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The species data is from Cook and Weisberg (1999a, pp. 285-286) and Johnson and Raven (1973). The response variable is the total number of species recorded on each of 29 islands in the Galápagos Archipelago. Predictors include area of island, areanear \( = \) the area of the closest island, the distance to the closest ... | A negative binomial regression suggested that only \( \log \left( \text{endem}\right) \) was needed in the model, and had a deviance of 26.12 on 27 degrees of freedom. The residual plot for this model was roughly ellipsoidal. The negative binomial GAM with \( \log \left( \text{endem}\right) \) had an \( \widehat{S} \) ... | Yes |
Lemma 1.4 (Riemann-Lebesgue). If \( f \in {L}^{1}\left( \mathbb{T}\right) \) then\n\n\[ \mathop{\lim }\limits_{{\left| k\right| \rightarrow \infty }}\widehat{f}\left( k\right) = 0 \] | Proof. Since \( {e}^{2\pi ix} \) has period 1,\n\n\[ \widehat{f}\left( k\right) = {\int }_{0}^{1}f\left( x\right) {e}^{-{2\pi ikx}}{dx} \]\n\n\[ = - {\int }_{0}^{1}f\left( x\right) {e}^{-{2\pi ik}\left( {x + 1/{2k}}\right) }{dx} \]\n\n\[ = - {\int }_{0}^{1}f\left( {x - 1/{2k}}\right) {e}^{-{2\pi ikx}}{dx}. \]\n\nHence,... | Yes |
Theorem 1.5. There exists a continuous function whose Fourier series diverges at a point. | Du Bois-Reymond constructed a function with this property, but we will show that one exists by applying the uniform boundedness principle, also known as the Banach-Steinhaus theorem. | No |
Lemma 1.6 (Uniform Boundedness Principle). Let \( X \) be a Banach space, \( Y \) a normed vector space, and let \( {\left\{ {T}_{a}\right\} }_{a \in A} \) be a family of bounded linear operators from \( X \) to \( Y \) . Then either \[ \mathop{\sup }\limits_{a}\begin{Vmatrix}{T}_{a}\end{Vmatrix} < \infty \] or there e... | A proof of this result can be found, for example, in Rudin [14, Chapter \( 5\rbrack \) . | No |
Lemma 1.7. \( {L}_{N} = \frac{4}{{\pi }^{2}}\log N + O\left( 1\right) \) . | Proof.\n\n\[ \n{L}_{N} = 2{\int }_{0}^{1/2}\left| \frac{\sin \left( {\pi \left( {{2N} + 1}\right) t}\right) }{\pi t}\right| {dt} + O\left( 1\right) \n\]\n\n\[ \n= 2{\int }_{0}^{N + 1/2}\left| \frac{\sin \left( {\pi t}\right) }{\pi t}\right| {dt} + O\left( 1\right) \n\]\n\n\[ \n= 2\mathop{\sum }\limits_{{k = 0}}^{{N - 1... | Yes |
Lemma 1.8. \( {S}_{N}f \) converges to \( f \) in \( {L}^{p} \) norm, \( 1 \leq p < \infty \), if and only if there exists \( {C}_{p} \) independent of \( N \) such that\n\n(1.7)\n\n\[ \n{\begin{Vmatrix}{S}_{N}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p} \n\] | Proof. The necessity of (1.7) follows from the uniform boundedness principle.\n\nTo see that it is sufficient, first note that if \( g \) is a trigonometric polynomial, then \( {S}_{N}g = g \) for \( N \geq \deg g \) . Therefore, since the trigonometric polynomials are dense in \( {L}^{p} \) (see Corollary 1.11), if \(... | Yes |
Theorem 1.9. The mapping \( f \mapsto \{ \widehat{f}\left( k\right) \} \) is an isometry from \( {L}^{2} \) to \( {\ell }^{2} \), that is,\n\n\[ \parallel f{\parallel }_{2}^{2} = \mathop{\sum }\limits_{{k = - \infty }}^{\infty }{\left| \widehat{f}\left( k\right) \right| }^{2} \] | Convergence in norm in \( {L}^{2} \) follows from this immediately. | No |
Theorem 1.10. If \( f \in {L}^{p},1 \leq p < \infty \), or if \( f \) is continuous and \( p = \infty \) , then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}{\begin{Vmatrix}{\sigma }_{N}f - f\end{Vmatrix}}_{p} = 0 \] | Proof. Since \( \int {F}_{N} = 1 \), by Minkowski’s inequality we have that\n\n\[ {\begin{Vmatrix}{\sigma }_{N}f - f\end{Vmatrix}}_{p} = {\int }_{-1/2}^{1/2}\parallel f\left( {\cdot - t}\right) - f\left( \cdot \right) {\parallel }_{p}{F}_{N}\left( t\right) {dt} \]\n\n\[ \leq {\int }_{\left| t\right| < \delta }\parallel... | Yes |
Theorem 1.13. The Fourier transform is a continuous map from \( \mathcal{S} \) to \( \mathcal{S} \) such that\n\n(1.21)\n\n\[{\int }_{{\mathbb{R}}^{n}}f\widehat{g} = {\int }_{{\mathbb{R}}^{n}}\widehat{f}g\]\n\nand\n\n(1.22)\n\n\[f\left( x\right) = {\int }_{{\mathbb{R}}^{n}}\widehat{f}\left( \xi \right) {e}^{{2\pi ix} \... | To prove Theorem 1.13 we need to compute the Fourier transform of a particular function. | No |
Lemma 1.14. If \( f\left( x\right) = {e}^{-\pi {\left| x\right| }^{2}} \) then \( \widehat{f}\left( \xi \right) = {e}^{-\pi {\left| \xi \right| }^{2}} \) . | Proof. We could prove this result directly by integrating in \( \mathbb{C} \), but we will give a different proof here. It is enough to prove this in one dimension, since in \( {\mathbb{R}}^{n}\widehat{f} \) is the product of \( n \) identical integrals.\n\nThe function \( f\left( x\right) = {e}^{-\pi {x}^{2}} \) is th... | Yes |
Theorem 1.17. The Fourier transform is a bounded linear bijection from \( {\mathcal{S}}^{\prime } \) to \( {\mathcal{S}}^{\prime } \) whose inverse is also bounded. | Proof. If \( {T}_{n} \rightarrow T \) in \( {\mathcal{S}}^{\prime } \), then for any \( f \in \mathcal{S} \) ,\n\n\[ \n{\widehat{T}}_{n}\left( f\right) = {T}_{n}\left( \widehat{f}\right) \rightarrow T\left( \widehat{f}\right) = \widehat{T}\left( f\right) .\n\]\n\nFurthermore, the Fourier transform has period 4, so its ... | Yes |
Theorem 1.18. The Fourier transform is an isometry on \( {L}^{2} \) ; that is, \( \widehat{f} \in \) \( {L}^{2} \) and \( \parallel \widehat{f}{\parallel }_{2} = \parallel f{\parallel }_{2} \) . Furthermore, | Proof. Given \( f, h \in \mathcal{S} \), let \( g = \overline{\widehat{h}} \), so that \( \widehat{g} = \bar{h} \) . Then by (1.21) we have that\n\n(1.23)\n\n\[{\int }_{{\mathbb{R}}^{n}}f\bar{h} = {\int }_{{\mathbb{R}}^{n}}\widehat{f}\overline{\widehat{h}}\]\n\nIf we let \( h = f \) then we get \( \parallel f{\parallel... | Yes |
Theorem 1.19 (Riesz-Thorin Interpolation). Let \( 1 \leq {p}_{0},{p}_{1},{q}_{0},{q}_{1} \leq \infty \) , and for \( 0 < \theta < 1 \) define \( p \) and \( q \) by\n\n\[ \n\frac{1}{p} = \frac{1 - \theta }{{p}_{0}} + \frac{\theta }{{p}_{1}},\;\frac{1}{q} = \frac{1 - \theta }{{q}_{0}} + \frac{\theta }{{q}_{1}}. \n\]\n\n... | The proof of this result uses the so-called \ | No |
Corollary 1.20 (Hausdorff-Young Inequality). If \( f \in {L}^{p},1 \leq p \leq 2 \), then \( \widehat{f} \in {L}^{{p}^{\prime }} \) and\n\n\[ \parallel \widehat{f}{\parallel }_{{p}^{\prime }} \leq \parallel f{\parallel }_{p} \] | Proof. Apply Theorem 1.19 using inequality (1.13), \( \parallel \widehat{f}{\parallel }_{\infty } \leq \parallel f{\parallel }_{1} \), and the Plancherel theorem, \( \parallel \widehat{f}{\parallel }_{2} = \parallel f{\parallel }_{2} \) . | No |
