sign indicates... | Indeed, by Theorem 2.5.16 we have that for some \( {\xi }_{n + 1} = h,{b}_{\lambda }\left( {{\xi }_{1}/h,\ldots ,{\xi }_{n}/h}\right) \) is in \( {\mathcal{M}}_{p}\left( {\mathbf{R}}^{n}\right) \) and hence so is \( {b}_{\lambda } \) by property (2.5.19). | Yes |
Lemma 2.6.5. Let \( k \geq 1 \) and suppose that \( {a}_{0},\ldots ,{a}_{k} \) are distinct real numbers. Let \( a = \min \left( {a}_{j}\right) \) and \( b = \max \left( {a}_{j}\right) \) and let \( f \) be a real-valued \( {\mathcal{C}}^{k - 1} \) function on \( \left\lbrack {a, b}\right\rbrack \) that is \( {\mathcal... | Proof. Suppose we could find a polynomial \( {p}_{k}\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{k}{b}_{j}{x}^{j} \) such that the function \[ \varphi \left( x\right) = f\left( x\right) - {p}_{k}\left( x\right) \] (2.6.8) satisfies \( \varphi \left( {a}_{m}\right) = 0 \) for all \( 0 \leq m \leq k \) . Since the ... | Yes |
Lemma 2.6.6. Let \( E \) be a measurable subset of \( \mathbf{R} \) with finite nonzero Lebesgue measure and let \( k \in {\mathbf{Z}}^{ + } \) . Then there exist \( {a}_{0},\ldots ,{a}_{k} \) in \( E \) such that for all \( \ell = 0,1,\ldots, k \) we have\n\n\[ \mathop{\prod }\limits_{\substack{{j = 0} \\ {j \neq \ell... | Proof. Given a measurable set \( E \) with finite measure, pick a compact subset \( {E}^{\prime } \) of \( E \) such that \( \left| {E \smallsetminus {E}^{\prime }}\right| < \delta \), for some \( \delta > 0 \) . For \( x \in \mathbf{R} \) define \( T\left( x\right) = \left| {\left( {-\infty, x}\right) \cap {E}^{\prime... | Yes |
Proposition 2.6.7. (a) Let \( u \) be a real-valued \( {\mathcal{C}}^{k} \) function, \( k \in {\mathbf{Z}}^{ + } \), that satisfies \( {u}^{\left( k\right) }\left( t\right) \geq 1 \) for all \( t \in \mathbf{R} \). Then the following estimate is valid for all \( \alpha > 0 \):\n\n\[ \left| {\{ t \in \mathbf{R} : }\rig... | Part (a): Let \( E = \{ t \in \mathbf{R} : \left| {u\left( t\right) }\right| \leq \alpha \} \). If \( \left| E\right| \) is nonzero, then by Lemma 2.6.6 there exist \( {a}_{0},{a}_{1},\ldots ,{a}_{k} \) in \( E \) such that for all \( \ell \) we have\n\n\[ {\left| E\right| }^{k} \leq {\left( 2e\right) }^{k}\mathop{\pro... | Yes |
Corollary 2.6.8. Let \( \left( {a, b}\right), u\left( t\right) ,\lambda > 0 \), and \( k \) be as in Proposition 2.6.7. Then for any function \( \psi \) on \( \left( {a, b}\right) \) with an integrable derivative and \( k \geq 2 \), we have\n\n\[ \left| {{\int }_{a}^{b}{e}^{{i\lambda u}\left( t\right) }\psi \left( t\ri... | Proof. Set\n\n\[ F\left( x\right) = {\int }_{a}^{x}{e}^{{i\lambda u}\left( t\right) }{dt} \]\n\nand use integration by parts to write\n\n\[ {\int }_{a}^{b}{e}^{{i\lambda u}\left( t\right) }\psi \left( t\right) {dt} = F\left( b\right) \psi \left( b\right) - {\int }_{a}^{b}F\left( t\right) {\psi }^{\prime }\left( t\right... | No |
The Bessel function of order \( m \) is defined as\n\n\[ \n{J}_{m}\left( r\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }{e}^{{ir}\sin \theta }{e}^{-{im\theta }}{d\theta }. \n\] | We use Corollary 2.6.8 to calculate the decay of the Bessel function \( {J}_{m}\left( r\right) \) as \( r \rightarrow \infty \) . Set\n\n\[ \n\varphi \left( \theta \right) = \sin \left( \theta \right) \n\]\n\nand note that \( {\varphi }^{\prime }\left( \theta \right) \) vanishes only at \( \theta = \pi /2 \) and \( {3\... | Yes |
Proposition 3.1.2. Let \( f, g \) be in \( {L}^{1}\left( {\mathbf{T}}^{n}\right) \) . Then for all \( m, k \in {\mathbf{Z}}^{n},\lambda \in \mathbf{C}, y \in {\mathbf{T}}^{n} \), and all multi-indices \( \alpha \) we have\n\n(1) \( \overset{⏜}{f + g}\left( m\right) = \widehat{f}\left( m\right) + \widehat{g}\left( m\rig... | Proof. The proof of properties (1)-(10) is rather easy and is left to the reader. We only sketch the proof of (9). We have\n\n\[ \widehat{f * g}\left( m\right) = {\int }_{{\mathbf{T}}^{n}}{\int }_{{\mathbf{T}}^{n}}f\left( {x - y}\right) g\left( y\right) {e}^{-{2\pi im} \cdot \left( {x - y}\right) }{e}^{-{2\pi im} \cdot... | No |
If the sequence \( {\left\{ {a}_{m}\right\} }_{m} \) has only one nonzero term, then the trigonometric polynomial of Definition 3.1.4 reduces to a trigonometric monomial, which has the form\n\n\[ P\left( x\right) = a{e}^{{2\pi i}\left( {{q}_{1}{x}_{1} + \cdots + {q}_{n}{x}_{n}}\right) } \] \n\nfor some \( q = \left( {{... | Let\n\n\[ P\left( x\right) = \mathop{\sum }\limits_{{\left| m\right| \leq N}}{a}_{m}{e}^{{2\pi im} \cdot x} = \mathop{\sum }\limits_{{\left| m\right| \leq N}}\widehat{P}\left( m\right) {e}^{{2\pi im} \cdot x} \] \n\nbe a trigonometric polynomial on \( {\mathbf{T}}^{n} \) and let \( \mu \) be a finite Borel measure on \... | No |
Proposition 3.1.10. The family of Fejér kernels \( {\left\{ {F}_{N}^{n}\right\} }_{N = 0}^{\infty } \) is an approximate identity on \( {\mathbf{T}}^{n} \) . | Proof. Since \( {F}_{N}^{n} \geq 0 \) we have that \( {\begin{Vmatrix}{F}_{N}^{n}\end{Vmatrix}}_{{L}^{1}} = {\int }_{{\mathbf{T}}^{n}}{F}_{N}^{n}{dx} \) . Also \( {\int }_{{\mathbf{T}}^{n}}{F}_{N}^{n}{dx} = 1 \), in view of identity (3.1.23). Thus properties (i) and (ii) of approximate identities (according to Definiti... | Yes |
Proposition 3.2.1. The set of trigonometric polynomials is dense in \( {L}^{p}\left( {\mathbf{T}}^{n}\right) \) for \( 1 \leq \) \( p < \infty \) . | Proof. Given \( f \) in \( {L}^{p}\left( {\mathbf{T}}^{n}\right) \) for \( 1 \leq p < \infty \), consider \( f * {F}_{N}^{n} \) . Clearly \( f * {F}_{N}^{n} \) is also a trigonometric polynomial. In view of Theorem 1.2.19 (1), \( f * {F}_{N}^{n} \) converges to \( f \) in \( {L}^{p} \) as \( N \rightarrow \infty \) . | Yes |
Corollary 3.2.2. (Weierstrass approximation theorem for trigonometric polynomials) Every continuous function on the torus is a uniform limit of trigonometric polynomials. | Proof. Since \( f \) is continuous on \( {\mathbf{T}}^{n} \), which is a compact set, Theorem 1.2.19 (2) gives that \( f * {F}_{N}^{n} \) converges uniformly to \( f \) as \( N \rightarrow \infty \) . But \( f * {F}_{N}^{n} \) is a trigonometric polynomial, and so we conclude that every continuous function on \( {\math... | Yes |
Proposition 3.2.4. If \( f, g \in {L}^{1}\left( {\mathbf{T}}^{n}\right) \) satisfy \( \widehat{f}\left( m\right) = \widehat{g}\left( m\right) \) for all \( m \) in \( {\mathbf{Z}}^{n} \), then \( f = {ga}.e \) . | Proof. By linearity of the problem, it suffices to assume that \( g = 0 \) . If \( \widehat{f}\left( m\right) = 0 \) for all \( m \in {\mathbf{Z}}^{n} \), Definition 3.2.3 implies that \( {F}_{N}^{n} * f = 0 \) for all \( N \in {\mathbf{Z}}^{ + } \) . In view of Proposition 3.1.10, the sequence \( {\left\{ {F}_{N}^{n}\... | Yes |
