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| In [[mathematics]], the '''Riesz–Thorin theorem''', often referred to as the '''Riesz–Thorin interpolation theorem''' or the '''Riesz–Thorin convexity theorem''' is a result about ''interpolation of operators''. It is named after [[Marcel Riesz]] and his student [[G. Olof Thorin]].
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| This theorem bounds the norms of linear maps acting between [[lp space|''L<sup>p</sup>'']] spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to ''L''<sup>2</sup> which is a [[Hilbert space]], or to ''L''<sup>1</sup> and ''L''<sup>∞</sup>. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The [[Marcinkiewicz theorem]] is similar but applies also to a class of non-linear maps.
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| ==Motivation==
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| Let <math>0< p_0 < p_1 \leq \infty</math> and if <math>\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}</math>. By splitting up the function <math>f \in L^{p_\theta}</math> as the product <math>|f| = |f|^{1-\theta}|f|^{\theta}</math> and applying [[Hölder's inequality]] to its <math>p_\theta</math>'s power, we obtain the follwing result, foundational in the study of
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| <math>L^p</math>-spaces:
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| <blockquote>'''Proposition (log-convexity of <math>L^p</math>-norms).''' If <math>0< p_0 < q_1 \leq \infty</math> and if <math>\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}</math> for some <math> 0 < \theta < 1</math>, then <math> \|f\|_{p_\theta} \leq \|f\|_{p_0}^{1-\theta}\|f\|_{p_1}^\theta</math> for all <math>f \in L^p \cap L^q</math>.</blockquote>
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| This result, whose name derives from the convexity of the map <math>p \mapsto \log \|f\|_p</math> on <math>[0,\infty]</math>, implies that <math>L^{p_0} \cap L_{p_1}</math> is included in <math>L^{p_\theta}</math>.
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| On the other hand, if we take the ''layer-cake decomposition'' <math>f = f\chi_{\{|f|>1\}} + f\chi_{\{|f| \leq 1\}}</math>, then we see that <math>f\chi_{\{|f|>1\}} \in L^{p_0}</math> and <math>f\chi_{\{|f| \leq 1\}} \in L^{p_1}</math>, whence we obtain the following result:
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| <blockquote>'''Proposition.''' If <math>0< p_0 < p_1 \leq \infty</math> and if <math>\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}</math> for some <math> 0 < \theta < 1</math>, then each <math>f \in L^{p_\theta}</math> can be written as a sum of <math>g \in L^{p_0}</math> and <math>h \in L^{p_1}</math>.</blockquote>
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| In particular, the above result implies that <math>L^{p_\theta}</math> is included in <math>L^{p_0} + L^{p_1}</math>, the [[Minkowski addition|sumset]] of <math>L^{p_0}</math> and <math>L^{p_1}</math> in the space of all measurable functions. Therefore, we have the following chain of inclusions:
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| <blockquote>'''Corollary.''' <math>L^{p_0} \cap L^{p_1} \subset L^{p_\theta} \subset L^{p_0} + L^{p_1}</math>.</blockquote>
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| In practice, we often encounter [[Operator (mathematics)|operators]] defined on the [[sumset]] <math>L^{p_0}+L^{p_1}</math>. For example, the [[Riemann-Lebesgue lemma]] shows that the [[Fourier transform]] maps <math>L^1(\mathbb{R}^d)</math> [[Bounded operator|boundedly]] into <math>L^\infty(\mathbb{R}^d)</math>, and [[Plancherel theorem|Plancherel's theorem]] shows that the Fourier transform maps <math>L^2(\mathbb{R}^d)</math> boundedly into itself, whence the Fourier transform <math>\mathcal{F}</math> extends to <math>(L^1+L^2)(\mathbb{R}^d)</math> by setting <math>\mathcal{F}(f_1+f_2) = \mathcal{F}_{L^1}(f_1) + \mathcal{F}_{L^2}(f_2)</math> for all <math>f_1 \in L^1(\mathbb{R}^d)</math> and <math>f_2 \in L^2(\mathbb{R}^d)</math>. It is therefore natural to investigate the behavior of such operators on the ''intermediate subspaces'' <math>L^{p_\theta}</math>.