Corollary 1.21 (Young’s Inequality). If \( f \in {L}^{p} \) and \( g \in {L}^{q} \), then \( f * g \in \) \( {L}^{r} \), where \( 1/r + 1 = 1/p + 1/q \), and\n\n\[ \parallel f * g{\parallel }_{r} \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{q} \] | Proof. If we fix \( f \in {L}^{p} \) we immediately get the inequalities\n\n\[ \parallel f * g{\parallel }_{p} \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{1} \]\n\nand\n\n\[ \parallel f * g{\parallel }_{\infty } \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{{p}^{\prime }} \]\n\nThe desired resul... | No |
Theorem 2.1. Let \( \\left\\{ {{\\phi }_{t} : t > 0}\\right\\} \) be an approximation of the identity. Then\n\n\[ \n\\mathop{\\lim }\\limits_{{t \\rightarrow 0}}{\\begin{Vmatrix}{\\phi }_{t} * f - f\\end{Vmatrix}}_{p} = 0 \n\]\n\nif \( f \\in {L}^{p},1 \\leq p < \\infty \), and uniformly (i.e. when \( p = \\infty \) ) ... | Proof. Because \( \\phi \) has integral 1,\n\n\[ \n{\\phi }_{t} * f\\left( x\\right) - f\\left( x\\right) = {\\int }_{{\\mathbb{R}}^{n}}\\phi \\left( y\\right) \\left\\lbrack {f\\left( {x - {ty}}\\right) - f\\left( x\\right) }\\right\\rbrack {dy}. \n\]\n\nGiven \( \\epsilon > 0 \), choose \( \\delta > 0 \) such that if... | Yes |
Theorem 2.2. Let \( \left\{ {T}_{t}\right\} \) be a family of linear operators on \( {L}^{p}\left( {X,\mu }\right) \) and define\n\n\[ \n{T}^{ * }f\left( x\right) = \mathop{\sup }\limits_{t}\left| {{T}_{t}f\left( x\right) }\right| \n\]\n\nIf \( {T}^{ * } \) is weak \( \left( {p, q}\right) \) then the set\n\n\[ \n\left\... | Proof. Let \( \left\{ {f}_{n}\right\} \) be a sequence of functions which converges to \( f \) in \( {L}^{p}\left( {X,\mu }\right) \) norm and such that \( {T}_{t}{f}_{n}\left( x\right) \) converges to \( {f}_{n}\left( x\right) \) almost everywhere. Then\n\n\[ \n\mu \left( \left\{ {x \in X : \mathop{\limsup }\limits_{{... | Yes |
Proposition 2.3. Let \( \phi : \lbrack 0,\infty ) \rightarrow \lbrack 0,\infty ) \) be differentiable, increasing and such that \( \phi \left( 0\right) = 0 \) . Then \[ {\int }_{X}\phi \left( \left| {f\left( x\right) }\right| \right) {d\mu } = {\int }_{0}^{\infty }{\phi }^{\prime }\left( \lambda \right) {a}_{f}\left( \... | To prove this it is enough to observe that the left-hand side is equivalent to \[ {\int }_{X}{\int }_{0}^{\left| f\left( x\right) \right| }{\phi }^{\prime }\left( \lambda \right) {d\lambda d\mu } \] and then change the order of integration. | Yes |
Theorem 2.4 (Marcinkiewicz Interpolation). Let \( \left( {X,\mu }\right) \) and \( \left( {Y,\nu }\right) \) be measure spaces, \( 1 \leq {p}_{0} < {p}_{1} \leq \infty \), and let \( T \) be a sublinear operator from \( {L}^{{p}_{0}}\left( {X,\mu }\right) + {L}^{{p}_{1}}\left( {X,\mu }\right) \) to the measurable funct... | Proof. Given \( f \in {L}^{p} \), for each \( \lambda > 0 \) decompose \( f \) as \( {f}_{0} + {f}_{1} \), where\n\n\[ \n{f}_{0} = f{\chi }_{\{ x : \left| {f\left( x\right) }\right| > {c\lambda }\} }\n\]\n\n\[ \n{f}_{1} = f{\chi }_{\{ x : \left| {f\left( x\right) }\right| \leq {c\lambda }\} }\n\]\n\nthe constant \( c \... | Yes |
Theorem 2.5. The operator \( M \) is weak \( \left( {1,1}\right) \) and strong \( \left( {p, p}\right) ,1 < p \leq \infty \) . | It is immediate from the definition that\n\n(2.7)\n\n\[ \parallel {Mf}{\parallel }_{\infty } \leq \parallel f{\parallel }_{\infty } \]\n\nso by the Marcinkiewicz interpolation theorem, to prove Theorem 2.5 it will be enough to prove that \( M \) is weak \( \left( {1,1}\right) \) . Here we will prove this when \( n = 1 ... | No |
Lemma 2.6. Let \( {\left\{ {I}_{\alpha }\right\} }_{\alpha \in A} \) be a collection of intervals in \( \mathbb{R} \) and let \( K \) be a compact set contained in their union. Then there exists a finite subcollection \( \left\{ {I}_{j}\right\} \) such that\n\n\[ K \subset \mathop{\bigcup }\limits_{j}{I}_{j},\;\text{ a... | Proof of Theorem 2.5 for \( n = 1 \) . Let \( {E}_{\lambda } = \{ x \in \mathbb{R} : {Mf}\left( x\right) > \lambda \} \) . If \( x \in {E}_{\lambda } \) then there exists an interval \( {I}_{x} \) centered at \( x \) such that\n\n(2.8)\n\n\[ \frac{1}{\left| {I}_{x}\right| }{\int }_{{I}_{x}}\left| f\right| > \lambda \]\... | Yes |
Proposition 2.7. Let \( \\phi \) be a function which is positive, radial, decreasing (as a function on \( \\left( {0,\\infty }\\right) \) ) and integrable. Then\n\n\[ \n\\mathop{\\sup }\\limits_{{t > 0}}\\left| {{\\phi }_{t} * f\\left( x\\right) }\\right| \\leq \\parallel \\phi {\\parallel }_{1}{Mf}\\left( x\\right) \n... | Proof. If we assume in addition to the given hypotheses that \( \\phi \) is a simple function, that is, it can be written as\n\n\[ \n\\phi \\left( x\\right) = \\mathop{\\sum }\\limits_{j}{a}_{j}{\\chi }_{{B}_{{r}_{j}}}\\left( x\\right) \n\]\n\nwith \( {a}_{j} > 0 \), then\n\n\[ \n\\phi * f\\left( x\\right) = \\mathop{\... | Yes |
Corollary 2.8. If \( \left| {\phi \left( x\right) }\right| \leq \psi \left( x\right) \) almost everywhere, where \( \psi \) is positive, radial, decreasing and integrable, then the maximal function \( \mathop{\sup }\limits_{t}\left| {{\phi }_{t} * f\left( x\right) }\right| \) is weak \( \left( {1,1}\right) \) and stron... | This is an immediate consequence of Proposition 2.7 and Theorem 2.5. | No |
Under the hypotheses of the previous corollary, if \( f \in {L}^{p} \) , \( 1 \leq p < \infty \), or if \( f \in {C}_{0} \), then\n\n\[ \mathop{\lim }\limits_{{t \rightarrow 0}}{\phi }_{t} * f\left( x\right) = \left( {\int \phi }\right) \cdot f\left( x\right) \text{ a.e. } \] | Proof. Since we have convergence for \( f \in \mathcal{S} \), by Theorem 2.2 we have convergence for \( f \in \overline{\mathcal{S}} = {L}^{p} \) (or \( f \in {C}_{0} \) if \( p = \infty \) ). The Poisson kernel (1.30) and the Gauss-Weierstrass kernel (1.31) are decreasing; the Féjer kernel (1.24) is not but \( {F}_{1}... | No |
The dyadic maximal function is weak \( \left( {1,1}\right) \) . | Fix \( f \in {L}^{1} \) ; we may assume that \( f \) is non-negative: if \( f \) is real, it can be decomposed into its positive and negative parts, and if it is complex, into its real and imaginary parts.\n\nNow let\n\n\[ \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > \lambda }\right\} = \mathop{\bigcup }... | Yes |
Theorem 2.11. Given a function \( f \) which is integrable and non-negative, and given a positive number \( \lambda \), there exists a sequence \( \left\{ {Q}_{j}\right\} \) of disjoint dyadic cubes such that\n\n(1) \( \;f\left( x\right) \leq \lambda \) for almost every \( x \notin \mathop{\bigcup }\limits_{j}{Q}_{j} \... | Proof. As in the proof of Theorem 2.10, form the sets \( {\Omega }_{k} \) and decompose each into disjoint dyadic cubes contained in \( {\mathcal{Q}}_{k} \) ; together, all of these cubes form the family \( \left\{ {Q}_{j}\right\} \) .\n\nPart (2) of the theorem is then just the weak \( \left( {1,1}\right) \) inequalit... | Yes |