Proposition 3.2.5. (Fourier inversion) Suppose that \( f \in {L}^{1}\left( {\mathbf{T}}^{n}\right) \) and that\n\n\[ \mathop{\sum }\limits_{{m \in {\mathbf{Z}}^{n}}}\left| {\widehat{f}\left( m\right) }\right| < \infty \]\n\nThen\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{m \in {\mathbf{Z}}^{n}}}\widehat{f}\left( ... | Proof. It is straightforward to check that both functions in (3.2.1) are well defined and have the same Fourier coefficients. Therefore, they must be almost everywhere equal by Proposition 3.2.4. Moreover, the function on the right in (3.2.1) is everywhere continuous. | Yes |
Proposition 3.2.7. The following are valid for \( f, g \in {L}^{2}\left( {\mathbf{T}}^{n}\right) \) :\n\n(1) (Plancherel's identity)\n\n\[ \parallel f{\parallel }_{{L}^{2}}^{2} = \mathop{\sum }\limits_{{m \in {\mathbf{Z}}^{n}}}{\left| \widehat{f}\left( m\right) \right| }^{2} \]\n\n(2) The function \( f\left( t\right) \... | Proof. (1) and (2) follow from the corresponding statements in Proposition 3.2.6. Notice that both sides of (3) converge by the Cauchy-Schwarz inequality. Parseval's relation (3) follows from polarization. By this we mean the following procedure. First replace \( f \) by \( f + g \) in (1) and expand the squares to obt... | Yes |
Theorem 3.2.8. (Poisson summation formula) Let \( f \) be a continuous function on \( {\mathbf{R}}^{n} \) which satisfies for some \( C,\delta > 0 \) and for all \( x \in {\mathbf{R}}^{n} \n\n\[ \left| {f\left( x\right) }\right| \leq C{\left( 1 + \left| x\right| \right) }^{-n - \delta }, \]\n\nand whose Fourier transfo... | Proof. Define a 1-periodic function on \( {\mathbf{T}}^{n} \) by setting\n\n\[ F\left( x\right) = \mathop{\sum }\limits_{{k \in {\mathbf{Z}}^{n}}}f\left( {x + k}\right) \]\n\nIt is straightforward to verify that \( \parallel F{\parallel }_{{L}^{1}\left( {\left\lbrack 0,1\right\rbrack }^{n}\right) } = \parallel f{\paral... | Yes |
We have seen earlier (Exercise 2.2.11) that the following identity gives the Fourier transform of the Poisson kernel in \( {\mathbf{R}}^{n} \) :\n\n\[ \n{\left( {e}^{-{2\pi }\left| x\right| }\right) }^{ \frown }\left( \xi \right) = \frac{\Gamma \left( \frac{n + 1}{2}\right) }{{\pi }^{\frac{n + 1}{2}}}\frac{1}{{\left( 1... | The Poisson summation formula yields the identity\n\n\[ \n\frac{\Gamma \left( \frac{n + 1}{2}\right) }{{\pi }^{\frac{n + 1}{2}}}\mathop{\sum }\limits_{{k \in {\mathbf{Z}}^{n}}}\frac{{\varepsilon }^{-n}}{{\left( 1 + \frac{{\left| k + x\right| }^{2}}{{\varepsilon }^{2}}\right) }^{\frac{n + 1}{2}}} = \mathop{\sum }\limits... | Yes |
Proposition 3.3.1. (Riemann-Lebesgue lemma) Given a function \( f \) in \( {L}^{1}\left( {\mathbf{T}}^{n}\right) \), we have that \( \left| {\widehat{f}\left( m\right) }\right| \rightarrow 0 \) as \( \left| m\right| \rightarrow \infty \) . | Proof. Given \( f \in {L}^{1}\left( {\mathbf{T}}^{n}\right) \) and \( \varepsilon > 0 \), let \( P \) be a trigonometric polynomial such that \( \parallel f - P{\parallel }_{{L}^{1}} < \varepsilon \) . If \( \left| m\right| > \operatorname{degree}\left( P\right) \), then \( \widehat{P}\left( m\right) = 0 \) and thus\n\... | Yes |
Lemma 3.3.2. Given a sequence of positive real numbers \( {\left\{ {a}_{m}\right\} }_{m = 0}^{\infty } \) that tends to zero as \( m \rightarrow \infty \), there exists a sequence \( {\left\{ {c}_{m}\right\} }_{m = 0}^{\infty } \) that satisfies\n\n\[ \n{c}_{m} \geq {a}_{m},\;{c}_{m} \downarrow 0,\;\text{ and }\;{c}_{m... | Proof. Let \( {k}_{0} = 0 \) and suppose that \( {a}_{m} \leq M \) for all \( m \geq 0 \) . Find \( {k}_{1} > {k}_{0} \) such that for \( m \geq {k}_{1} \) we have \( {a}_{m} \leq M/2 \) . Now find \( {k}_{2} > {k}_{1} + \frac{{k}_{1} - {k}_{0}}{2} \) such that for \( m \geq {k}_{2} \) we have \( {a}_{m} \leq M/4 \) . ... | Yes |
Given a convex decreasing sequence \( {\left\{ {c}_{m}\right\} }_{m = 0}^{\infty } \) of positive real numbers satisfying \( \mathop{\lim }\limits_{{m \rightarrow \infty }}{c}_{m} = 0 \) and a fixed integer \( s \geq 0 \), we have that\n\n\[ \mathop{\sum }\limits_{{r = 0}}^{\infty }\left( {r + 1}\right) \left( {{c}_{r ... | Proof. We begin by observing the validity of the telescoping sum\n\n\[ \mathop{\sum }\limits_{{r = 0}}^{N}\left( {r + 1}\right) \left( {{c}_{r + s} + {c}_{r + s + 2} - 2{c}_{r + s + 1}}\right) \]\n\n\[ = {c}_{s} - \left( {N + 1}\right) \left( {{c}_{s + N + 1} - {c}_{s + N + 2}}\right) - {c}_{s + N + 1}. \]\n\nTo show t... | Yes |
Theorem 3.3.4. Let \( {\left( {d}_{m}\right) }_{m \in {\mathbf{Z}}^{n}} \) be a sequence of positive real numbers with \( {d}_{m} \rightarrow 0 \) as \( \left| m\right| \rightarrow \infty \) . Then there exists a function \( f \in {L}^{1}\left( {\mathbf{T}}^{n}\right) \) such that \( \widehat{f}\left( m\right) \geq {d}... | Proof. We are given a sequence of positive numbers \( {\left\{ {a}_{m}\right\} }_{m \in \mathbf{Z}} \) that converges to zero as \( \left| m\right| \rightarrow \infty \) and we would like to find an integrable function on \( {\mathbf{T}}^{1} \) with \( \widehat{f}\left( m\right) \geq {a}_{m} \) for all \( m \in \mathbf... | Yes |
Theorem 3.3.9. Let \( s \in \mathbf{Z} \) with \( s \geq 0 \) .\n\n(a) Suppose that \( {\partial }^{\alpha }f \) exist and are integrable for all \( \left| \alpha \right| \leq s \) . Then\n\n\[ \left| {\widehat{f}\left( m\right) }\right| \leq {\left( \frac{\sqrt{n}}{2\pi }\right) }^{s}\frac{\mathop{\max }\limits_{{\lef... | Proof. Fix \( m \in {\mathbf{Z}}^{n} \smallsetminus \{ 0\} \) and pick a \( j \) such that \( \left| {m}_{j}\right| = \mathop{\sup }\limits_{{1 \leq k \leq n}}\left| {m}_{k}\right| \) . Then clearly \( {m}_{j} \neq 0 \) . Integrating by parts \( s \) times with respect to the variable \( {x}_{j} \), we obtain\n\n\[ \wi... | Yes |
Proposition 3.3.14. If \( f \) is in \( {BV}\left( {\mathbf{T}}^{1}\right) \), then\n\n\[ \left| {\widehat{f}\left( m\right) }\right| \leq \frac{\operatorname{Var}\left( f\right) }{{2\pi }\left| m\right| } \]\n\nwhenever \( m \neq 0 \) . | Proof. Integration by parts gives\n\n\[ \widehat{f}\left( m\right) = {\int }_{{\mathbf{T}}^{1}}f\left( x\right) {e}^{-{2\pi imx}}{dx} = {\int }_{{\mathbf{T}}^{1}}\frac{{e}^{-{2\pi imx}}}{-{2\pi im}}{df}, \]\n\nwhere the boundary terms vanish because of periodicity. The conclusion follows from the fact that the norm of ... | Yes |