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| To this end, we go back to our example and note that the Fourier transform on the sumset <math>L^1 + L^2</math> was obtained by taking the sum of two instantiations of the same operator, namely <math>\mathcal{F}_{L^1}:L^1(\mathbb{R}^d) \to L^\infty(\mathbb{R}^d)</math> and <math>\mathcal{F}_{L^2}:L^2(\mathbb{R}^d) \to L^2(\mathbb{R}^d)</math>. These really are the ''same'' operator, in the sense that they agree on the subspace <math>(L^1 \cap L^2)(\mathbb{R}^d)</math>. Since the intersection contains [[simple function]]s, it is dense in both <math>L^1(\mathbb{R}^d)</math> and <math>L^2(\mathbb{R}^d)</math>. Densely-defined continuous functions admit unique extensions, and so we are justified in considering <math>\mathcal{F}_{L^1}</math> and <math>\mathcal{F}_{L^2}</math>
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| to be ''the same''.
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| Therefore, the problem of studying operators on the sumset <math>L^{p_0} + L^{p_1}</math> essentially reduces to the study of operators that map two natural domain spaces, <math>L^{p_0}</math> and <math>L^{q_0}</math>, boundedly to two target spaces: <math>L^{q_0}</math> and <math>L^{q_1}</math>, respectively. Since such operators map the sumset space <math>L^{p_0} + L^{p_1}</math> to <math>L^{q_0} + L^{q_1}</math>, it is natural to expect that these operators map the intermediate space <math>L^{p_\theta}</math> to the corresponding intermediate space<math>L^{q_\theta}</math>.
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| ==Statement of the Theorem==
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| There are several ways to state the Riesz-Thorin interpolation theorem;<ref>Stein and Weiss (1971) and Grafakos (2010) use operators on simple functions, and Muscalu and Schlag (2013) uses operators on generic dense subsets of the intersection <math>L^{p_0} \cap L^{p_1}</math>. In contrast, Duoanddikoetxea (2001), Tao (2010), and Stein and Shakarchi (2011) use the sumset formulation, which we adopt in this section.</ref> to be consistent with the notations in the previous section, we shall use the sumset formulation.
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| <blockquote>'''Riesz-Thorin interpolation theorem'''. Let <math>(\Omega_1,\Sigma_1,\mu_1)</math> and <math>(\Omega_2,\Sigma_2,\mu_2)</math> be [[Σ-finite measure|<math>\sigma</math>-finite measure]] spaces, <math>1 \leq p_0 \leq p_1 \leq \infty</math>, <math>1 \leq q_0 \leq q_1 \leq \infty</math>, and <math>T:L^{p_0}(\mu_1) + L^{p_1}(\mu_1) \to L^{q_0}(\mu_2) + L^{q_1}(\mu_2)</math> be a [[linear operator]] that maps <math>L^{p_0}(\mu_1)</math> and <math>L^{p_1}(\mu_1)</math> [[Bounded operator|boundedly]] into <math>L^{q_0}(\mu_2)</math> and <math>L^{q_1}(\mu_2)</math>, respectively. If <math>0 < \theta < 1</math>, <math>\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}</math>, and <math>\frac{1}{q_\theta} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}</math>, then <math>T</math> maps <math>L^{p_\theta}(\mu_1)</math> boundedly into <math>L^{q_\theta}(\mu_2)</math> and satisfies the [[operator norm]] estimate <math>\|T\|_{L^{p_\theta} \to L^{q_\theta}} \leq \|T\|^{1-\theta}_{L^{p_0} \to L^{q_0}} \|T\|^{\theta}_{L^{p_1} \to L^{q_1}}</math>. </blockquote>
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| In other words, if <math>T</math> is simultaneously of '''type''' <math>(p_0,q_0)</math> and of type <math>(p_1,q_1)</math>, then <math>T</math> is of type <math>(p_\theta,q_\theta)</math> for all <math>0 < \theta < 1</math>. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, '''the Riesz diagram''' of <math>T</math> is the collection of all points <math>\left(\frac{1}{p},\frac{1}{q}\right)</math> in the unit square <math>[0,1] \times [0,1]</math> such that <math>T</math> is of type <math>(p,q)</math>. The interpolation theorem states that the Riesz diagram of <math>T</math> is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram.