Lemma 2.12. If \( f \) is a non-negative function, then\n\n\[ \left| \left\{ {x \in {\mathbb{R}}^{n} : {M}^{\prime }f\left( x\right) > {4}^{n}\lambda }\right\} \right| \leq {2}^{n}\left| \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > \lambda }\right\} \right| .\n\] | Proof of Lemma 2.12. As before, we form the decomposition\n\n\[ \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > \lambda }\right\} = \mathop{\bigcup }\limits_{j}{Q}_{j}.\n\]\n\nLet \( 2{Q}_{j} \) be the cube with the same center as \( {Q}_{j} \) and whose sides are twice as long. To complete the proof it wil... | Yes |
Corollary 2.13 (Lebesgue Differentiation Theorem). If \( f \in {L}_{\mathrm{{loc}}}^{1}\left( {\mathbb{R}}^{n}\right) \) then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow {0}^{ + }}}\frac{1}{\left| {B}_{r}\right| }{\int }_{{B}_{r}}f\left( {x - y}\right) {dy} = f\left( x\right) \text{ a.e. } \] | From this we see that \( \left| {f\left( x\right) }\right| \leq {Mf}\left( x\right) \) almost everywhere. The same is true if we replace \( M \) by \( {M}^{\prime } \) or \( {M}^{\prime \prime } \) . | No |
Proposition 2.14. If \( f \in {L}^{1} \) and is not identically 0, then \( {Mf} \notin {L}^{1} \) . | The proof is simple: since \( f \) is not identically 0, there exists \( R > 0 \) such that\n\n\[{\int }_{{B}_{R}}\left| f\right| \geq \epsilon > 0\]\n\nNow if \( \left| x\right| > R,{B}_{R} \subset B\left( {x,2\left| x\right| }\right) \), so\n\n\[{Mf}\left( x\right) \geq \frac{1}{{\left( 2\left| x\right| \right) }^{n}... | Yes |
Theorem 2.15. If \( B \) is a bounded subset of \( {\mathbb{R}}^{n} \), then\n\n\[{\int }_{B}{Mf} \leq 2\left| B\right| + C{\int }_{{\mathbb{R}}^{n}}\left| f\right| {\log }^{ + }\left| f\right|\]\n\nwhere \( {\log }^{ + }t = \max \left( {\log t,0}\right) \). | Proof.\n\n\[{\int }_{B}{Mf} \leq 2{\int }_{0}^{\infty }\left| {\{ x \in B : {Mf}\left( x\right) > {2\lambda }\} }\right| {d\lambda }\]\n\n\[ \leq 2\left| B\right| + 2{\int }_{1}^{\infty }\left| {\{ x \in B : {Mf}\left( x\right) > {2\lambda }\} }\right| {d\lambda }\]\n\nDecompose \( f \) as \( {f}_{1} + {f}_{2} \), wher... | Yes |
Theorem 2.16. If \( w \) is a non-negative, measurable function and \( 1 < p < \) \( \infty \), then there exists a constant \( {C}_{p} \) such that\n\n\[{\int }_{{\mathbb{R}}^{n}}{Mf}{\left( x\right) }^{p}w\left( x\right) {dx} \leq {C}_{p}{\int }_{{\mathbb{R}}^{n}}{\left| f\left( x\right) \right| }^{p}{Mw}\left( x\rig... | Proof. It will suffice to show that \( \parallel {Mf}{\parallel }_{{L}^{\infty }\left( w\right) } \leq \parallel f{\parallel }_{{L}^{\infty }\left( {Mw}\right) } \) and that the weak \( \left( {1,1}\right) \) inequality holds; the strong \( \left( {p, p}\right) \) inequality then follows from the Marcinkiewicz interpol... | Yes |
Theorem 2.19. Let \( {\left\{ {B}_{j}\right\} }_{j \in \mathcal{J}} \) be a collection of balls in \( {\mathbb{R}}^{n} \) . Then there exists an at most countable subcollection of disjoint balls \( \left\{ {B}_{k}\right\} \) such that\n\n\[ \mathop{\bigcup }\limits_{{j \in \mathcal{J}}}{B}_{j} \subset \mathop{\bigcup }... | The second is due independently to A. Besicovitch and A. P. Morse; for a proof and further references, see the book by M. de Guzmán (Differentiation of Integrals in \( {\mathbb{R}}^{n} \), Lecture Notes in Math. 481, Springer-Verlag, Berlin, 1985). | No |
Proposition 3.1. In \( {\mathcal{S}}^{\prime },\mathop{\lim }\limits_{{t \rightarrow 0}}{Q}_{t} = \frac{1}{\pi } \) p. v. \( \frac{1}{x} \) . | Proof. For each \( \epsilon > 0 \), the functions \( {\psi }_{\epsilon }\left( x\right) = {x}^{-1}{\chi }_{\{ \left| x\right| > \epsilon \} } \) are bounded and define tempered distributions. It follows at once from the definition that in \( {\mathcal{S}}^{\prime } \) ,\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarro... | Yes |
Theorem 3.3. Given \( f \in {L}^{p},1 \leq p < \infty \), then\n\n(3.7)\n\n\[ \n{Hf}\left( x\right) = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{H}_{\epsilon }f\left( x\right) \;\text{ a.e. }x \in \mathbb{R}.\n\] | Since we know that (3.7) holds for some subsequence \( \left\{ {{H}_{{\epsilon }_{k}}f}\right\} \), we only need to show that \( \lim {H}_{\epsilon }f\left( x\right) \) exists for almost every \( x \) . By Theorem 2.2 (and the remarks following it) it will suffice to show that the maximal operator\n\n\[ \n{H}^{ * }f\le... | Yes |
Theorem 3.4. \( {H}^{ * } \) is strong \( \left( {p, p}\right) ,1 < p < \infty \), and weak \( \left( {1,1}\right) \) . | To prove this we need a lemma which is referred to as Cotlar's inequality.\n\nLemma 3.5. If \( f \in | No |
Lemma 3.5. If \( f \in \mathcal{S} \) then \( {H}^{ * }f\left( x\right) \leq M\left( {Hf}\right) \left( x\right) + {CMf}\left( x\right) \) . | Proof. It will suffice to prove this inequality for each \( {H}_{\epsilon } \) with a constant independent of \( \epsilon \) .\n\nFix a function \( \phi \in \mathcal{S}\left( \mathbb{R}\right) \) which is non-negative, even, decreasing on \( \left( {0,\infty }\right) \), supported on \( \{ x \in \mathbb{R} : \left| x\r... | Yes |
Proposition 3.6. There exists a constant \( {C}_{p},1 < p < \infty \), such that for all \( a \) and \( b, - \infty \leq a < b \leq \infty \), \[ {\begin{Vmatrix}{S}_{a, b}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p} \] | For an application of this result, let \( a = - R, b = R \) . Then \( {S}_{a, b} \) is the partial sum operator \( {S}_{R} \) introduced in Chapter 1: \( {S}_{R}f = {D}_{R} * f \), where \( {D}_{R} \) is the Dirichlet kernel. Hence, \[ {\begin{Vmatrix}{S}_{R}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p} \... | No |
Corollary 3.8. If \( m \) is a function of bounded variation on \( \mathbb{R} \), then \( m \) is a multiplier on \( {L}^{p},1 < p < \infty \) . | Proof. Since \( m \) is of bounded variation, the limit of \( m\left( t\right) \) as \( t \rightarrow - \infty \) exists, so by adding a constant to \( m \) if necessary we may assume that this limit equals 0 . Furthermore, we may assume \( m \) is normalized so that it is right continuous at each \( x \in \mathbb{R} \... | Yes |
Proposition 3.9. If \( m \) is a multiplier on \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \), then the functions defined by \( m\left( {\xi + a}\right), a \in {\mathbb{R}}^{n}, m\left( {\lambda \xi }\right) ,\lambda > 0 \), and \( m\left( {\rho \xi }\right) ,\rho \in O\left( n\right) \) (orthogonal transformations), are ... | The proof of this result follows at once from properties (1.16), (1.17) and (1.18) of the Fourier transform. | No |
Theorem 3.11. Let \( B \) and \( C \) be two open balls such that \( \bar{B} \subset C \) .\n\n(1) If \( f{\log }^{ + }\left| f\right| \in {L}^{1}\left( C\right) \), then \( {Hf} \in {L}^{1}\left( B\right) \) .\n\n(2) Conversely, if \( f \geq 0 \) and \( {Hf} \in {L}^{1}\left( C\right) \), then \( f{\log }^{ + }\left| ... | The first part of the theorem is due to A. P. Calderón and A. Zygmund (see the paper cited above). It can be sharpened using Orlicz spaces (cf. Chapter 2, Section 8.4, and Zygmund [21, Chapter 4]). The second part is due to E. M. Stein (Note on the class \( L\log L \), Studia Math. 32 (1969),305- 310). The hypothesis \... | No |
