Riesz–Thorin theorem

Template:Short description Template:About In mathematical analysis, 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.

This theorem bounds the norms of linear maps acting between Template:Math spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to Template:Math which is a Hilbert space, or to Template:Math and Template:Math. 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.

Motivation

First we need the following definition:

Definition. Let Template:Math be two numbers such that Template:Math. Then for Template:Math define Template:Math by: Template:Math.

By splitting up the function Template:Math in Template:Math as the product Template:Math and applying Hölder's inequality to its Template:Math power, we obtain the following result, foundational in the study of Template:Math-spaces:

This result, whose name derives from the convexity of the map Template:Math on Template:Math, implies that Template:Math.

On the other hand, if we take the layer-cake decomposition Template:Math, then we see that Template:Math and Template:Math, whence we obtain the following result:

In particular, the above result implies that Template:Math is included in Template:Math, the sumset of Template:Math and Template:Math in the space of all measurable functions. Therefore, we have the following chain of inclusions:

In practice, we often encounter operators defined on the sumset Template:Math. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps Template:Math boundedly into Template:Math, and Plancherel's theorem shows that the Fourier transform maps Template:Math boundedly into itself, hence the Fourier transform <math>\mathcal{F}</math> extends to Template:Math by setting <math display="block">\mathcal{F}(f_1+f_2) = \mathcal{F}_{L^1}(f_1) + \mathcal{F}_{L^2}(f_2)</math> for all Template:Math and Template:Math. It is therefore natural to investigate the behavior of such operators on the intermediate subspaces Template:Math.

To this end, we go back to our example and note that the Fourier transform on the sumset Template:Math was obtained by taking the sum of two instantiations of the same operator, namely <math display="block">\mathcal{F}_{L^1}:L^1(\mathbf{R}^d) \to L^\infty(\mathbf{R}^d), </math> <math display="block">\mathcal{F}_{L^2}:L^2(\mathbf{R}^d) \to L^2(\mathbf{R}^d).</math>

These really are the same operator, in the sense that they agree on the subspace Template:Math. Since the intersection contains simple functions, it is dense in both Template:Math and Template:Math. Densely defined continuous operators admit unique extensions, and so we are justified in considering <math>\mathcal{F}_{L^1}</math> and <math>\mathcal{F}_{L^2}</math> to be the same.

Therefore, the problem of studying operators on the sumset Template:Math essentially reduces to the study of operators that map two natural domain spaces, Template:Math and Template:Math, boundedly to two target spaces: Template:Math and Template:Math, respectively. Since such operators map the sumset space Template:Math to Template:Math, it is natural to expect that these operators map the intermediate space Template:Math to the corresponding intermediate space Template:Math.

Statement of the theorem

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 Template: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.

Template:Math theorem \|T\|^{\theta}_{L^{p_1} \to L^{q_1}}.</math>|Template:EquationRef}} }}

In other words, if Template:Mvar is simultaneously of type Template:Math and of type Template:Math, then Template:Mvar is of type Template:Math for all Template:Math. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of Template:Mvar is the collection of all points Template:Math in the unit square Template:Math such that Template:Mvar is of type Template:Math. The interpolation theorem states that the Riesz diagram of Template:Mvar is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram.

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 Template:Math and Template: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>

Proof

We will first prove the result for simple functions and eventually show how the argument can be extended by density to all measurable functions.

Simple functions

By symmetry, let us assume <math display="inline">p_0 < p_1</math> (the case <math display="inline">p_0 = p_1</math> trivially follows from (Template:EquationNote)). Let <math display="inline">f</math> be a simple function, that is <math display="block">f = \sum_{j=1}^m a_j \mathbf{1}_{A_j}</math> for some finite <math display="inline">m\in\mathbb{N}</math>, <math display="inline">a_j = \left\vert a_j\right\vert\mathrm{e}^{\mathrm{i}\alpha_j} \in \mathbb{C}</math> and <math display="inline">A_j\in\Sigma_1</math>, <math display="inline">j=1,2,\dots,m</math>. Similarly, let <math display="inline">g</math> denote a simple function <math display="inline">\Omega_2 \to \mathbb{C}</math>, namely <math display="block">g = \sum_{k=1}^n b_k \mathbf{1}_{B_k}</math> for some finite <math display="inline">n\in\mathbb{N}</math>, <math display="inline">b_k = \left\vert b_k\right\vert\mathrm{e}^{\mathrm{i}\beta_k} \in \mathbb{C}</math> and <math display="inline">B_k\in\Sigma_2</math>, <math display="inline">k=1,2,\dots,n</math>.