Proposition 3.4.2. Let \( {x}_{0} \in {\mathbf{T}}^{1} \) and let \( f \) be a complex-valued function on \( {\mathbf{T}}^{1} \) . Suppose that the left and right limits of \( f \) exist as \( x \rightarrow {x}_{0} \) and that the partial sums (Dirichlet means) \( \left( {{D}_{N} * f}\right) \left( {x}_{0}\right) \) co... | Proof. If \( \left( {{D}_{N} * f}\right) \left( {x}_{0}\right) \rightarrow L\left( {x}_{0}\right) \) as \( N \rightarrow \infty \), then\n\n\[ \left( {{F}_{N} * f}\right) \left( {x}_{0}\right) = \frac{\left( {{D}_{0} * f}\right) \left( {x}_{0}\right) + \left( {{D}_{1} * f}\right) \left( {x}_{0}\right) + \cdots + \left(... | Yes |
On \( \left( {-1/2,1/2}\right) \) let \( f\left( t\right) = t \) and \( f\left( {1/2}\right) = f\left( {-1/2}\right) = {1000} \) . Then \( f \) is discontinuous at the point \( - 1/2 \equiv 1/2 \) but it has left and right limits at this point: | \[ \mathop{\lim }\limits_{{t \rightarrow - \frac{1}{2} + }}f\left( t\right) = - \frac{1}{2}\;\mathop{\lim }\limits_{{t \rightarrow \frac{1}{2} - }}f\left( t\right) = \frac{1}{2}. \] | Yes |
Proposition 3.4.6. (a) (duBois Reymond) There exists a continuous function \( f \) on \( {\mathbf{T}}^{1} \) whose partial sums diverge at a point. Precisely, for some point \( {x}_{0} \in {\mathbf{T}}^{1} \) we have\n\n\[ \mathop{\limsup }\limits_{{N \rightarrow \infty }}\left| {\mathop{\sum }\limits_{\substack{{m \in... | We now prove part (a) using functional analysis. For a constructive proof, see Exercise 3.4.7. Let \( C\left( {\mathbf{T}}^{1}\right) \) be the Banach space of all continuous functions on the circle equipped with the \( {L}^{\infty } \) norm. Consider the continuous linear functionals\n\n\[ f \rightarrow {T}_{N}\left( ... | No |
Theorem 3.4.7. (Dini) Let \( f \) be an integrable function on \( {\mathbf{T}}^{1} \), let \( {t}_{0} \) be a point on \( {\mathbf{T}}^{1} \) for which \( f\left( {t}_{0}\right) \) is defined and assume that\n\n\[{\int }_{\left| t\right| \leq \frac{1}{2}}\frac{\left| f\left( t + {t}_{0}\right) - f\left( {t}_{0}\right) ... | Proof. Since the one-dimensional result is contained in the multidimensional one, we prove the latter. Replacing \( f\left( x\right) \) by \( f\left( {x + a}\right) - f\left( a\right) \), we may assume that \( a = 0 \) and \( f\left( a\right) = 0 \) . Using identities (3.1.15) and (3.1.14), we can write\n\n\[ \left( {{... | Yes |
Corollary 3.4.8. (a) (Riemann’s principle of localization) Let \( f \) be an integrable function on \( {\mathbf{T}}^{1} \) that vanishes on an open interval \( I \) . Then \( {D}_{N} * f \) converges to zero on the interval \( I \) . | Proof. (a) Let \( {t}_{0} \in I \) . If \( f \) vanishes on \( I \), condition (3.4.9) holds, since the function \( t \mapsto f\left( {t + {t}_{0}}\right) - f\left( {t}_{0}\right) \) vanishes on \( - {t}_{0} + I \), which is an interval containing the origin, and is integrable outside \( - {t}_{0} + I \) . Thus \( \lef... | Yes |
Corollary 3.4.9. Let \( a \in {\mathbf{T}}^{n} \) and suppose that \( f \in {L}^{1}\left( {\mathbf{T}}^{n}\right) \) satisfies\n\n\[ \left| {f\left( x\right) - f\left( a\right) }\right| \leq C{\left| {x}_{1} - {a}_{1}\right| }^{{\varepsilon }_{1}}\cdots {\left| {x}_{n} - {a}_{n}\right| }^{{\varepsilon }_{n}} \]\n\nfor ... | Proof. Note that condition (3.4.10) holds. | No |
Corollary 3.4.10. (Dirichlet) If \( f \) is defined on \( {\mathbf{T}}^{1} \) and is a differentiable function at a point \( a \) in \( {\mathbf{T}}^{1} \), then \( \left( {{D}_{N} * f}\right) \left( a\right) \rightarrow f\left( a\right) \) . | Proof. There exists a \( \delta > 0 \) (say less than \( 1/2 \) ) such that \( \left| {f\left( x\right) - f\left( a\right) }\right| /\left| {x - a}\right| \) is bounded by \( \left| {{f}^{\prime }\left( a\right) }\right| + 1 \) for \( \left| {x - a}\right| \leq \delta \) . Also \( \left| {f\left( x\right) - f\left( a\r... | Yes |
Corollary 3.5.2. Suppose that a function \( f \) on \( {\mathbf{T}}^{1} \) is continuous and there is a constant \( M > 0 \) such that \( \left| {\widehat{f}\left( m\right) }\right| \leq M{\left| m\right| }^{-1} \) for all \( m \in {\mathbf{Z}}^{ + } \smallsetminus \{ 0\} \) . Then the Fourier series of \( f \) converg... | Proof. The Fejér means \( {\left\{ {F}_{N}\right\} }_{N = 0}^{\infty } \) are an approximate identity on \( {\mathbf{T}}^{n} \) (Proposition 3.1.10) and so \( {F}_{N} * f \) converge uniformly to \( f \) on \( {\mathbf{T}}^{1} \) as \( N \rightarrow \infty \) in view of Theorem 1.2.19 (2). Moreover, we have \( \left| m... | Yes |
Theorem 3.5.5. Suppose that \( f \) is a function of bounded variation on the circle \( {\mathbf{T}}^{1} \) . Then the partial sums of the Fourier series of \( f \) are uniformly bounded, in particular, we have\n\n\[ \mathop{\sup }\limits_{{{t}_{0} \in {\mathbf{T}}^{1}}}\mathop{\sup }\limits_{{N \in {\mathbf{Z}}^{ + }}... | Proof. We take \( \delta = 1/2 \) in the proof of the preceding theorem. For a point \( {t}_{0} \in {\mathbf{T}}^{1} \) , let \( {F}_{{t}_{0}}\left( t\right) = \frac{f\left( {{t}_{0} - t}\right) + f\left( {{t}_{0} + t}\right) }{2} \) . We have that\n\n\[ \left( {f * {D}_{N}}\right) \left( {t}_{0}\right) = {\int }_{{\ma... | No |
Theorem 3.5.7. (a) Let \( h \) be defined in (3.5.11). Then the set of accumulation points of sets of the form \( {\left\{ \left( h * {D}_{N}\right) \left( {t}_{N}\right) \right\} }_{N \in {\mathbf{Z}}^{ + }} \), where \( {t}_{N} \in \left\lbrack {0,1/2}\right\rbrack \), is the interval\n\n\[ \left\lbrack {0,\frac{\ope... | Proof. (a) Since \( h \geq 0 \) on \( \left( {0,\frac{1}{2}}\right\rbrack \) and \( \left( {h * {D}_{N}}\right) \left( t\right) \rightarrow \frac{1}{2} - t \) for \( 0 < t \leq \frac{1}{2} \) we have that all accumulation points of sequences \( \left( {h * {D}_{N}}\right) \left( {t}_{N}\right) \) are nonnegative. We sh... | Yes |
Corollary 3.6.3. (Weierstrass) There exists a continuous function on the circle that is nowhere differentiable. | Proof. Consider the 1-periodic function\n\n\[ f\left( t\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }{2}^{-k}{e}^{{2\pi i}{3}^{k}t}. \]\n\nSince this series converges absolutely and uniformly, \( f \) is a continuous function. If \( f \) were differentiable at a point, then by Proposition 3.6.2 we would have that ... | Yes |