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| The interpolation theorem was originally stated and proved by [[Marcel Riesz]] in 1927.<ref>Riesz (1927). The proof makes use of convexity results in the theory of bilinear forms. For this reason, many classical references such as Stein and Weiss (1971) refer to the Riesz-Thorin interpolation theorem as the '''Riesz convexity theorem'''.</ref> The 1927 paper establishes the theorem only for the ''lower triangle'' of the Riesz diagram, viz., with the restriction that <math>p_0 \leq q_0</math> and <math>p_1 \leq q_1</math>. [[Olof Thorin]] extended the interpolation theorem to the entire square, removing the lower-triangle restriction. The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis.<ref>Thorin (1948)</ref>
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| ==Sketch of Proof==
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| The proof of the Riesz-Thorin interpolation theorem relies crucially on the [[Hadamard three-lines theorem]] to establish the requisite bounds. By the [[Lp space#Dual spaces|characterization of the dual spaces of <math>L^p</math>-spaces]], we see that
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| : <math> \|Tf\|_{q_\theta} = \sup_{\|g\|_{p_\theta} \leq 1} \left| \int (Tf)g \, d\mu_2\right|</math>.
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| By suitably defining variants <math>f_z</math> and <math>g_z</math> of <math>f</math> and <math>g</math> for each <math>z \in \mathbb{C}</math>, we obtain the [[entire function]]
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| : <math> \phi(z) = \int (Tf_z)g_z \, d\mu_2</math>
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| whose value at <math>z=\theta</math> is <math>\int (Tf)g</math>. We can then use the hypotheses to establish upper bounds of <math>\Phi</math> on the lines <math>\mathrm{Re} z = 0</math> and <math>\mathrm{Re} z = 1</math>, whence the [[Hadamard three-lines theorem]] establishes the interpolated bound of <math>\Phi</math> on the line <math>\mathrm{Re} z = \theta</math>. It now suffices to check that the bound at <math>z = \theta</math> is what we wanted.
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| ==Interpolation of Analytic Families of Operators==
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| The proof outline presented in the above section readily generalizes to the case in which the operator <math>T</math> is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function
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| :<math>\varphi(z) = \int (T_z f_z)g_z \, d\mu_2</math>,
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| from which we obtain the following theorem of [[Elias Stein]], published in his 1956 thesis:<ref>Stein (1956). As [[Charles Fefferman]] points out in his essay in Fefferman, Fefferman, Wainger (1995), the proof of Stein interpolation theorem is essentially that of the Riesz-Thorin theorem with the letter <math>z</math> added to the operator. To compensate for this, a stronger version of the [[Hadamard three-lines theorem]], due to [[Isidore Isaac Hirschman, Jr.]], is used to establish the desired bounds. See Stein and Weiss (1971) for a detailed proof, and [http://terrytao.wordpress.com/2011/05/03/steins-interpolation-theorem/ a blog post of Tao] for a high-level exposition of the theorem.</ref>
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| <blockquote>
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| '''Stein interpolation theorem'''. Let <math>(\Omega_1,\Sigma_1,\mu_1)</math> and <math>(\Omega_2,\Sigma_2,\mu_2)</math> be [[Σ-finite measure|<math>\sigma</math>-finite measure]] spaces, <math>1 \leq p_0 \leq p_1 \leq \infty</math>, and <math>1 \leq q_0 \leq q_1 \leq \infty</math>. We take a collection of linear operators <math>\{T_z\}_{0 \leq \mathrm{Re} z \leq 1}</math> on the space of simple functions in <math>L^1(\mu_1)</math> into the space of all <math>\mu_2</math>-measurable functions on <math>\Omega_2</math> such that the mapping
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| :<math> z \mapsto \int (T_zf)g \, d\mu_2</math>
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| is continuous on the strip <math>S = \{z \in \mathbb{C} : 0 \leq \mathrm{Re} z \leq 1\}</math> and holomorphic in the interior of <math>S</math> for all simple functions <math>f</math> and <math>g</math>. We assume also that the operators satisfy the uniform bound
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| :<math> \sup_{z \in S} e^{-k|\mathrm{Im} z|} \left| \int (T_zf)g \, \mu_2 \right| < \infty</math>
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| for some constant <math>k < \pi</math>.