Theorem 3.12. Given a linear operator \( T \) which is translation invariant and bounded from \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \) to \( {L}^{q}\left( {\mathbb{R}}^{n}\right) \) for any pair \( \left( {p, q}\right) ,1 \leq p, q \leq \infty \), then there exists a unique tempered distribution \( K \) such that \(... | If such a \( T \) is bounded from \( {L}^{p} \) to \( {L}^{q} \) and not zero, then \( p \) cannot be greater than \( q \) . An elegant and very simple proof of this fact is due to L. Hörmander (Estimates for translation invariant operators in \( {L}^{p} \) -spaces, Acta Math. 104 (1960),93-139). First note that \( \ma... | Yes |
Theorem 3.14. Given \( 1 \leq p \leq \infty \), suppose the operator \( T \) defined by \( {Tf} = K * f \) is bounded on \( {L}^{p}\left( \mathbb{R}\right) \) . If \( \widehat{K} \) is continuous at each point of \( \mathbb{Z} \) and \( \lambda \left( k\right) = \widehat{K}\left( k\right) \), then the operator \( {T}_{... | For a proof see Stein and Weiss [18, Chapter 6]. | No |
Proposition 4.1. A necessary condition for the limit in (4.1) or (4.2) to exist is that \( \Omega \) have zero average on \( {S}^{n - 1} \) . | Proof. Let \( f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) be such that \( f\left( x\right) = 1 \) for any \( \left| x\right| \leq 2 \) . Then for \( \left| x\right| < 1 \)\n\n\[ \n{Tf}\left( x\right) = {\int }_{\left| y\right| > 1}\frac{\Omega \left( {y}^{\prime }\right) }{{\left| y\right| }^{n}}f\left( {x - y}\... | Yes |
Proposition 4.3. If \( T \) is a tempered distribution which is homogeneous of degree \( a \), then its Fourier transform is homogeneous of degree \( - n - a \) . | Proof. By Definition 1.16 and property (1.18), if \( \phi \in \mathcal{S} \) then\n\n\[ \widehat{T}\left( {\phi }_{\lambda }\right) = T\left( {\widehat{\phi }\left( {\lambda \cdot }\right) }\right) = {\lambda }^{-n}T\left( {\widehat{\phi }}_{{\lambda }^{-1}}\right) = {\lambda }^{-n - a}T\left( \widehat{\phi }\right) = ... | Yes |
Theorem 4.4. If \( \Omega \) is an integrable function on \( {S}^{n - 1} \) with zero average, then the Fourier transform of p. v. \( \Omega \left( {x}^{\prime }\right) /{\left| x\right| }^{n} \) is a homogeneous function of degree 0 given by\n\n\[ m\left( \xi \right) = {\int }_{{S}^{n - 1}}\Omega \left( u\right) \left... | Proof. By Proposition 4.3 the Fourier transform is homogeneous of degree 0 ; therefore, we may now assume that \( \left| \xi \right| = 1 \) . Since \( \Omega \) has zero average,\n\n\[ m\left( \xi \right) = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{\int }_{\epsilon < \left| y\right| < 1/\epsilon }\frac{\Omega \l... | Yes |
Corollary 4.5. Given a function \( \Omega \) with zero average on \( {S}^{n - 1} \), suppose that \( {\Omega }_{o} \in {L}^{1}\left( {S}^{n - 1}\right) \) and \( {\Omega }_{e} \in {L}^{q}\left( {S}^{n - 1}\right) \) for some \( q > 1 \) . Then the Fourier transform of p. v. \( \Omega \left( {x}^{\prime }\right) /{\left... | From (4.4) one can easily find an integrable function \( \Omega \) such that \( m \) is not bounded. Nevertheless, in Corollary 4.5 we can substitute the weaker hypothesis \( {\Omega }_{e} \in L\log L\left( {S}^{n - 1}\right) \), that is,\n\n\[ \n{\int }_{{S}^{n - 1}}\left| {{\Omega }_{e}\left( u\right) }\right| {\log ... | Yes |
Corollary 4.7. If \( \Omega \in {L}^{1}\left( {S}^{n - 1}\right) \) then \( {M}_{\Omega } \) is bounded on \( {L}^{p}\left( {\mathbb{R}}^{n}\right) ,1 < p \leq \) \( \infty \) . | If we let \( \Omega \left( u\right) = 1 \) for all \( u \), then \( {M}_{\Omega } \) becomes the Hardy-Littlewood maximal function in \( {\mathbb{R}}^{n} \) ; thus the method of rotations shows it is bounded on \( {L}^{p}\left( {\mathbb{R}}^{n}\right), p > 1 \), starting from the one-dimensional case. Further, it is in... | No |
Corollary 4.9. With the same hypotheses as before, the operator \( {T}^{ * } \) defined by (4.6) is strong \( \left( {p, p}\right) ,1 < p < \infty \) . In particular, given \( f \in {L}^{p} \), the limit (4.1) holds for almost every \( x \in {\mathbb{R}}^{n} \) . | To prove (4.8) one could use Theorem 4.4 directly; instead we will argue as follows. With equality in the sense of distributions, \[ \frac{\partial }{\partial {x}_{j}}{\left| x\right| }^{-n + 1} = \left( {1 - n}\right) \text{ p. v. }\frac{{x}_{j}}{{\left| x\right| }^{n + 1}} \] so if we apply (4.3) we get \[ {\left( \t... | Yes |
Lemma 4.11. The kernel \( {\widetilde{K}}_{j} \) defined in Lemma 4.10 satisfies\n\n\[ \n{\int }_{{S}^{n - 1}}\left| {{\widetilde{K}}_{j}\left( u\right) }\right| {d\sigma }\left( u\right) \leq {C}_{q}\parallel \Omega {\parallel }_{q} \n\]\n\nFurthermore, if \( {\widetilde{K}}_{j,\epsilon }\left( x\right) = {\widetilde{... | Proof. By the homogeneity of \( {\widetilde{K}}_{j} \) ,\n\n\[ \n{\int }_{{S}^{n - 1}}\left| {{\widetilde{K}}_{j}\left( u\right) }\right| {d\sigma }\left( u\right) = \frac{1}{\log 2}{\int }_{1 < \left| x\right| < 2}\left| {{\widetilde{K}}_{j}\left( x\right) }\right| {dx} \n\]\n\n\[ \n\leq \frac{1}{\log 2}{\int }_{1 < \... | Yes |
Theorem 4.12. Let \( \Omega \) be a function on \( {S}^{n - 1} \) with zero average such that its odd part is in \( {L}^{1}\left( {S}^{n - 1}\right) \) and its even part is in \( {L}^{q}\left( {S}^{n - 1}\right) \) for some \( q > 1 \) . Then the singular integral \( T \) in (4.1) is bounded on \( {L}^{p}\left( {\mathb... | Proof. By Corollary 4.8 we may assume \( \Omega \) is even. Further, by arguing as we did for the Hilbert transform, it will suffice to establish the \( {L}^{p} \) inequality for functions \( f \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . For such \( f,{Tf} = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}... | Yes |
Theorem 4.13. If \( m \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) is a homogeneous function of degree 0, and \( {T}_{m} \) is the operator defined by \( {\left( {T}_{m}f\right) }^{ \frown } = m\widehat{f} \), then there exist \( a \in \mathbb{C} \) and \( \Omega \in {C}^{\infty }\left( {S... | Since any homogeneous function of degree 0 is the sum of a constant and a homogeneous function of degree 0 with zero average on \( {S}^{n - 1} \), Theorem 4.13 is an immediate consequence of the following lemma. | No |
Theorem 4.15. The set \( \mathcal{A} \) of operators defined by Theorem 4.13 is a commutative algebra. An element of \( \mathcal{A} \) is invertible if and only if \( m \) is never zero on \( {S}^{n - 1} \) . | Proof. To see that \( \mathcal{A} \) is an algebra, it is enough to note that given \( {m}_{1} \) and \( {m}_{2} \) with associated operators \( {T}_{{m}_{1}} \) and \( {T}_{{m}_{2}},{T}_{{m}_{1}} \circ {T}_{{m}_{2}} = {T}_{{m}_{1}{m}_{2}} \). Since the identity has the function 1 as its multiplier, \( {T}_{m} \) is in... | Yes |