Theorem 3.6.6. Let \( 1 < {\lambda }_{1} < {\lambda }_{2} < {\lambda }_{3} < \cdots \) be a lacunary sequence of integers with constant \( A > 1 \) . Set \( \Lambda = \left\{ {{\lambda }_{k} : k \in {\mathbf{Z}}^{ + }}\right\} \) . Then there exists a constant \( C\left( A\right) \) such that for all \( f \in {L}^{\inf... | Proof. Let us assume first that \( A \geq 3 \) . Also fix \( f \in {L}^{\infty }\left( {\mathbf{T}}^{1}\right) \) . We consider the Riesz product\n\n\[ {P}_{N}\left( x\right) = \mathop{\prod }\limits_{{j = 1}}^{N}\left( {1 + \cos \left( {{2\pi }{\lambda }_{j}x + {2\pi }{\gamma }_{j}}\right) }\right) ,\]\n\nwhere \( {\g... | Yes |
Corollary 3.6.7. Let \( \Lambda = \left\{ {{\lambda }_{k} : k \in {\mathbf{Z}}^{ + }}\right\} \) be a lacunary set and let \( f \) be a bounded function on the circle that satisfies \( \widehat{f}\left( k\right) = 0 \) when \( k \in \mathbf{Z} \smallsetminus \Lambda \) . Then \( f \) is almost everywhere equal to the a... | Proof. It follows from Theorem 3.6.6 that if \( \widehat{f}\left( k\right) = 0 \) when \( k \in \mathbf{Z} \smallsetminus \Lambda \), then we have that \( f \in A\left( {\mathbf{T}}^{1}\right) \) . Applying the inversion result in Proposition 3.2.5 we obtain that \( f \) is almost everywhere equal to a continuous funct... | Yes |
Proposition 3.6.9. The following assertions are equivalent for a subset \( E \) of \( \mathbf{Z} \) .\n\n(1) There is a constant \( K \) such that for all trigonometric polynomials \( P \) with \( \widehat{P} \) supported in \( E \) we have\n\n\[ \mathop{\sum }\limits_{{m \in \mathbf{Z}}}\left| {\widehat{P}\left( m\rig... | Proof. Suppose that (1) holds. Given \( f \) in \( {L}^{\infty }\left( {\mathbf{T}}^{1}\right) \) with \( \widehat{f} \) is supported in \( E \), write\n\n\[ \left( {f * {F}_{N}}\right) \left( x\right) = \mathop{\sum }\limits_{{m = - N}}^{N}\left( {1 - \frac{\left| m\right| }{N + 1}}\right) \widehat{f}\left( m\right) {... | Yes |
Every lacunary set is a Sidon set. Indeed, suppose that \( E \) is a lacunary set with constant \( A \) . If \( f \) is a continuous function which satisfies (3.6.24), then Theorem 3.6.6 gives that | \[ \mathop{\sum }\limits_{{m \in \Lambda }}\left| {\widehat{f}\left( m\right) }\right| \leq C\left( A\right) \parallel f{\parallel }_{{L}^{\infty }} < \infty \] hence \( f \) has an absolutely convergent Fourier series. | No |
Then for all \( f \in {L}^{p}\left( {\mathbf{T}}^{n}\right) \) the sequence \( {S}_{R}\left( f\right) \) converges in \( {L}^{p} \) as \( R \rightarrow \infty \) if and only if there exists a constant \( K < \infty \) such that\n\n\[ \mathop{\sup }\limits_{{R > 0}}{\begin{Vmatrix}{S}_{R}\end{Vmatrix}}_{{L}^{p} \rightar... | Proof. If \( {S}_{R}\left( f\right) \) converges in \( {L}^{p} \), then \( {\begin{Vmatrix}{S}_{R}\left( f\right) \end{Vmatrix}}_{{L}^{p}} \leq {C}_{f} \) for some constant \( {C}_{f} \) that depends on \( f \in {L}^{p}\left( {\mathbf{T}}^{n}\right) \) . Moreover, each \( {S}_{R} \) is a bounded operator from \( {L}^{p... | Yes |
We investigate the one-dimensional case in some detail. We take \( n = 1 \), and we define \( a\left( {m, N}\right) = 1 \) for all \( - N \leq m \leq N \), and zero otherwise. Then \( {S}_{N}\left( f\right) = {S}_{N}\left( f\right) = {D}_{N} * f \), where \( {D}_{N} \) is the Dirichlet kernel. Clearly, the expressions ... | This reasoning, however, allows us to deduce that for some function \( g \in {L}^{1}\left( {\mathbf{T}}^{1}\right) \) , \( {S}_{N}\left( g\right) \) may not converge in \( {L}^{1} \) . This is also a consequence of the proof of Theorem 4.2.1; see (4.2.13). Note that since the Fejér kernel \( {F}_{M} \) has \( {L}^{1} \... | Yes |
Theorem 4.2.1. There exists an integrable function on the circle \( {\mathbf{T}}^{1} \) whose Fourier series diverges almost everywhere. | Proof. The proof of this theorem is a bit involved, and we need a sequence of lemmas, which we prove first. | No |
Lemma 4.2.4. For each \( 0 < M < \infty \) there exists a trigonometric polynomial \( {g}_{M} \) and a measurable subset \( {A}_{M} \) of \( {\mathbf{T}}^{1} \) with measure \( \left| {A}_{M}\right| > 1 - {2}^{-M} \) such that \( {\begin{Vmatrix}{g}_{M}\end{Vmatrix}}_{{L}^{1}} = 1 \) , and such that | Proof. Given an \( M \in {\mathbf{Z}}^{ + } \), we pick an integer \( N\left( M\right) \) such that \( c\log N\left( M\right) > {2}^{M + 2} \) , where \( c \) is as in (4.2.2), and we also pick the measure \( {\mu }_{N\left( M\right) } \), which satisfies (4.2.2). By Fatou's lemma we have | No |
Theorem 4.3.1. Suppose that \( T \) is a linear operator that commutes with translations and maps \( {L}^{p}\left( {\mathbf{T}}^{n}\right) \) to \( {L}^{q}\left( {\mathbf{T}}^{n}\right) \) for some \( 1 \leq p, q \leq \infty \) . Then there exists a bounded sequence \( {\left\{ {a}_{m}\right\} }_{m \in {\mathbf{Z}}^{n}... | Proof. Consider the functions \( {e}_{m}\left( x\right) = {e}^{{2\pi im} \cdot x} \) defined on \( {\mathbf{T}}^{n} \) for \( m \) in \( {\mathbf{Z}}^{n} \) . Since \( T \) commutes with translations, for every \( h \in {\mathbf{T}}^{n} \) there is a subset \( {F}_{h} \) of \( {\mathbf{T}}^{n} \) of full measure such t... | Yes |
A linear operator \( T \) that commutes with translations maps \( {L}^{2}\left( {\mathbf{T}}^{n}\right) \) to itself if and only if there exists a sequence \( {\left\{ {a}_{m}\right\} }_{m \in {\mathbf{Z}}^{n}} \) in \( {\ell }^{\infty }\left( {\mathbf{Z}}^{n}\right) \) such that\n\n\[ T\left( f\right) \left( x\right) ... | The existence of such a sequence is guaranteed by Theorem 4.3.1, which also gives \( {\begin{Vmatrix}{\left\{ {a}_{m}\right\} }_{m}\end{Vmatrix}}_{{\ell }^{\infty }} \leq \parallel T{\parallel }_{{L}^{2} \rightarrow {L}^{2}} \) . Conversely, any operator of the form (4.3.3) satisfies\n\n\[ \parallel T\left( f\right) {\... | Yes |
Lemma 4.3.9. Let \( T \) be the operator on \( {\mathbf{R}}^{n} \) whose multiplier is \( b\left( \xi \right) \), and let \( S \) be the operator on \( {\mathbf{T}}^{n} \) whose multiplier is the sequence \( \{ b\left( m\right) {\} }_{m \in {\mathbf{Z}}^{n}} \) . Assume that \( b\left( \xi \right) \) is regulated at ev... | Proof. It suffices to prove the required assertion for \( P\left( x\right) = {e}^{{2\pi im} \cdot x} \) and \( Q\left( x\right) = \) \( {e}^{{2\pi ik} \cdot x}, k, m \in {\mathbf{Z}}^{n} \), since the general case follows from this case by linearity. In view of Parseval's relation (Proposition 3.2.7 (3)), we have\n\n\[... | Yes |
Theorem 4.3.10. Suppose that \( b\left( \xi \right) \) is a bounded function defined on \( {\mathbf{R}}^{n} \) which is Riemann integrable over any cube. Suppose that the sequences \( {\left\{ b\left( \frac{m}{R}\right) \right\} }_{m \in {\mathbf{Z}}^{n}} \) are in \( {\mathcal{M}}_{p}\left( {\mathbf{Z}}^{n}\right) \) ... | Proof. Suppose that \( f \) and \( g \) are smooth functions with compact support on \( {\mathbf{R}}^{n} \) . Then there is an \( {R}_{0} > 0 \) such that for \( R \geq {R}_{0} \), the functions \( x \mapsto f\left( {Rx}\right) \) and \( x \mapsto g\left( {Rx}\right) \) are supported in \( {\left\lbrack -1/2,1/2\right\... | Yes |