Note that, since we are assuming <math display="inline">\Omega_1</math> and <math display="inline">\Omega_2</math> to be <math display="inline">\sigma</math>-finite metric spaces, <math display="inline">f\in L^{r}(\mu_1)</math> and <math display="inline">g\in L^r(\mu_2)</math> for all <math display="inline">r \in [1, \infty]</math>. Then, by proper normalization, we can assume <math display="inline">\lVert f\rVert_{p_\theta}= 1</math> and <math display="inline">\lVert g\rVert_{q_\theta'}=1</math>, with <math display="inline">q_\theta' = q_\theta(q_\theta-1)^{-1}</math> and with <math display="inline">p_\theta</math>, <math display="inline">q_\theta</math> as defined by the theorem statement.

Next, we define the two complex functions <math display="block">\begin{aligned} u: \mathbb{C}&\to \mathbb{C}& v: \mathbb{C}&\to \mathbb{C}\\

 z &\mapsto u(z)=\frac{1-z}{p_0} + \frac{z}{p_1} &
 z &\mapsto v(z)=\frac{1-z}{q_0} + \frac{z}{q_1}.\end{aligned}</math> Note that, for <math display="inline">z=\theta</math>, <math display="inline">u(\theta) = p_\theta^{-1}</math> and <math display="inline">v(\theta) = q_\theta^{-1}</math>. We then extend <math display="inline">f</math> and <math display="inline">g</math> to depend on a complex parameter <math display="inline">z</math> as follows: <math display="block">\begin{aligned}

f_z &= \sum_{j=1}^m \left\vert a_j\right\vert^{\frac{u(z)}{u(\theta)}} \mathrm{e}^{\mathrm{i}\alpha_j} \mathbf{1}_{A_j} \\ g_z &= \sum_{k=1}^n \left\vert b_k\right\vert^{\frac{1-v(z)}{1-v(\theta)}} \mathrm{e}^{\mathrm{i} \beta_k} \mathbf{1}_{B_k}\end{aligned}</math> so that <math display="inline">f_\theta = f</math> and <math display="inline">g_\theta = g</math>. Here, we are implicitly excluding the case <math display="inline">q_0 = q_1 = 1</math>, which yields <math display="inline">v\equiv 1</math>: In that case, one can simply take <math display="inline">g_z=g</math>, independently of <math display="inline">z</math>, and the following argument will only require minor adaptations.

Let us now introduce the function <math display="block">\Phi(z) = \int_{\Omega_2} (T f_z) g_z \,\mathrm{d}\mu_2 = \sum_{j=1}^m \sum_{k=1}^n \left\vert a_j\right\vert^{\frac{u(z)}{u(\theta)}}

 \left\vert b_k\right\vert^{\frac{1-v(z)}{1-v(\theta)}} \gamma_{j,k}</math> where <math display="inline">\gamma_{j,k} = \mathrm{e}^{\mathrm{i}(\alpha_j + \beta_k)} \int_{\Omega_2} (T \mathbf{1}_{A_j})

\mathbf{1}_{B_k} \,\mathrm{d}\mu_2</math> are constants independent of <math display="inline">z</math>. We readily see that <math display="inline">\Phi(z)</math> is an entire function, bounded on the strip <math display="inline">0 \le \operatorname{\mathbb{R}e}z \le 1</math>. Then, in order to prove (Template:EquationNote), we only need to show that Template:NumBlk &&\text{and} & \left\vert\Phi(1 + \mathrm{i}y)\right\vert &\le \|T\|_{L^{p_1} \to L^{q_1}}\end{aligned}</math>|Template:EquationRef}} for all <math display="inline">f_z</math> and <math display="inline">g_z</math> as constructed above. Indeed, if (Template:EquationNote) holds true, by Hadamard three-lines theorem, <math display="block">\left\vert\Phi(\theta + \mathrm{i}0)\right\vert = \biggl\vert\int_{\Omega_2} (Tf) g \,\mathrm{d}\mu_2\biggr\vert \le \|T\|_{L^{p_0} \to L^{q_0}}^{1-\theta} \|T\|_{L^{p_1} \to L^{q_1}}^\theta</math> for all <math display="inline">f</math> and <math display="inline">g</math>. This means, by fixing <math display="inline">f</math>, that <math display="block">\sup_g \biggl\vert\int_{\Omega_2} (Tf) g \,\mathrm{d}\mu_2\biggr\vert \le \|T\|_{L^{p_0} \to L^{q_0}}^{1-\theta} \|T\|_{L^{p_1} \to L^{q_1}}^\theta</math> where the supremum is taken with respect to all <math display="inline">g</math> simple functions with <math display="inline">\lVert g\rVert_{q_\theta'} = 1</math>. The left-hand side can be rewritten by means of the following lemma.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In our case, the lemma above implies <math display="block">\lVert Tf\rVert_{q_\theta} \le \|T\|_{L^{p_0} \to L^{q_0}}^{1-\theta} \|T\|_{L^{p_1} \to L^{q_1}}^\theta</math> for all simple function <math display="inline">f</math> with <math display="inline">\lVert f\rVert_{p_\theta} = 1</math>. Equivalently, for a generic simple function, <math display="block">\lVert Tf\rVert_{q_\theta} \le \|T\|_{L^{p_0} \to L^{q_0}}^{1-\theta} \|T\|_{L^{p_1} \to L^{q_1}}^\theta \lVert f\rVert_{p_\theta}.</math>