Theorem 4.16. Let \( \Omega \left( {x, y}\right) \) be a function which is homogeneous of degree 0 in \( y \) and such that:\n\n(1) \( \Omega \left( {x, - y}\right) = - \Omega \left( {x, y}\right) \) ;\n\n(2) \( {\Omega }^{ * }\left( u\right) = \mathop{\sup }\limits_{x}\left| {\Omega \left( {x, u}\right) }\right| \in {... | Proof. Because \( \Omega \) is odd, we can argue as in the proof of Corollary 4.8 (where the kernel does not depend on \( x \) ) to get\n\n(4.21)\n\n\[ \n{Tf}\left( x\right) = \frac{\pi }{2}{\int }_{{S}^{n - 1}}\Omega \left( {x, u}\right) {H}_{u}f\left( x\right) {d\sigma }\left( u\right) ,\;f \in \mathcal{S}.\n\]\n\nTh... | Yes |
Theorem 4.17. If in the statement of Theorem 4.16 we replace (2) by\n\n(4.22)\n\n\[ \mathop{\sup }\limits_{x}{\left( {\int }_{{S}^{n - 1}}{\left| \Omega \left( x, u\right) \right| }^{q}d\sigma \left( u\right) \right) }^{q} = {B}_{q} < \infty \]\n\nfor some \( q,1 < q < \infty \), then \( T \) is bounded on \( {L}^{p}\l... | Proof. Apply Hölder’s inequality with exponents \( q \) and \( {q}^{\prime } \) to get\n\n(4.23)\n\n\[ \left| {{Tf}\left( x\right) }\right| \leq \frac{\pi }{2}{B}_{q}{\left( {\int }_{{S}^{n - 1}}{\left| {H}_{u}f\left( x\right) \right| }^{{q}^{\prime }}d\sigma \left( u\right) \right) }^{1/{q}^{\prime }}.\n\]\n\nIf we ra... | Yes |
Theorem 4.18. Let \( 0 < a < n,1 \leq p < n/a \), and define \( q \) by (4.26). Then (4.25) holds. If \( p = 1 \) then \( {I}_{a} \) satisfies the weak \( \left( {1, n/\left( {n - a}\right) }\right) \) inequality | For a proof, see Stein [15, Chapter 5]. Another proof uses the following inequality due to L. Hedberg (On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510):\n\n(4.27)\n\n\[ {I}_{a}f\left( x\right) \leq {C}_{a}\parallel f{\parallel }_{p}^{{ap}/n}{Mf}{\left( x\right) }^{1 - {ap}/n},\;1 \leq p <... | Yes |
Theorem 5.1 (Calderón-Zygmund). Let \( K \) be a tempered distribution in \( {\mathbb{R}}^{n} \) which coincides with a locally integrable function on \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) and is such that\n\n(5.1)\n\n\[ \left| {\widehat{K}\left( \xi \right) }\right| \leq A \]\n\n(5.2)\n\n\[ {\int }_{\left| x\ri... | We will show that these inequalities are true for \( f \in \mathcal{S} \), but they can be extended to arbitrary \( f \in {L}^{p} \) as we did for the Hilbert transform. Condition (5.2) is usually referred to as the Hörmander condition; in practice it is often deduced from another, stronger, condition called the gradie... | Yes |
Proposition 5.2. The Hörmander condition (5.2) holds if for every \( x \neq 0 \)\n\n(5.3)\n\n\[ \left| {\nabla K\left( x\right) }\right| \leq \frac{C}{{\left| x\right| }^{n + 1}} \] | This follows from the mean value theorem; details are left to the reader. | No |
Proposition 5.3. If \( \Omega \) satisfies\n\n(5.5)\n\n\[ \n{\int }_{0}^{1}\frac{{\omega }_{\infty }\left( t\right) }{t}{dt} < \infty \n\]\n\nthen the kernel \( \Omega \left( {x}^{\prime }\right) /{\left| x\right| }^{n} \) satisfies (5.2).\n\nCondition (5.5) is referred to as a Dini-type condition. | Proof.\n\n\[ \n\left| {K\left( {x - y}\right) - K\left( x\right) }\right| = \left| {\frac{\Omega \left( {\left( x - y\right) }^{\prime }\right) }{{\left| x - y\right| }^{n}} - \frac{\Omega \left( {x}^{\prime }\right) }{{\left| x\right| }^{n}}}\right| \n\]\n\n\[ \n\leq \frac{\left| \Omega \left( {\left( x - y\right) }^{... | Yes |
Corollary 5.4. If \( \Omega \) is a function defined on \( {S}^{n - 1} \) with zero integral and satisfying (5.5), then the operator\n\n\[ \n{Tf}\left( x\right) = \text{ p. v. }{\int }_{{\mathbb{R}}^{n}}\frac{\Omega \left( {y}^{\prime }\right) }{{\left| y\right| }^{n}}f\left( {x - y}\right) {dy} \]\n\n is strong \( \le... | Since (5.5) implies that \( \Omega \) is bounded, we already had the strong \( \left( {p, p}\right) \) inequality via the method of rotations. Nevertheless, we now also have the weak \( \left( {1,1}\right) \) inequality. | Yes |
Proposition 5.5. Let \( K \in {L}_{\text{loc }}^{1}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) be such that\n\n(5.6)\n\n\[ \left| {{\int }_{a < \left| x\right| < b}K\left( x\right) {dx}}\right| \leq A,\;0 < a < b < \infty \]\n\n(5.7)\n\n\[ {\int }_{a < \left| x\right| < {2a}}\left| {K\left( x\right) }\rig... | Proof. If we fix \( \xi \) then whenever \( \epsilon < {\left| \xi \right| }^{-1} < R \) ,\n\n\[ \widehat{{K}_{\epsilon, R}}\left( \xi \right) = {\int }_{\epsilon < \left| x\right| < R}K\left( x\right) {e}^{-{2\pi ix} \cdot \xi }{dx} \]\n\n\[ = {\int }_{\epsilon < \left| x\right| < {\left| \xi \right| }^{-1}}K\left( x\... | Yes |
Corollary 5.6. If \( K \) satisfies the hypotheses of Proposition 5.5, then\n\n\[ \n{\begin{Vmatrix}{K}_{\epsilon, R} * f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p},\;1 < p < \infty ,\n\]\n\nand\n\n\[ \n\left| \left\{ {x \in {\mathbb{R}}^{n} : \left| {{K}_{\epsilon, R} * f\left( x\right) }\right| > \lamb... | When \( p = 2 \) this result follows immediately from Proposition 5.5; for the rest note that (5.8) implies the Hörmander condition,\n\n\[ \n{\int }_{\left| x\right| > 2\left| y\right| }\left| {{K}_{\epsilon, R}\left( {x - y}\right) - {K}_{\epsilon, R}\left( x\right) }\right| {dx} \leq C\n\]\n\nwith \( C \) independent... | No |
Given a function \( K \) which satisfies condition (5.7), the tempered distribution \( \mathrm{p}.\mathrm{v}.K \) exists if and only if\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{\int }_{\epsilon < \left| x\right| < 1}K\left( x\right) {dx} \]\n\nexists. | Proof. Suppose that the tempered distribution exists. If we fix \( \phi \in \mathcal{S} \) which is identically 1 on \( B\left( {0,1}\right) \), then\n\n\[ \text{ p. v. }K\left( \phi \right) = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{\int }_{\epsilon < \left| x\right| < 1}K\left( x\right) {dx} + {\int }_{\left|... | Yes |
Example 5.9 (An application to \( K\left( x\right) = {\left| x\right| }^{-n - {it}} \) ). The function \( {\left| x\right| }^{-n - {it}} \) is locally integrable on \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) and satisfies the hypotheses of Proposition 5.5. | In fact,\n\n\[ \left| {{\int }_{a < \left| x\right| < b}\frac{dx}{{\left| x\right| }^{n + {it}}}}\right| = \left| {\left| {S}^{n - 1}\right| \frac{{b}^{it} - {a}^{-{it}}}{-{it}}}\right| \leq \frac{2}{t}\left| {S}^{n - 1}\right| ,\] \n\n\[ {\int }_{a < \left| x\right| < {2a}}\frac{dx}{{\left| x\right| }^{n}} = \left| {S... | Yes |