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| Assume that <math>T_z</math> maps <math>L^{p_0}(\mu_1)</math> [[Bounded operator|boundedly]] to <math>L^{q_0}(\mu_2)</math> whenever <math>\mathrm{Re} z = 0</math>, and that <math>T_z</math> maps <math>L^{p_1}(\mu_1)</math> boundedly to <math>L^{q_1}(\mu_2)</math> whenever <math>\mathrm{Re} z = 1</math>. If the operator norms satisfy the uniform bound
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| : <math> \sup_{\mathrm{Re} z = 0 \mbox{ or } 1} e^{-k|\mathrm{Im}z|} \log\|T_z\| < \infty</math>,
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| then, for each <math>0 < \theta < 1</math>, the operator <math>T_\theta</math> maps <math>L^{p_\theta}(\mu_1)</math> boundedly into <math>L^{q_\theta}(\mu_2)</math>.
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| </blockquote>
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| The theory of [[Hardy space#Real Hardy spaces for Rn|real Hardy spaces]] and the [[Bounded mean oscillation|space of bounded mean oscillations]] permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space <math>H^1(\mathbb{R}^d)</math> and the space <math>BMO</math> of bounded mean oscillations; this is a result of [[Charles Fefferman]] and [[Elias Stein]].<ref>Fefferman and Stein (1972)</ref>
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| ==Applications==
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| ===Hausdorff−Young inequality===
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| {{main|Hausdorff−Young inequality}}
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| We have seen in the [[Riesz-Thorin theorem#Motivation|first section]] that the [[Fourier transform]] <math>\mathcal{F}</math> maps <math>L^1(\mathbb{R}^d)</math> boundedly into <math>L^\infty(\mathbb{R}^d)</math> and <math>L^2(\mathbb{R}^d)</math> into itself. A similar argument shows that the [[Fourier series|Fourier series operator]] that transforms periodic functions <math>f:\mathbb{T} \to \mathbb{C}</math> into functions <math>\hat{f}:\mathbb{Z} \to \mathbb{C}</math> whose values are the Fourier coefficients
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| : <math>\hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx</math>
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| maps <math>L^1(\mathbb{T})</math> boundedly into <math>l^\infty(\mathbb{Z})</math> and <math>L^2(\mathbb{T})</math> into <math>l^2(\mathbb{Z})</math>. The Riesz-Thorin interpolation theorem now implies that
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| : <math>\|\mathcal{F}f\|_{L^{p'}(\mathbb{R}^d)} \leq \|f\|_{L^p(\mathbb{R}^d)}</math> and <math>\|\hat{f}\|_{l^{p'}(\mathbb{Z})} \leq \|f\|_{L^p(\mathbb{T})}</math>,
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| where <math>1 \leq p \leq 2</math> and <math>\frac{1}{p} + \frac{1}{p'} = 1</math>. This is the [[Hausdorff–Young inequality]].
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| The Hausdorff-Young inequality can also be established for the [[Locally compact abelian group|Fourier transform on locally compact abelian groups]]. We also note that the norm estimate of 1 is not optimal. See [[Hausdorff-Young inequality|the main article]] for references.