Theorem 4.3.12. Let \( b \) be a function defined on \( {\mathbf{R}}^{n} \). Suppose that \( b \) is bounded, regulated, Riemann integrable over any cube, and assume that for all \( \xi \in {\mathbf{R}}^{n} \) the function \( t \mapsto b\left( {\xi /t}\right) \) has only countably many discontinuities on \( {\mathbf{R}... | Proof. Let \( \mathcal{F} = \left\{ {{t}_{1},\ldots ,{t}_{k}}\right\} \) be a finite subset of \( {\mathbf{R}}^{ + } \). We prove the claimed equivalences for the maximal operators\n\n\[ {M}_{b}^{\mathcal{F}}\left( G\right) \left( x\right) = \mathop{\sup }\limits_{{t \in \mathcal{F}}}\left| {{S}_{b, t}\left( G\right) \... | Yes |
Theorem 4.3.15. For every \( 1 < p < \infty \) there exists a finite constant \( {C}_{p} \) such that for all \( f \in {\mathcal{C}}_{0}^{\infty }\left( \mathbf{R}\right) \) we have\n\n\[ \n{\begin{Vmatrix}{\mathcal{C}}_{* * }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( \mathbf{R}\right) } \leq {C}_{p}\parallel f{\par... | As a consequence of Theorem 4.3.14, we obtain that for any \( F \in {L}^{p}\left( {\mathbf{T}}^{1}\right) \), we have\n\n\[ \n\mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\sum }\limits_{{\left| m\right| \leq N}}\widehat{F}\left( m\right) {e}^{2\pi imx} = F\left( x\right)\n\]\n\nfor almost every \( x \in \left... | No |
Proposition 4.4.3. Let \( k > 0 \) be fixed and let \( f \) be in \( {\mathcal{C}}^{\infty }\left( {\mathbf{R}}^{n}\right) \) . Assume that \( f \) is 1-periodic function in each variable. Then the heat equation\n\n\[ \n\frac{\partial }{\partial t}F\left( {x, t}\right) = k{\Delta }_{x}F\left( {x, t}\right) \;t \in \lef... | Proof. Since \( f \) is \( {\mathcal{C}}^{\infty } \), the series in (4.4.14) is rapidly convergent in \( m \) and thus it gives a continuous function on \( \lbrack 0,\infty ) \times {\mathbf{R}}^{n} \) . Moreover, the series can be differentiated term by term in the variable \( t > 0 \), and thus it produces a \( {\ma... | Yes |
The sequence \( \{ k\sqrt{2} - \left\lbrack {k\sqrt{2}}\right\rbrack {\} }_{k = 0}^{\infty } \) is equidistributed on \( {\mathbf{T}}^{1} \). | We check this by verifying condition (c) of Theorem 4.5.6. Indeed if \( m \in \mathbf{Z} \smallsetminus \{ 0\} \) then\n\n\[\n\mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{k = 0}}^{{N - 1}}{e}^{{2\pi im}\left( {k\sqrt{2} - \left\lbrack {k\sqrt{2}}\right\rbrack }\right) } = \mathop{\l... | Yes |
We examine the sequence of the first digits of powers of 2. Consider the following sequence of numbers defined for \( m = 1,2,\ldots \)\n\n\[ \n{d}_{m} = \text{first digit of}{2}^{m}\text{.}\n\]\n\nFor instance we have \( {d}_{1} = 2,{d}_{2} = 4,{d}_{3} = 8,{d}_{4} = 1,{d}_{5} = 3,\ldots \) .\n\nFix an integer \( k \in... | The crucial observation is that the first digit of \( {2}^{m} \) is equal to \( k \) if and only if there is a nonnegative integer \( s \) such that\n\n\[ \nk{10}^{s} \leq {2}^{m} < \left( {k + 1}\right) {10}^{s}.\n\]\n\nTaking logarithms with base 10 we obtain\n\n\[ \ns + {\log }_{10}\left( k\right) \leq m{\log }_{10}... | Yes |
For the characteristic function \( {\chi }_{\left\lbrack a, b\right\rbrack } \) of an interval \( \left\lbrack {a, b}\right\rbrack \) we show that \[ H\left( {\chi }_{\left\lbrack a, b\right\rbrack }\right) \left( x\right) = \frac{1}{\pi }\log \frac{\left| x - a\right| }{\left| x - b\right| }.\] | Let us verify this identity. Pick \( \varepsilon < \min \left( {\left| {x - a}\right| ,\left| {x - b}\right| }\right) \) . To show (5.1.6) consider the three cases \( 0 < x - b, x - a < 0 \), and \( x - b < 0 < x - a \) . In the first two cases, (5.1.6) follows immediately. In the third case we have \[ H\left( {\chi }_... | Yes |
Let \( {\log }^{ + }x = \log x \) when \( x \geq 1 \) and zero otherwise. Observe that the calculation in the previous example actually gives\n\n\[ \n{H}^{\left( \varepsilon \right) }\left( {\chi }_{\left\lbrack a, b\right\rbrack }\right) \left( x\right) = \left\{ \begin{array}{ll} \frac{1}{\pi }{\log }^{ + }\frac{\lef... | We now give an alternative characterization of the Hilbert transform using the Fourier transform. To achieve this we need to compute the Fourier transform of the distribution \( {W}_{0} \) defined in (5.1.1). Fix a Schwartz function \( \varphi \) on \( \mathbf{R} \) . Then\n\n\[ \n\left\langle {\widehat{{W}_{0}},\varph... | Yes |
Theorem 5.1.5. Let \( 1 \leq p < \infty \) . For any \( f \in {L}^{p}\left( \mathbf{R}\right) \) we have\n\n\[ f * {Q}_{\varepsilon } - {H}^{\left( \varepsilon \right) }\left( f\right) \rightarrow 0 \]\n\nin \( {L}^{p} \) and almost everywhere as \( \varepsilon \rightarrow 0 \) . Moreover, for \( \varphi \) in \( \math... | Proof. We see that\n\n\[ \left( {{Q}_{\varepsilon } * f}\right) \left( x\right) - \frac{1}{\pi }{\int }_{\left| t\right| \geq \varepsilon }\frac{f\left( {x - t}\right) }{t}{dt} = \frac{1}{\pi }\left( {f * {\psi }_{\varepsilon }}\right) \left( x\right) ,\]\n\nwhere \( {\psi }_{\varepsilon }\left( x\right) = {\varepsilon... | Yes |
Proposition 5.1.16. The Riesz transforms satisfy\n\n\[ \n- I = \mathop{\sum }\limits_{{j = 1}}^{n}{R}_{j}^{2},\;\text{ on }{L}^{2}\left( {\mathbf{R}}^{n}\right) , \n\]\n\n(5.1.46)\n\nwhere \( I \) is the identity operator. | Proof. Use the Fourier transform and the identity \( \mathop{\sum }\limits_{{j = 1}}^{n}{\left( -i{\xi }_{j}/\left| \xi \right| \right) }^{2} = - 1 \) to obtain that \( \mathop{\sum }\limits_{{j = 1}}^{n}{R}_{j}^{2}\left( f\right) = - f \) for any \( f \) in \( {L}^{2}\left( {\mathbf{R}}^{n}\right) \) . | Yes |
Proposition 5.1.17. For \( \varphi \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) and \( 1 \leq j, k \leq n \) we have\n\n\[ \n{\partial }_{j}{\partial }_{k}\varphi \left( x\right) = - {R}_{j}{R}_{k}{\Delta \varphi }\left( x\right) \n\]\n\n(5.1.47)\n\nfor all \( x \in {\mathbf{R}}^{n} \) . | Proof. We verify the claimed identity by taking Fourier transforms. We have\n\n\[ \n{\left( {\partial }_{j}{\partial }_{k}\varphi \right) }^{ \frown }\left( \xi \right) = \left( {{2\pi i}{\xi }_{j}}\right) \left( {{2\pi i}{\xi }_{k}}\right) \widehat{\varphi }\left( \xi \right) \n\]\n\n\[ \n= - \left( {-\frac{i{\xi }_{j... | Yes |