Proof of (Template:EquationNote)

Let us now prove that our claim (Template:EquationNote) is indeed certain. The sequence <math display="inline">(A_j)_{j=1}^m</math> consists of disjoint subsets in <math display="inline">\Sigma_1</math> and, thus, each <math display="inline">\xi\in \Omega_1</math> belongs to (at most) one of them, say <math display="inline">A_{\hat{\jmath}}</math>. Then, for <math display="inline">z=\mathrm{i}y</math>, <math display="block">\begin{aligned} \left\vert f_{\mathrm{i}y}(\xi)\right\vert &= \left\vert \left\vert a_{\hat{\jmath}}\right\vert^\frac{u(\mathrm{i}y)}{u(\theta)} \right\vert \\

 &= \left\vert \exp\biggl(\log\left\vert a_{\hat{\jmath}}\right\vert\frac{p_\theta}{p_0}\biggr)
    \exp\biggl(-\mathrm{i}y \log\left\vert a_{\hat{\jmath}}\right\vert p_\theta\biggl(\frac{1}{p_0}
         - \frac{1}{p_1} \biggr) \biggr) \right\vert \\
 &= \left\vert a_{\hat{\jmath}}\right\vert^{\frac{p_\theta}{p_0}} \\
 & = \left\vert f(\xi)\right\vert^{\frac{p_\theta}{p_0}}\end{aligned}</math> which implies that <math display="inline">\lVert f_{\mathrm{i}y}\rVert_{p_0} \le

\lVert f\rVert_{p_\theta}^{\frac{p_\theta}{p_0}}</math>. With a parallel argument, each <math display="inline">\zeta \in \Omega_2</math> belongs to (at most) one of the sets supporting <math display="inline">g</math>, say <math display="inline">B_{\hat{k}}</math>, and <math display="block">\left\vert g_{\mathrm{i}y}(\zeta)\right\vert = \left\vert b_{\hat{k}}\right\vert^{\frac{1-1/q_0}{1-1/q_\theta}} = \left\vert g(\zeta)\right\vert^{\frac{1-1/q_0}{1-1/q_\theta}} = \left\vert g(\zeta)\right\vert^{\frac{q_\theta'}{q_0'}} \implies \lVert g_{\mathrm{i}y}\rVert_{q_0'} \le \lVert g\rVert_{q_\theta'}^{\frac{q_\theta'}{q_0'}}.</math>

We can now bound <math display="inline">\Phi(\mathrm{i}y)</math>: By applying Hölder’s inequality with conjugate exponents <math display="inline">q_0</math> and <math display="inline">q_0'</math>, we have <math display="block">\begin{aligned} \left\vert\Phi(\mathrm{i}y)\right\vert &\le \lVert T f_{\mathrm{i}y}\rVert_{q_0} \lVert g_{\mathrm{i}y}\rVert_{q_0'} \\

 &\le \|T\|_{L^{p_0} \to L^{q_0}} \lVert f_{\mathrm{i}y}\rVert_{p_0} \lVert g_{\mathrm{i}y}\rVert_{q_0'} \\
 &= \|T\|_{L^{p_0} \to L^{q_0}} \lVert f\rVert_{p_\theta}^{\frac{p_\theta}{p_0}}
   \lVert g\rVert_{q_\theta'}^{\frac{q_\theta'}{q_0'}} \\
 &= \|T\|_{L^{p_0} \to L^{q_0}}.\end{aligned}</math>

We can repeat the same process for <math display="inline">z=1+\mathrm{i}y</math> to obtain <math display="inline">\left\vert f_{1+\mathrm{i} y}(\xi)\right\vert = \left\vert f(\xi)\right\vert^{p_\theta/p_1}</math>, <math display="inline">\left\vert g_{1+\mathrm{i}y}(\zeta)\right\vert = \left\vert g(\zeta)\right\vert^{q_\theta'/q_1'}</math> and, finally, <math display="block">\left\vert\Phi(1+\mathrm{i}y)\right\vert \le \|T\|_{L^{p_1} \to L^{q_1}} \lVert f_{1+\mathrm{i}y}\rVert_{p_1} \lVert g_{1+\mathrm{i}y}\rVert_{q_1'} = \|T\|_{L^{p_1} \to L^{q_1}}.</math>

Extension to all measurable functions in L^p_θ

So far, we have proven that Template:NumBlk \lVert f\rVert_{p_\theta}</math>|Template:EquationRef}} when <math display="inline">f</math> is a simple function. As already mentioned, the inequality holds true for all <math display="inline">f\in L^{p_\theta}(\Omega_1)</math> by the density of simple functions in <math display="inline">L^{p_\theta}(\Omega_1)</math>.