Theorem 5.10. Let \( T \) be a bounded operator on \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \), and let \( K \) be a function on \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \smallsetminus \Delta \) such that if \( f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) has compact support then\n\n\[ \n{Tf}\left( x\right) = {\in... | The given representation holds only for functions in \( {L}^{2} \) of compact support, but this suffices for the proof since it is only applied to the functions \( {b}_{j} \) gotten from the Calderón-Zygmund decomposition. condition (5.10) is used to prove the weak \( \left( {1,1}\right) \) inequality for \( T \), and ... | Yes |
Lemma 5.15. If \( T \) is a Calderón-Zygmund operator then for any \( \nu ,0 < \) \( \nu \leq 1 \), and for any \( f \in {C}_{c}^{\infty } \), (5.18) \[ {T}^{ * }f\left( x\right) \leq {C}_{\nu }\left( {M\left( {\left| Tf\right| }^{\nu }\right) {\left( x\right) }^{1/\nu } + {Mf}\left( x\right) }\right) . \] | To prove this we need the following result due to Kolmogorov. Lemma 5.16. Given an op | No |
Lemma 5.16. Given an operator \( S \) which is weak \( \left( {1,1}\right) ,\nu ,0 < \nu < 1 \), and a set \( E \) of finite measure, there exists a constant \( C \) depending only on \( \nu \) such that\n\n\[{\int }_{E}{\left| Sf\left( x\right) \right| }^{\nu }{dx} \leq C{\left| E\right| }^{1 - \nu }\parallel f{\paral... | Proof. By (2.1) and the weak \( \left( {1,1}\right) \) inequality,\n\n\[{\int }_{E}{\left| Sf\left( x\right) \right| }^{\nu }{dx} = \nu {\int }_{0}^{\infty }{\lambda }^{\nu - 1}\left| {\{ x \in E : }\right| {Sf}\left( x\right) \left| { > \lambda \} }\right| {d\lambda }\n\n\leq \nu {\int }_{0}^{\infty }{\lambda }^{\nu -... | Yes |
Theorem 5.17. Let \( T \) be a bounded operator from \( {L}^{r}\left( A\right) \) to \( {L}^{r}\left( B\right) \) for some \( r,1 < r < \infty \), with associated kernel \( K \) . If \( K \) satisfies\n\n(5.20)\n\n\[ \n{\int }_{\left| {x - y}\right| > 2\left| {y - z}\right| }\parallel K\left( {x, y}\right) - K\left( {x... | The proof of Theorem 5.17 initially follows the outline of the scalar result: inequality (5.20) together with the boundedness from \( {L}^{r}\left( A\right) \) to \( {L}^{r}\left( B\right) \) yields the weak \( \left( {1,1}\right) \) inequality; then the Marcinkiewicz interpolation theorem (which can easily be shown to... | No |
Proposition 6.2. Let \( T \) be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.10). Then there exists a constant \( C \) such that, given any atom a, \[ \parallel {Ta}{\parallel }_{1} \leq C \] | Proof. Since \( a \in {L}^{2},{Ta} \) is well defined. Let \( {Q}^{ * } \) be the cube with the same center as \( Q,{c}_{Q} \), and side length \( 2\sqrt{n} \) times larger. Then, since \( T \) is bounded on \( {L}^{2} \), \[ {\int }_{{Q}^{ * }}\left| {{Ta}\left( x\right) }\right| {dx} \leq {\left| {Q}^{ * }\right| }^{... | Yes |
\[ \frac{1}{2}\parallel f{\parallel }_{ * } \leq \mathop{\sup }\limits_{Q}\mathop{\inf }\limits_{{a \in \mathbb{C}}}\frac{1}{\left| Q\right| }{\int }_{Q}\left| {f\left( x\right) - a}\right| {dx} \leq \parallel f{\parallel }_{ * } \] | The second inequality in (6.2) is immediate. To prove the first, note that for all \( a \) ,\n\n\[ {\int }_{Q}\left| {f\left( x\right) - {f}_{Q}}\right| {dx} \leq {\int }_{Q}\left| {f\left( x\right) - a}\right| {dx} + {\int }_{Q}\left| {a - {f}_{Q}}\right| {dx} \leq 2{\int }_{Q}\left| {f\left( x\right) - a}\right| {dx}... | Yes |
Theorem 6.6. Let \( T \) be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.11). Then if \( f \) is a bounded function of compact support, \( {Tf} \in {BMO} \) and\n\n\[ \parallel {Tf}{\parallel }_{ * } \leq C\parallel f{\parallel }_{\infty } \] | Proof. Fix a cube \( Q \) in \( {\mathbb{R}}^{n} \) with center \( {c}_{Q} \), and let \( {Q}^{ * } \) be the cube centered at \( {c}_{Q} \) whose side length is \( 2\sqrt{n} \) times that of \( Q \) . Decompose \( f \) as \( f = {f}_{1} + {f}_{2} \) , where \( {f}_{1} = f{\chi }_{{Q}^{ * }} \) .\n\nLet \( a = T{f}_{2}... | Yes |
Let \( f\left( x\right) = \operatorname{sgn}\left( x\right) \) ; we will find the image of \( f \) under the Hilbert transform, whose kernel is \( K\left( {x, y}\right) = {\left\lbrack \pi \left( x - y\right) \right\rbrack }^{-1} \) . | Let \( \left| x\right| < a/2 \) ; then\n\n\[ \n{\pi Hf}\left( x\right) = \text{ p. v. }{\int }_{-a}^{a}\frac{\operatorname{sgn}\left( y\right) }{x - y}{dy} + \mathop{\lim }\limits_{{N \rightarrow \infty }}{\int }_{-N}^{-a} + {\int }_{a}^{N}\left( {\frac{1}{x - y} + \frac{1}{y}}\right) \operatorname{sgn}\left( y\right) ... | Yes |
Lemma 6.10. If \( f \in {L}^{{p}_{0}} \) for some \( {p}_{0},1 \leq {p}_{0} < \infty \), then for all \( \gamma > 0 \) and \( \lambda > 0 \)\n\n\[ \left| \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > {2\lambda },{M}^{\# }f\left( x\right) \leq {\gamma \lambda }}\right\} \right| \leq {2}^{n}\gamma \left| \l... | Proof. Without loss of generality we may assume that \( f \) is non-negative. Fix \( \lambda ,\gamma > 0 \) . If we form the Calderón-Zygmund decomposition of \( f \) at height \( \lambda \), then the set \( \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > \lambda }\right\} \) can be written as the union of ... | Yes |
For all \( p,1 < p < \infty \) , \[ \parallel f{\parallel }_{*, p} = \mathop{\sup }\limits_{Q}{\left( \frac{1}{\left| Q\right| }{\int }_{Q}{\left| f\left( x\right) - {f}_{Q}\right| }^{p}dx\right) }^{1/p} \] is a norm on BMO equivalent to \( \parallel \cdot {\parallel }_{ * } \) . | Proof. It will suffice to prove that \( \parallel f{\parallel }_{*, p} \leq {C}_{p}\parallel f{\parallel }_{ * } \) since the reverse inequality is immediate. But by Theorem 6.11 \[ {\int }_{Q}{\left| f\left( x\right) - {f}_{Q}\right| }^{p}{dx} = {\int }_{0}^{\infty }p{\lambda }^{p - 1}\left| \left\{ {x \in Q : \left| ... | Yes |
Corollary 6.13. Given \( f \in {BMO} \), there exists \( \lambda > 0 \) such that for any cube \( Q \) ,\n\n\[ \n\frac{1}{\left| Q\right| }{\int }_{Q}{e}^{\lambda \left| {f\left( x\right) - {f}_{Q}}\right| }{dx} < \infty .\n\] | Further, the proof of Corollary 6.12 (with \( p = 1 \) ) can be readily adapted to show that the converse of the John-Nirenberg inequality (Theorem 6.11) holds. | No |
Theorem 6.16. Given a sublinear operator \( T \), suppose that \( T \) maps \( {H}^{1} \) into \( {L}^{1} \) and for some \( {p}_{1},1 < {p}_{1} \leq \infty, T \) is weak \( \left( {{p}_{1},{p}_{1}}\right) \) . Then for all \( p \) , \( 1 < p < {p}_{1}, T \) is bounded on \( {L}^{p} \) . | This result can be generalized to the other \( {H}^{p} \) spaces. For details and references, see García-Cuerva and Rubio de Francia [6, Chapter 3]. | No |
Theorem 6.17. If \( f \in {L}^{n/a} \) then \( {I}_{a}f \in {BMO} \) . | The proof of this theorem follows at once from an inequality relating the fractional integral, the sharp maximal function and the fractional maximal function. | No |