Proposition 5.2.3. Let \( n \geq 2 \) and \( \Omega \in {L}^{1}\left( {\mathbf{S}}^{n - 1}\right) \) have mean value zero. Then the Fourier transform of \( {W}_{\Omega } \) is a (finite a.e.) function given by the formula \[ \widehat{{W}_{\Omega }}\left( \xi \right) = {\int }_{{\mathbf{S}}^{n - 1}}\Omega \left( \theta ... | Remark 5.2.4. We need to show that the function of \( \xi \) on the right in (5.2.8) is well defined and finite for almost all \( \xi \) in \( {\mathbf{R}}^{n} \) . Write \( \xi = \left| \xi \right| {\xi }^{\prime } \) where \( {\xi }^{\prime } \in {\mathbf{S}}^{n - 1} \) and notice that \[ \log \frac{1}{\left| \xi \cd... | Yes |
Lemma 5.2.5. Let a be a nonzero real number. Then for \( 0 < \varepsilon < N < \infty \) we have\n\n\[ \mathop{\lim }\limits_{\substack{{\varepsilon \rightarrow 0} \\ {N \rightarrow \infty } }}{\int }_{\varepsilon }^{N}\frac{\cos \left( {ra}\right) - \cos \left( r\right) }{r}{dr} = \log \frac{1}{\left| a\right| }, \] | Proof. We first prove (5.2.10) and (5.2.11). By the fundamental theorem of calculus we write\n\n\[ {\int }_{\varepsilon }^{N}\frac{\cos \left( {ra}\right) - \cos \left( r\right) }{r}{dr} = {\int }_{\varepsilon }^{N}\frac{\cos \left( {r\left| a\right| }\right) - \cos \left( r\right) }{r}{dr} \]\n\n\[ = - {\int }_{\varep... | Yes |
Let \( \Omega \in {L}^{1}\left( {\mathbf{S}}^{n - 1}\right) \) have mean value zero. Then for almost all \( {\xi }^{\prime } \) in \( {\mathbf{S}}^{n - 1} \) the integral\n\n\[ \n{\int }_{{\mathbf{S}}^{n - 1}}\Omega \left( \theta \right) \log \frac{1}{\left| {\xi }^{\prime } \cdot \theta \right| }{d\theta } \n\]\n\n(5.... | Proof. To obtain the absolute convergence of the integral in (5.2.15) we integrate over \( {\xi }^{\prime } \in {\mathbf{S}}^{n - 1} \) and we apply Fubini’s theorem. The assertion concerning the boundedness of \( {T}_{\Omega } \) on \( {L}^{2} \) is an immediate consequence of Proposition 5.2.3 and Theorem 2.5.10. | No |
Corollary 5.2.8. The Riesz transforms \( {R}_{j} \) and the maximal Riesz transforms \( {R}_{j}^{\left( *\right) } \) are bounded on \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) for \( 1 < p < \infty \) . | Proof. The assertion follows from the fact that the Riesz transforms have odd kernels. Since the kernel of \( {R}_{j} \) decays like \( {\left| x\right| }^{-n} \) near infinity, it follows that \( {R}_{j}^{\left( *\right) }\left( f\right) \) is well defined for \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) . Since ... | Yes |
Corollary 5.2.12. Let \( \Omega \) be as in Theorem 5.2.11. Then for \( 1 < p < \infty \) and \( f \) in \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) the functions \( {T}_{\Omega }^{\left( \varepsilon, N\right) }\left( f\right) \) converge to \( {T}_{\Omega }\left( f\right) \) in \( {L}^{p} \) and almost everywhere as \... | Proof. The a.e. convergence is a consequence of Theorem 2.1.14. The \( {L}^{p} \) convergence is a consequence of the Lebesgue dominated convergence theorem since for \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) we have that \( \left| {{T}_{\Omega }^{\left( \varepsilon, N\right) }\left( f\right) }\right| \leq {T}_... | Yes |
Theorem 5.3.1. 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 }_{{L... | Proof. Decompose \( {\mathbf{R}}^{n} \) into a mesh of disjoint dyadic cubes of the same size such that\n\n\[ \left| Q\right| \geq \frac{1}{\alpha }\parallel f{\parallel }_{{L}^{1}} \]\n\nfor every cube \( Q \) in the mesh. Call these cubes of zero generation. Subdivide each cube of zero generation into \( {2}^{n} \) c... | Yes |
Theorem 5.3.4. (Cotlar’s inequality) Let \( 0 < {A}_{1},{A}_{2},{A}_{3} < \infty \) and suppose that \( K \) is defined on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) and satisfies the size condition,\n\n\[ \left| {K\left( x\right) }\right| \leq {A}_{1}{\left| x\right| }^{-n},\;x \neq 0, \]\n\n(5.3.19)\n\nthe smoothne... | Proof. Let \( \varphi \) be a radially decreasing smooth function with integral 1 supported in the ball \( B\left( {0,1/2}\right) \) . For a function \( g \) and \( \varepsilon > 0 \) we use the notation \( {g}_{\varepsilon }\left( x\right) = \) \( {\varepsilon }^{-n}g\left( {{\varepsilon }^{-1}x}\right) \) . For a dis... | Yes |
Theorem 5.3.5. Let \( K\left( x\right) \) be function on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) satisfying (5.3.4) with constant \( {A}_{1} < \infty \) and Hörmander’s condition (5.3.12) with constant \( {A}_{2} < \infty \) . Suppose that the operator \( {T}^{\left( * * \right) } \) as defined in (5.3.18) maps \(... | Proof. The proof of this theorem is only a little more involved than the proof of Theorem 5.3.3. We fix an \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) function \( f \) . We apply the Calderón-Zygmund decomposition of \( f \) at height \( {\gamma \alpha } \) for some \( \gamma ,\alpha > 0 \) . We then write \( f = g + b... | Yes |
Corollary 5.3.6. The maximal Hilbert transform \( {H}^{\left( *\right) } \) and the maximal Riesz transforms \( {R}_{j}^{\left( *\right) } \) are weak type \( \left( {1,1}\right) \) . Secondly, \( \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{H}^{\left( \varepsilon \right) }\left( f\right) \) and \( \mathop{\lim ... | Proof. Since the kernels \( 1/x \) on \( \mathbf{R} \) and \( {x}_{j}/{\left| x\right| }^{n} \) on \( {\mathbf{R}}^{n} \) satisfy (5.3.10), the first statement in the corollary is an immediate consequence of Theorem 5.3.5. The second statement follows from Theorem 2.1.14 and Corollary 5.2.8, since these limits exist fo... | Yes |
Corollary 5.3.7. Under the hypotheses of Theorem 5.3.5, \( {T}^{\left( * * \right) } \) maps \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) to itself for \( 1 < p < 2 \) with norm | \[ {\begin{Vmatrix}{T}^{\left( * * \right) }\end{Vmatrix}}_{{L}^{p} \rightarrow {L}^{p}} \leq \frac{{C}_{n}\left( {{A}_{1} + {A}_{2} + B}\right) }{p - 1}, \] where \( {C}_{n} \) is some dimensional constant. | Yes |
Theorem 5.4.1. Assume that \( K \) satisfies (5.4.1),(5.4.2), and (5.4.3), and let \( W \) be a tempered distribution of the form (5.3.7) that coincides with \( K \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) . Then we have\n\n\[ \mathop{\sup }\limits_{{0 < \varepsilon < N < \infty }}\mathop{\sup }\limits_{{\xi \i... | Proof. Let us set \( {K}^{\left( \varepsilon, N\right) }\left( x\right) = K\left( x\right) {\chi }_{\varepsilon < \left| x\right| < N} \) . Estimate (5.4.4) implies that for all \( f \) in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ {\begin{Vmatrix}f * {K}^{\left( {\delta }_{j}, j\right) }\end{Vmatrix... | Yes |