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| ===Convolution operators===
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| {{main|Young's_inequality#Young.27s_inequality_for_convolutions}}
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| Let ''f'' be a fixed integrable function and let ''T'' be the operator of convolution with ''f'', i.e., for each function ''g'' we have
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| :<math>Tg = f * g.</math>
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| It is well known that ''T'' is bounded from ''L''<sup>1</sup> to ''L''<sup>1</sup> and it is trivial that it is bounded from ''L''<sup>∞</sup> to ''L''<sup>∞</sup> (both bounds are by <math>\|f\|_1</math>). Therefore the Riesz–Thorin theorem gives
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| :<math>\|f*g\|_p\leq \|f\|_1\|g\|_p.</math>
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| We take this inequality and switch the role of the operator and the operand, or in other words, we think of ''S'' as the operator of convolution with ''g'', and get that ''S'' is bounded from ''L''<sup>1</sup> to ''L<sup>p</sup>''. Further, since ''g'' is in ''L<sup>p</sup>'' we get, in view of Hölder's inequality, that ''S'' is bounded from ''L<sup>q</sup>'' to ''L''<sup>∞</sup>, where again 1/''p'' + 1/''q'' = 1. So interpolating we get
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| :<math>\|f*g\|_s\leq \|f\|_r\|g\|_p</math>
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| where the connection between ''p'', ''r'' and ''s'' is | |
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| :<math>\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.</math>
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| ===The Hilbert transform===
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| {{main|Hilbert transform}}
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| The '''Hilbert transform''' of <math>f:\mathbb{R} \to \mathbb{C}</math> is given by
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| : <math> \mathcal{H}f(x) = \frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^\infty \frac{f(x-t)}{t} \, dt = \left(\frac{1}{\pi} \, \mathrm{p.v.} \frac{1}{t} \ast f\right)(x)</math>,
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| where p.v. indicates the [[Cauchy principal value]] of the integral. The Hilbert transform is a [[Multiplier (Fourier analysis)|Fourier multiplier operator]] with a particularly simple multiplier:
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| : <math> \widehat{\mathcal{H}f}(\xi) = -i \, \mathrm{sgn}(\xi) \hat{f}(\xi)</math>.
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| It follows from the [[Plancherel theorem]] that the Hilbert transform maps <math>L^2(\mathbb{R})</math> boundedly into itself.
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| Nevertheless, the Hilbert transform is is not bounded on <math>L^1(\mathbb{R})</math> or <math>L^\infty(\mathbb{R})</math>, and so we cannot use the Riesz-Thorin interpolation theorem directly.
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| To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions <math>\chi_{(-1,1)}(x)</math> and <math>\chi_{(0,1)}(x)- \chi_{(0,1)}(-x)</math>.
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| We can show, however, that
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| : <math>(\mathcal{H}f)^2 = f^2 + 2\mathcal{H}(f\mathcal{H}f)</math>
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| for all [[Schwartz space|Schwartz functions]] <math>f:\mathbb{R} \to \mathbb{C}</math>, and this identity can be used in conjunction with the [[Cauchy–Schwarz inequality]] to show that the Hilbert transform maps <math>L^{2^n}(\mathbb{R}^d)</math> boundedly into itself for all <math>n \geq 2</math>.
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| Interpolation now establishes the bound
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| :<math> \|\mathcal{H}f\|_p \leq A_p \|f\|_p</math>
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| for all <math>2 \leq p < \infty </math>, and the [[Self-adjoint operator|self-adjointness]] of the Hilbert transform can be used to carry over these bounds to the <math>1 < p \leq 2 </math> case.
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| ==Comparison with the Real Interpolation Method==
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| While the Riesz-Thorin interpolation theorem and its variants are powerful tools that yield a clean estimate on the interpolated operator norms, they suffer from numerous defects: some minor, some more severe. Note first that the complex-analytic nature of the proof of the Riesz-Thorin interpolation theorem forces the scalar field to be <math>\mathbb{C}</math>. For extended-real-valued functions, this restriction can be bypassed by redefining the function to be finite everywhere---possible, as every integrable function must be finite almost everywhere. A more serious disadvantage is that, in practice, many operators, such as the [[Hardy-Littlewood maximal operator]] and the [[Calderón-Zygmund theory|Calderón-Zygmund operators]], do not have good endpoint estimates.<ref>[[Elias Stein]] is quoted for saying that interesting operators in [[harmonic analysis]] are rarely bounded on <math>L^1</math> and <math>L^\infty</math>.</ref> In the case of the Hilbert transform in the previous section, we were able to bypass this problem by explicitly computing the norm estimates at several midway points. This is cumbersome and is often not possible in more general scenarios. Since many such operators satisfy the '''weak-type estimates'''
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| : <math> \mu\left(\{x : Tf(x) > \alpha\}\right) \leq \left( \frac{C_{p,q} \|f\|_p}{\alpha} \right)^q </math>,
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| real interpolation theorems such as the [[Marcinkiewicz interpolation theorem]] are better-suited for them. Furthermore, a good number of important operators, such as the [[Hardy-Littlewood maximal operator]], are only [[Sublinear function|sublinear]]. This is not a hindrance to applying real interpolation methods, but complex interpolation methods are ill-equipped to handle non-linear operators. On the other hand, real interpolation methods, compared to complex interpolation methods, tend to produce worse estimates on the intermediate operator norms and do not behave as well off the diagonal in the Riesz diagram. The off-diagonal versions of the Marcinkiewicz interpolation theorem require the formalism of [[Lorentz space]]s and do not necessarily produce norm estimates on the <math>L^p</math>-spaces.