Theorem 6.18. There exist positive constants \( {C}_{1} \) and \( {C}_{2} \) such that for all locally integrable \( f \) ,\n\n\[ \n{C}_{1}{M}_{a}f\left( x\right) \leq {M}^{\# }\left( {{I}_{a}f}\right) \left( x\right) \leq {C}_{2}{M}_{a}f\left( x\right) .\n\] | Both of these results are due to D. R. Adams (A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778). | No |
Theorem 6.19. Given \( 0 < a < n \) and any cube \( Q \), then there exist constants \( {C}_{1} \) and \( {C}_{2} \) such that if \( f \in {L}^{n/a}\left( Q\right) \), \[ \frac{1}{\left| Q\right| }{\int }_{Q}\exp \left( {\left| \frac{{I}_{a}f}{{C}_{1}\parallel f{\parallel }_{{L}^{n/a}\left( Q\right) }}\right| }^{{p}^{\... | Theorem 6.19 was proved by N. S. Trudinger (On imbeddings into Or-licz spaces and some applications, J. Math. Mech. 17 (1967), 473-483) when \( a = 1 \) and in general by R. S. Strichartz (A note on Trudinger’s extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841- 842). Sharp constants have been f... | No |
Theorem 6.21. Given a singular integral \( T \) and \( b \in {BMO} \), then\n\n\[ \left| \left\{ {x \in {\mathbb{R}}^{n} : \left| {\left\lbrack {b, T}\right\rbrack f\left( x\right) }\right| > \lambda }\right\} \right| \leq C\parallel b{\parallel }_{ * }{\int }_{{\mathbb{R}}^{n}}\frac{\left| f\left( y\right) \right| }{\... | Theorem 6.21 is due to C. Pérez (Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), 163-185). | No |
For \( 1 \leq p < \infty \), the weak \( \left( {p, p}\right) \) inequality\n\n\[ w\left( \left\{ {x \in {\mathbb{R}}^{n} : {Mf}\left( x\right) > \lambda }\right\} \right) \leq \frac{C}{{\lambda }^{p}}{\int }_{{\mathbb{R}}^{n}}{\left| f\left( x\right) \right| }^{p}w\left( x\right) {dx} \]\n\nholds if and only if \( w \... | Proof. We proved the necessity of the \( {A}_{p} \) condition above.\n\nTo prove sufficiency, we first consider the case \( p = 1 \) . This case is a corollary to Theorem 2.16. The right-hand side of the weak \( \left( {1,1}\right) \) inequality (2.11) contains \( {Mw} \), and since \( w \in {A}_{1} \), by (7.5) we can... | Yes |
Theorem 7.3. If \( 1 < p < \infty \) then \( M \) is bounded on \( {L}^{p}\left( w\right) \) if and only if \( w \in {A}_{p} \) . | Since the strong \( \left( {p, p}\right) \) inequality implies the weak \( \left( {p, p}\right) \) inequality, necessity follows from Theorem 7.1. Sufficiency would follow from the above interpolation argument if we could show that given \( w \in {A}_{p} \), there exists \( q \) , \( 1 < q < p \), such that \( w \in {A... | No |
Lemma 7.5. Let \( w \in {A}_{p},1 \leq p < \infty \) . Then for every \( \alpha ,0 < \alpha < 1 \), there exists \( \beta ,0 < \beta < 1 \), such that given a cube \( Q \) and \( S \subset Q \) . with \( \left| S\right| \leq \alpha \left| Q\right| \) , \( w\left( S\right) \leq {\beta w}\left( Q\right) \) . | Proof. If we replace \( S \) by \( Q \smallsetminus S \) in inequality (7.3) we get\n\n\[ w\left( Q\right) {\left( 1 - \frac{\left| S\right| }{\left| Q\right| }\right) }^{p} \leq C\left( {w\left( Q\right) - w\left( S\right) }\right) .\n\]\n\nIf \( \left| S\right| \leq \alpha \left| Q\right| \) then\n\n\[ w\left( S\righ... | Yes |
(1) \( {A}_{p} = \mathop{\bigcup }\limits_{{q < p}}{A}_{q};1 < p < \infty \) . | Proof. (1) If \( w \in {A}_{p} \) then by Proposition 7.2, \( {w}^{1 - {p}^{\prime }} \in {A}_{{p}^{\prime }} \) . Therefore, it also satisfies the reverse Hölder inequality for some \( \epsilon > 0 \) :\n\n(7.11)\n\n\[{\left( \frac{1}{\left| Q\right| }{\int }_{Q}{w}^{\left( {1 - {p}^{\prime }}\right) \left( {1 + \epsi... | No |
(1) Let \( f \in {L}_{\text{loc }}^{1}\left( {\mathbb{R}}^{n}\right) \) be such that \( {Mf}\left( x\right) < \infty \) a.e. If \( 0 \leq \delta < 1 \), then \( w\left( x\right) = {Mf}{\left( x\right) }^{\delta } \) is an \( {A}_{1} \) weight whose \( {A}_{1} \) constant depends only on \( \delta \) . | Proof. (1) It will suffice to show that there exists a constant \( C \) such that for every \( f \), every cube \( Q \) and almost every \( x \in Q \) ,\n\n\[ \frac{1}{\left| Q\right| }{\int }_{Q}{\left( Mf\right) }^{\delta } \leq {CMf}{\left( x\right) }^{\delta } \]\n\nFix \( Q \) and decompose \( f \) as \( f = {f}_{... | Yes |
Lemma 7.9. If \( T \) is a Calderón-Zygmund operator, then for each \( s > 1 \) , \[ {M}^{\# }\left( {Tf}\right) \left( x\right) \leq {C}_{s}M\left( {\left| f\right| }^{s}\right) {\left( x\right) }^{1/s}, \] where \( {M}^{\# } \) is the sharp maximal operator (6.1). | Proof. Fix \( s > 1 \) . Given \( x \) and a cube \( Q \) containing it, by Proposition 6.5 it will suffice to find a constant \( a \) such that \[ \frac{1}{\left| Q\right| }{\int }_{Q}\left| {{Tf}\left( y\right) - a}\right| {dy} \leq {CM}\left( {\left| f\right| }^{s}\right) {\left( x\right) }^{1/s}. \] As in the proof... | Yes |
Lemma 7.10. Let \( w \in {A}_{p},1 \leq {p}_{0} \leq p < \infty \) . If \( f \) is such that \( {M}_{d}f \in \) \( {L}^{{p}_{0}}\left( w\right) \), then\n\n\[{\int }_{{\mathbb{R}}^{n}}{\left| {M}_{d}f\right| }^{p}w \leq C{\int }_{{\mathbb{R}}^{n}}{\left| {M}^{\# }f\right| }^{p}w\]\n\nwhere \( {M}_{d} \) is the dyadic m... | Proof. This is a weighted version of Lemma 6.9, and the proof is almost exactly the same. It will suffice to prove a good- \( \lambda \) inequality which is a weighted analogue of Lemma 6.10: for some \( \delta > 0 \),\n\n\[w\left( \left\{ {x \in {\mathbb{R}}^{n} : {M}_{d}f\left( x\right) > {2\lambda },{M}^{\# }f\left(... | Yes |
Theorem 7.11. If \( T \) is a Calderón-Zygmund operator, then for any \( w \in \) \( {A}_{p},1 < p < \infty, T \) is bounded on \( {L}^{p}\left( w\right) \) . | Proof. Fix \( w \in {A}_{p} \) . We may assume that \( f \) is a bounded function of compact support since the set of such functions is dense in \( {L}^{p}\left( w\right) \) . By Corollary 7.6, we can find \( s > 1 \) such that \( w \in {A}_{p/s} \) . Therefore, since \( {Tf}\left( x\right) \leq {M}_{d}\left( {Tf}\righ... | Yes |
Corollary 7.13. If \( T \) is a Calderón-Zygmund operator, then for \( 1 < p < \) \( \infty ,{T}^{ * } \) is bounded on \( {L}^{p}\left( w\right) \) if \( w \in {A}_{p} \), and \( {T}^{ * } \) is weak \( \left( {1,1}\right) \) with respect to \( w \) if \( w \in {A}_{1} \) . | Proof. Corollary 7.13 is a weighted version of Theorem 5.14 and its proof depends on Cotlar’s inequality (5.18): for \( 0 < \nu \leq 1 \) ,\n\n\[ \n{T}^{ * }f\left( x\right) \leq {C}_{\nu }\left( {M\left( {\left| Tf\right| }^{\nu }\right) {\left( x\right) }^{1/\nu } + {Mf}\left( x\right) }\right) .\n\]\n\nWhen \( 1 < p... | Yes |
Theorem 7.14. For \( 1 < p < \infty ,{M}_{s} \) is bounded on \( {L}^{p}\left( w\right) \) if and only if \( w \in {A}_{p}^{ * } \) . | Theorem 7.14 is a consequence of the corresponding result for the Hardy-Littlewood maximal function. For \( 1 \leq i \leq n \), define \( {M}_{i} \) to be the maximal operator restricted to the \( i \) -th variable:\n\n\[ \n{M}_{i}f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {Mf}\left( {{x}_{1},\ldots ,{x}_{i - 1},\cdot ... | Yes |