Example 5.4.2. Let \( \tau \) be a nonzero real number and let \( K\left( x\right) = \frac{1}{{\left| x\right| }^{n + {i\tau }}} \) defined for \( x \in {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) . For a sequence \( {\delta }_{k} \downarrow 0 \) and \( \varphi \) a Schwartz function on \( {\mathbf{R}}^{n} \), define\n\n... | Take, for example, \( {\delta }_{k} = {e}^{-{2\pi k}/\tau } \) . For this sequence \( {\delta }_{k} \), observe that\n\n\[ {\int }_{{\delta }_{k} \leq \left| x\right| \leq 1}\frac{1}{{\left| x\right| }^{n + {i\tau }}}{dx} = {\omega }_{n - 1}\frac{1 - {\left( {e}^{-{2\pi k}/\tau }\right) }^{-{i\tau }}}{-{i\tau }} = 0, \... | Yes |
For any \( 0 < r < \infty \), define constants\n\n\[ \n{A}_{r} = {\left( \frac{\Gamma \left( \frac{r + 1}{2}\right) }{{\pi }^{\frac{r + 1}{2}}}\right) }^{\frac{1}{r}}\;\text{ and }\;{B}_{r} = {\left( \frac{\Gamma \left( {\frac{r}{2} + 1}\right) }{{\pi }^{\frac{r}{2}}}\right) }^{\frac{1}{r}}.\n\]\n\nThen for any \( {\la... | Proof. Dividing both sides of (5.5.8) by \( {\left( {\lambda }_{1}^{2} + \cdots + {\lambda }_{n}^{2}\right) }^{\frac{1}{2}} \), we reduce things to the situation in which \( {\lambda }_{1}^{2} + \cdots + {\lambda }_{n}^{2} = 1 \) . Let \( {e}_{1} = {\left( 1,0,\ldots ,0\right) }^{t} \) be the standard basis column unit... | Yes |
On the real line consider the intervals \( {I}_{j} = \left\lbrack {{b}_{j},\infty }\right) \) for \( j \in \mathbf{Z} \) . Let \( {T}_{j} \) be the operator given by multiplication on the Fourier transform by the characteristic function of \( {I}_{j} \) . Then we have the following two inequalities:\n\n\[{\begin{Vmatri... | To prove these, first observe that the operator \( T = \frac{1}{2}\left( {I + {iH}}\right) \) is given on the Fourier transform by multiplication by the characteristic function of the half-axis \( \lbrack 0,\infty ) \) [precisely, the Fourier multiplier of \( T \) is equal to 1 on the set \( \left( {0,\infty }\right) \... | Yes |
Corollary 5.5.4. Let \( \\left( {X,\\mu }\\right) \) and \( \\left( {Y,\\nu }\\right) \) be \( \\sigma \) -finite measure spaces. Suppose that \( T \) is a linear bounded operator from \( {L}^{p}\\left( X\\right) \) to \( {L}^{p}\\left( Y\\right) \) with norm \( A \) for some \( 1 < p < \\infty \) . Let \( r \) be a nu... | Proof. Using Exercise 5.5.2 we interpolate between the trivial bound \( {L}^{p}\\left( {X,{\\ell }^{p}}\\right) \\rightarrow \) \( {L}^{p}\\left( {Y,{\\ell }^{p}}\\right) \) and the bound \( {L}^{p}\\left( {X,{\\ell }^{2}}\\right) \\rightarrow {L}^{p}\\left( {Y,{\\ell }^{2}}\\right) \), which follows from Theorem 5.5.1... | No |
Proposition 5.5.6. Let \( \mathcal{B} \) be a Banach space and \( \left( {X,\mu }\right) \) a \( \sigma \) -finite measure space. (a) The set \( \left\{ {\mathop{\sum }\limits_{{j = 1}}^{m}{\chi }_{{E}_{j}}{u}_{j} : {u}_{j} \in \mathcal{B},{E}_{j} \subseteq X}\right. \) are pairwise disjoint and \( \left. {\mu \left( {... | Proof. If \( F \in {L}^{p}\left( {X,\mathcal{B}}\right) \) for \( 0 < p \leq \infty \), then \( F \) is \( \mathcal{B} \) -measurable; thus there exists \( {X}_{0} \subseteq X \) satisfying \( \mu \left( {X \smallsetminus {X}_{0}}\right) = 0 \) and \( F\left\lbrack {X}_{0}\right\rbrack \subseteq {\mathcal{B}}_{0} \), w... | Yes |
Let \( \mathcal{B} = {\ell }^{r} \) for some \( 1 \leq r < \infty \) . Then a measurable function \( F : X \rightarrow \mathcal{B} \) is just a sequence \( {\left\{ {f}_{j}\right\} }_{j} \) of measurable functions \( {f}_{j} : X \rightarrow \mathbf{C} \) . The space \( {L}^{p}\left( {X,{\ell }^{r}}\right) \) consists o... | The space \( {L}^{p}\left( X\right) \otimes {\ell }^{r} \) is the set of all finite sums \[ \mathop{\sum }\limits_{{j = 1}}^{m}\left( {{a}_{j1},{a}_{j2},{a}_{j3},\ldots }\right) {g}_{j} \] where \( {g}_{j} \in {L}^{p}\left( X\right) \) and \( \left( {{a}_{j1},{a}_{j2},{a}_{j3},\ldots }\right) \in {\ell }^{r}, j = 1,\ld... | Yes |
Theorem 5.6.1. Let \( {\mathcal{B}}_{1} \) and \( {\mathcal{B}}_{2} \) be Banach spaces. Suppose that \( \overrightarrow{K}\left( x\right) \) satisfies (5.6.1),(5.6.2), and (5.6.3) for some \( A > 0 \) and \( {\overrightarrow{K}}_{0} \in L\left( {{\mathcal{B}}_{1},{\mathcal{B}}_{2}}\right) \) . Let \( \overrightarrow{T... | Proof. Although \( \overrightarrow{T} \) is defined on the entire \( {L}^{1}\left( {{\mathbf{R}}^{n},{\mathcal{B}}_{1}}\right) \cap {L}^{r}\left( {{\mathbf{R}}^{n},{\mathcal{B}}_{1}}\right) \), it will be convenient to work with its restriction to a smaller dense subspace of \( {L}^{1}\left( {{\mathbf{R}}^{n},{\mathcal... | Yes |
Corollary 5.6.2. Let \( A, B > 0 \) and let \( {W}_{j} \) be a sequence of tempered distributions on \( {\mathbf{R}}^{n} \) whose Fourier transforms are uniformly bounded functions (i.e., \( \left| \widehat{{W}_{j}}\right| \leq B \) ). Suppose that for each \( j,{W}_{j} \) coincides with a function \( {K}_{j} \) on \( ... | Proof. Let \( {T}_{j} \) be the operator given by convolution with the distribution \( {W}_{j} \) . Clearly \( {T}_{j} \) is \( {L}^{2} \) bounded with norm at most \( B \) . It follows from Theorem 5.3.3 that the \( {T}_{j} \) ’s are of weak type \( \left( {1,1}\right) \) and also bounded on \( {L}^{r} \) with bounds ... | Yes |
Proposition 5.6.4. Let let \( 1 < p, r < \infty \) and let \( {\mathcal{B}}_{1} \) and \( {\mathcal{B}}_{2} \) be two Banach spaces. Suppose that \( \overrightarrow{T} \) given by (5.6.4) is a bounded linear operator from \( {L}^{r}\left( {{\mathbf{R}}^{n},{\mathcal{B}}_{1}}\right) \) to \( {L}^{r}\left( {{\mathbf{R}}^... | Proof. Let us denote by \( {\ell }^{r}\left( {\mathcal{B}}_{1}\right) \) the Banach space of all \( {\mathcal{B}}_{1} \)-valued sequences \( {\left\{ {u}_{j}\right\} }_{j} \) that satisfy \[ {\begin{Vmatrix}{\left\{ {u}_{j}\right\} }_{j}\end{Vmatrix}}_{{\ell }^{r}\left( {\mathcal{B}}_{1}\right) } = {\left( \mathop{\sum... | Yes |
Corollary 5.6.5. Let \( \Phi \) be an integrable function on \( {\mathbf{R}}^{n} \) that satisfies (5.6.18). Then there exist dimensional constants \( {C}_{n} \) and \( {C}_{n}^{\prime } \) such that for all \( 1 < p, r < \infty \) the following vector-valued inequalities are valid:\n\n\[ \n{\begin{Vmatrix}{\left( \mat... | Proof. We set \( {\mathcal{B}}_{1} = \mathbf{C} \) and \( {\mathcal{B}}_{2} = {L}^{\infty }\left( {\mathbf{R}}^{ + }\right) \) . We use estimate (5.6.21) as a starting point in Proposition 5.6.4, which immediately yields the required conclusions (5.6.22) and (5.6.23). | Yes |