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| ==Mityagin's theorem==
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| B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of [[Sequence space|spaces of sequences]] with [[Schauder basis|unconditional bases]] (cf. below).
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| Assume <math>\|A\|_{\ell_1 \to \ell_1} \leq M</math>, <math>\|A\|_{\ell_\infty \to \ell_\infty} \leq M</math>. Then <math>\|A\|_{X \to X} \leq M</math> for any unconditional Banach space of sequences ''X'' (that is, for any <math>(x_i) \in X</math> and any <math>(\epsilon_i) \in \{ -1, +1 \}^\infty</math>, <math>\| (\epsilon_i x_i) \|_X = \| (x_i) \|_X </math>).
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| The proof is based on the [[Krein–Milman theorem]].
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| == See also ==
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| * [[Marcinkiewicz interpolation theorem]]
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| * [[Interpolation space]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|authorlink2=Jacob T. Schwartz|title=Linear operators, Parts I and II|publisher=Wiley-Interscience|year=1958}}.
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| * {{Citation | last1=Fefferman |first1=Charles | last2=Stein | first2=Elias M. | title=<math>H^p</math> Spaces of Several variables | year=1972 | volume=129 | journal=Acta Mathematica | pages=137–193}}
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| * {{citation|first1=I.M.|last1=Glazman|first2=Yu.I.|last2=Lyubich|title=Finite-dimensional linear analysis: a systematic presentation in problem form''|publisher=The M.I.T. Press|publication-place=Cambridge, Mass.|year=1974}}. Translated from the Russian and edited by G. P. Barker and G. Kuerti.
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| * {{citation|mr=0717035|first=L.|last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer |year=1983|ISBN=3-540-12104-8 }}.
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| * {{citation|first=B.S.|last=Mitjagin [Mityagin]|title=An interpolation theorem for modular spaces'' (Russian)|journal=Mat. Sb. (N.S.)|volume=66|issue=108|year=1965|pages=473–482}}.
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| * {{Citation | last1=Thorin | first1=G. O. | title=Convexity theorems generalizing those of M. Riesz and Hadamard with some applications | mr=0025529 | year=1948 | journal=Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=9 | pages=1–58}}
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| * {{Citation |last1= Riesz | first1=Marcel | title=Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires | year=1927 | volume=49 | journal=Acta Mathematica | pages=465–497}}
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| * {{Citation | last1=Stein | first1=Elias M. | title=Interpolation of Linear Operators | year=1956 | volume=83 | journal=Trans. Amer. Math. Soc. | pages=482–492}}
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| * {{Citation | last1=Stein |first1=Elias M. | last2=Shakarchi | first2=Rami | title=Functional Analysis: Introduction to Further Topics in Analysis | year=2011 | publisher=Princeton University Press}}
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| * {{Citation | last1=Stein |first1=Elias M. | last2=Weiss | first2=Guido | title=Introduction to Fourier Analysis on Euclidean Spaces | year=1971 | publisher=Princeton University Press}}
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| {{DEFAULTSORT:Riesz-Thorin theorem}}
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| [[Category:Theorems in harmonic analysis]]
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| [[Category:Theorems in Fourier analysis]]
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| [[Category:Theorems in functional analysis]]
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