Theorem 7.15. Let \( 1 < p < \infty \) . Then \( w \in {A}_{p}^{ * } \) if and only if there exist \( {w}_{0},{w}_{1} \in {A}_{1}^{ * } \) such that \( w = {w}_{0}{w}_{1}^{1 - p} \) . | This theorem was first proved by J. L. Rubio de Francia; see the paper cited in Section 5.1. | No |
Theorem 7.16. Given a basis \( \mathcal{B} \), a weight \( w \) and \( p,1 < p < \infty \), let \( \sigma = \) \( {w}^{1 - {p}^{\prime }} \) . Then the following are equivalent:\n\n(1) \( {M}_{\mathcal{B}} \) is bounded on both \( {L}^{p}\left( w\right) \) and \( {L}^{{p}^{\prime }}\left( \sigma \right) \) ;\n\n(2) \( ... | The boundedness of \( {M}_{\mathcal{B}, w} \) and \( {M}_{\mathcal{B},\sigma } \) depends on the geometry of the basis \( \mathcal{B} \) . If \( \mathcal{B} \) is the basis of rectangles then the boundedness of these two operators was proved by R. Fefferman (Strong differentiation with respect to measures, Amer. J. Mat... | No |
Theorem 7.18. Given a pair of weights \( \left( {u, v}\right) \) and \( p,1 < p < \infty \), the following are equivalent:\n\n(1) \( M \) is a bounded operator from \( {L}^{p}\left( v\right) \) to \( {L}^{p}\left( u\right) \) ;\n\n(2) given any cube \( Q \), \n\n(7.18)\n\n\[ \n{\int }_{Q}M{\left( {v}^{1 - {p}^{\prime }... | Theorem 7.18 was first proved by E. Sawyer (A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11). A simpler version of this proof was given by D. Cruz-Uribe (New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J. 7 (2000), 33-42). A very... | No |
Theorem 8.1. If \( T \) is a convolution operator which is bounded on \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) and whose associated kernel \( K \) satisfies the Hörmander condition (5.2), then for any \( r,1 < r < \infty \), and \( p,1 < p < \infty \), \[ {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{j}{\left| T{f}... | Proof. We use Theorem 5.17 with \( A = B = {\ell }^{r} \) . The first inequality is immediate when \( p = r \) since \( T \) is bounded on \( {L}^{r}\left( {\mathbb{R}}^{n}\right) ,1 < r < \infty \) . Now consider the vector-valued operator which associates to each sequence \( \left\{ {f}_{j}\right\} \) the sequence \(... | Yes |
Corollary 8.2. Let \( \\left\\{ {I}_{j}\\right\\} \) be a sequence of intervals on the real line, finite or infinite, and let \( \\left\\{ {S}_{j}\\right\\} \) be the sequence of operators defined by \( {\\left( {S}_{j}f\\right) }^{ \\frown }\\left( \\xi \\right) = \) \( {\\chi }_{{I}_{j}}\\left( \\xi \\right) \\wideha... | Proof. If \( {I}_{j} = \\left( {{a}_{j},{b}_{j}}\\right) \) then by (3.9) we have the formula\n\n\[ {S}_{j}{f}_{j} = \\frac{i}{2}\\left( {{M}_{{a}_{j}}H{M}_{-{a}_{j}}{f}_{j} - {M}_{{b}_{j}}H{M}_{-{b}_{j}}{f}_{j}}\\right) ,\]\n\nwith the obvious modifications if the interval is unbounded. The desired result now follows ... | No |
Theorem 8.3. Let \( \\left\\{ {T}_{j}\\right\\} \) be a sequence of linear operators which are bounded on \( {L}^{2}\\left( w\\right) \) for any \( w \\in {A}_{2} \) with constants that are uniform in \( j \) and which depend only on the \( {A}_{2} \) constant of \( w \) . Then for all \( p,1 < p < \\infty \) , | Proof. When \( p = 2 \) this result is immediate. Now suppose that \( p > 2 \) . Then there exists a function \( u \\in {L}^{{\\left( p/2\\right) }^{\\prime }} \) with norm 1 such that\n\n\[{\\begin{Vmatrix}{\\left( \\mathop{\\sum }\\limits_{j}{\\left| {T}_{j}{f}_{j}\\right| }^{2}\\right) }^{1/2}\\end{Vmatrix}}_{p}^{2}... | Yes |
Theorem 8.4 (Littlewood-Paley). Let \( f \in {L}^{p}\left( \mathbb{R}\right) ,1 < p < \infty \) . Then there exist positive constants \( {c}_{p} \) and \( {C}_{p} \) such that\n\n\[{c}_{p}\parallel f{\parallel }_{p} \leq {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{j}{\left| {S}_{j}f\right| }^{2}\right) }^{1/2}\end{V... | We will prove Theorem 8.4 as a consequence of a similar result, but one where the operators are defined using smooth functions instead of characteristic functions of intervals. Let \( \psi \in \mathcal{S}\left( \mathbb{R}\right) \) be non-negative, have support in \( 1/2 \leq \left| \xi \right| \leq 4 \) and be equal t... | Yes |
Theorem 8.5. Let \( f \in {L}^{p}\left( \mathbb{R}\right) ,1 < p < \infty \) . Then there exists a constant \( {C}_{p} \) such that\n\n\[{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{j}{\left| {\widetilde{S}}_{j}f\right| }^{2}\right) }^{1/2}\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p}\] | Proof. If \( \widehat{\Psi } = \psi \) and \( {\Psi }_{j}\left( x\right) = {2}^{j}\Psi \left( {{2}^{j}x}\right) \), then \( {\widehat{\Psi }}_{j} = {\psi }_{j} \) and \( {\widetilde{S}}_{j}f = {\Psi }_{j} * f \) . Therefore, given the vector-valued operator which maps \( f \) to the sequence \( \left\{ {{\widetilde{S}}... | Yes |
Theorem 8.6. Given \( \psi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) with \( \psi \left( 0\right) = 0 \), let \( {S}_{j}, j \in \mathbb{Z} \), be the operator defined by \( {\left( {S}_{j}f\right) }^{ \frown }\left( \xi \right) = \psi \left( {{2}^{-j}\xi }\right) \widehat{f}\left( \xi \right) \) . Then for \( 1 ... | Since \( \psi \in \mathcal{S} \) and \( \psi \left( 0\right) = 0 \), it satisfies\n\n\[ \mathop{\sum }\limits_{j}{\left| \psi \left( {2}^{-j}\xi \right) \right| }^{2} \leq C \]\n\nTherefore, the proof of the first part of Theorem 8.6 is exactly the same as the proof of Theorem 8.5, while the proof of the second part is... | Yes |
Theorem 8.7. Let \( f \in {L}^{p}\left( {\mathbb{R}}^{2}\right) ,1 < p < \infty \) . Then there exist positive constants \( {c}_{p} \) and \( {C}_{p} \) such that\n\n\[ \n{c}_{p}\parallel f{\parallel }_{p} \leq {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j, k}}{\left| {S}_{j}^{1}{S}_{k}^{2}f\right| }^{2}\right) }^{... | Proof. By the same argument as in the proof of Theorem 8.5, and with the same notation as there, we have that\n\n(8.9)\n\n\[ \n{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j, k}}{\left| {\widetilde{S}}_{j}{f}_{k}\right| }^{2}\right) }^{1/2}\end{Vmatrix}}_{p} \leq {C}_{p}{\begin{Vmatrix}{\left( \mathop{\sum }\limits_... | Yes |
Proposition 8.8. If \( a > n/2 \) and \( g \in {L}_{a}^{2}\left( {\mathbb{R}}^{n}\right) \) then \( \widehat{g} \in {L}^{1} \) ; in particular, \( g \) is continuous and bounded. | Proof. Since \( {\left( 1 + {\left| \xi \right| }^{2}\right) }^{a/2}\widehat{g}\left( \xi \right) = h\left( \xi \right) \in {L}^{2} \), \[ {\int }_{{\mathbb{R}}^{n}}\left| {\widehat{g}\left( \xi \right) }\right| {d\xi } \leq {\left( {\int }_{{\mathbb{R}}^{n}}{\left| h\left( \xi \right) \right| }^{2}d\xi \right) }^{1/2}... | Yes |
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