Theorem 6.1.2. (Littlewood-Paley theorem) Suppose that \( \Psi \) is an integrable \( {\mathcal{C}}^{1} \) function on \( {\mathbf{R}}^{n} \) with mean value zero that satisfies\n\n\[ \left| {\Psi \left( x\right) }\right| + \left| {\nabla \Psi \left( x\right) }\right| \leq B{\left( 1 + \left| x\right| \right) }^{-n - 1... | Proof. We first prove (6.1.4) when \( p = 2 \) . Using Plancherel’s theorem, we see that (6.1.4) is a consequence of the inequality\n\n\[ \mathop{\sum }\limits_{j}{\left| \widehat{\Psi }\left( {2}^{-j}\xi \right) \right| }^{2} \leq {C}_{n}{B}^{2} \]\n\nfor some \( {C}_{n} < \infty \) . Because of (6.1.3), Fourier inver... | Yes |
Proposition 6.1.4. Let \( \Psi \) be an integrable \( {\mathcal{C}}^{1} \) function on \( {\mathbf{R}}^{n} \) with mean value zero that satisfies (6.1.3) and let \( {\Delta }_{j} \) be the Littlewood-Paley operator associated with \( \Psi \) . Then there exists a constant \( {C}_{n} < \infty \) such that for all \( 1 <... | Proof. We introduce Banach spaces \( {\mathcal{B}}_{1} = \mathbf{C} \) and \( {\mathcal{B}}_{2} = {\ell }^{2} \) and for \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) define an operator\n\n\[ \n\overrightarrow{T}\left( f\right) = {\left\{ {\Delta }_{k}\left( f\right) \right\} }_{k \in \mathbf{Z}}\n\]\n\nIn the proo... | Yes |
Theorem 6.1.5. There exists a constant \( {C}_{1} \) such that for all \( 1 < p < \infty \) and all \( f \) in \( {L}^{p}\left( \mathbf{R}\right) \) we have\n\n\[ \frac{\parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }}{{C}_{1}{\left( p + \frac{1}{p - 1}\right) }^{2}} \leq {\begin{Vmatrix}{\left( \mathop... | Proof. Pick a Schwartz function \( \psi \) on the line whose Fourier transform is supported in the set \( {2}^{-1} \leq \left| \xi \right| \leq {2}^{2} \) and is equal to 1 on the set \( 1 \leq \left| \xi \right| \leq 2 \) . Let \( {\Delta }_{j} \) be the Littlewood-Paley operator associated with \( \psi \) . Observe t... | Yes |
For a Schwartz function \( \psi \) on the line with integral zero we define the operator\n\n\[ \n{\Delta }_{\mathbf{j}}\left( f\right) \left( x\right) = {\left( \widehat{\psi }\left( {2}^{-{j}_{1}}{\xi }_{1}\right) \cdots \widehat{\psi }\left( {2}^{-{j}_{n}}{\xi }_{n}\right) \widehat{f}\left( \xi \right) \right) }^{ \v... | We first prove (6.1.28). Note that if \( \mathbf{j} = \left( {{j}_{1},\ldots ,{j}_{n}}\right) \in {\mathbf{Z}}^{n} \), then the operator \( {\Delta }_{\mathbf{j}} \) is equal to\n\n\[ \n{\Delta }_{\mathbf{j}}\left( f\right) = {\Delta }_{{j}_{1}}^{\left( {j}_{1}\right) }\cdots {\Delta }_{{j}_{n}}^{\left( {j}_{n}\right) ... | Yes |
Example 6.1.8. Pick a Schwartz function \( \zeta \) whose Fourier transform is positive and supported in the interval \( \left| \xi \right| \leq 1/4 \) . Let \( N \) be a large integer and let\n\n\[ \n{f}_{j}\left( x\right) = {e}^{2\pi ijx}\zeta \left( x\right) \n\]\n\nThen\n\n\[ \n{\widehat{f}}_{j}\left( \xi \right) =... | \[ \n{\begin{Vmatrix}\mathop{\sum }\limits_{{j = 0}}^{N}{f}_{j}\end{Vmatrix}}_{{L}^{p}}^{p} = {\int }_{\mathbf{R}}{\left| \frac{{e}^{{2\pi i}\left( {N + 1}\right) x} - 1}{{e}^{2\pi ix} - 1}\right| }^{p}{\left| \zeta \left( x\right) \right| }^{p}{dx} \n\]\n\n\[ \n\geq c{\int }_{\left| x\right| < \frac{1}{10}{\left( N + ... | Yes |
A similar idea illustrates the necessity of the \( {\ell }^{2} \) norm in (6.1.4). To see this, let \( \Psi \) and \( {\Delta }_{j} \) be as in Example 6.1.9. Let us fix \( 1 < p < \infty \) and \( q < 2 \) . We show that the inequality\n\n\[ \n{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j \in \mathbf{Z}}}{\left| {... | Take \( f = \mathop{\sum }\limits_{{j = 3}}^{N}{f}_{j} \), where the \( {f}_{j} \) are as in (6.1.34) and \( N \geq 3 \) . Then the left-hand side of (6.1.35) is bounded from below by \( \parallel \varphi {\parallel }_{{L}^{p}}{\left( N - 2\right) }^{1/q} \), while the right-hand side is bounded above by \( \parallel \... | Yes |
For \( 1 < p < \infty \) and \( 2 < q < \infty \), the inequality\n\n\[ \parallel g{\parallel }_{{L}^{p}} \leq {C}_{p, q}{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j \in \mathbf{Z}}}{\left| {\Delta }_{j}\left( g\right) \right| }^{q}\right) }^{\frac{1}{q}}\end{Vmatrix}}_{{L}^{p}} \]\n\ncannot hold even under assump... | Let \( {\Delta }_{j} \) be as in Example 6.1.9. Let us suppose that (6.1.36) did hold for some \( q > 2 \) for these \( {\Delta }_{j} \) ’s. Then the self-adjointness of the \( {\Delta }_{j} \) ’s and duality would give\n\n\[ {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{k \in \mathbf{Z}}}{\left| {\Delta }_{k}\left( ... | Yes |
Proposition 6.2.1. Let \( m \in {L}^{\infty }\left( {\mathbf{R}}^{n}\right) \) and let \( {m}_{\mathbf{j}} = m{\chi }_{{R}_{\mathbf{j}}} \) . Then \( m \) lies in \( {\mathcal{M}}_{p}\left( {\mathbf{R}}^{n}\right) \), that is, for some \( {c}_{p} \) we have\n\n\[ \n{\begin{Vmatrix}{\left( \widehat{f}m\right) }^{ \vee }... | Proof. Suppose that \( m \in {\mathcal{M}}_{p}\left( {\mathbf{R}}^{n}\right) \) . Exercise 5.6.1 gives the first inequality below\n\n\[ \n{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{\mathbf{j} \in {\mathbf{Z}}^{n}}}{\left| {\left( {\chi }_{{R}_{\mathbf{j}}}m{\widehat{f}}_{\mathbf{j}}\right) }^{ \vee }\right| }^{2}\... | No |
Theorem 6.2.2. (Marcinkiewicz multiplier theorem) Let \( m : \mathbf{R} \rightarrow \mathbf{R} \) be a bounded function that is \( {\mathcal{C}}^{1} \) in every dyadic set \( \left( {{2}^{j},{2}^{j + 1}}\right) \bigcup \left( {-{2}^{j + 1}, - {2}^{j}}\right) \) for \( j \in \mathbf{Z} \) . Assume that the derivative \(... | Proof. Since the function \( m \) has an integrable derivative on \( \left( {{2}^{j},{2}^{j + 1}}\right) \), it has bounded variation in this interval and hence it is a difference of two increasing functions. Therefore, \( m \) has left and right limits at the points \( {2}^{j} \) and \( {2}^{j + 1} \), and by redefini... | No |
Theorem 6.2.4. Let \( m \) be a bounded function on \( {\mathbf{R}}^{n} \) such that for all \( \alpha = \left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) \) with \( \left| {\alpha }_{1}\right| ,\ldots ,\left| {\alpha }_{n}\right| \leq 1 \) the derivatives \( {\partial }^{\alpha }m \) are continuous up to the boundar... | Proof. We prove this theorem only in dimension \( n = 2 \), since the general case presents no substantial differences but only some notational inconvenience. We decompose the given function \( m \) as\n\n\[ m\left( \xi \right) = {m}_{+ + }\left( \xi \right) + {m}_{- + }\left( \xi \right) + {m}_{+ - }\left( \xi \right)... | Yes |
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