Formally, let <math display="inline">f\in L^{p_\theta}(\Omega_1)</math> and let <math display="inline">(f_n)_n</math> be a sequence of simple functions such that <math display="inline">\left\vert f_n\right\vert \le \left\vert f\right\vert</math>, for all <math display="inline">n</math>, and <math display="inline">f_n \to f</math> pointwise. Let <math display="inline">E=\{x\in \Omega_1: \left\vert f(x)\right\vert > 1\}</math> and define <math display="inline">g = f \mathbf{1}_E</math>, <math display="inline">g_n = f_n \mathbf{1}_E</math>, <math display="inline">h = f - g = f \mathbf{1}_{E^\mathrm{c}}</math> and <math display="inline">h_n = f_n - g_n</math>. Note that, since we are assuming <math display="inline">p_0 \le p_\theta \le p_1</math>, <math display="block">\begin{aligned} \lVert f\rVert_{p_\theta}^{p_\theta} &= \int_{\Omega_1} \left\vert f\right\vert^{p_\theta} \,\mathrm{d}\mu_1 \ge \int_{\Omega_1} \left\vert f\right\vert^{p_\theta} \mathbf{1}_{E} \,\mathrm{d}\mu_1 \ge \int_{\Omega_1} \left\vert f \mathbf{1}_{E}\right\vert^{p_0} \,\mathrm{d}\mu_1 = \int_{\Omega_1} \left\vert g\right\vert^{p_0} \,\mathrm{d}\mu_1 = \lVert g\rVert_{p_0}^{p_0} \\ \lVert f\rVert_{p_\theta}^{p_\theta} &= \int_{\Omega_1} \left\vert f\right\vert^{p_\theta} \,\mathrm{d}\mu_1 \ge \int_{\Omega_1} \left\vert f\right\vert^{p_\theta} \mathbf{1}_{E^\mathrm{c}} \,\mathrm{d}\mu_1 \ge \int_{\Omega_1} \left\vert f \mathbf{1}_{E^\mathrm{c}}\right\vert^{p_1} \,\mathrm{d}\mu_1 = \int_{\Omega_1} \left\vert h\right\vert^{p_1} \,\mathrm{d}\mu_1 = \lVert h\rVert_{p_1}^{p_1}\end{aligned}</math> and, equivalently, <math display="inline">g\in L^{p_0}(\Omega_1)</math> and <math display="inline">h\in L^{p_1}(\Omega_1)</math>.

Let us see what happens in the limit for <math display="inline">n\to\infty</math>. Since <math display="inline">\left\vert f_n\right\vert \le \left\vert f\right\vert</math>, <math display="inline">\left\vert g_n\right\vert \le \left\vert g\right\vert</math> and <math display="inline">\left\vert h_n\right\vert \le \left\vert h\right\vert</math>, by the dominated convergence theorem one readily has <math display="block">\begin{aligned} \lVert f_n\rVert_{p_\theta} &\to \lVert f\rVert_{p_\theta} & \lVert g_n\rVert_{p_0} &\to \lVert g\rVert_{p_0} & \lVert h_n\rVert_{p_1} &\to \lVert h\rVert_{p_1}.\end{aligned}</math> Similarly, <math display="inline">\left\vert f - f_n\right\vert \le 2\left\vert f\right\vert</math>, <math display="inline">\left\vert g-g_n\right\vert \le 2\left\vert g\right\vert</math> and <math display="inline">\left\vert h - h_n\right\vert \le 2\left\vert h\right\vert</math> imply <math display="block">\begin{aligned} \lVert f - f_n\rVert_{p_\theta} &\to 0 & \lVert g - g_n\rVert_{p_0} &\to 0 & \lVert h - h_n\rVert_{p_1} &\to 0\end{aligned}</math> and, by the linearity of <math display="inline">T</math> as an operator of types <math display="inline">(p_0, q_0)</math> and <math display="inline">(p_1, q_1)</math> (we have not proven yet that it is of type <math display="inline">(p_\theta, q_\theta)</math> for a generic <math display="inline">f</math>) <math display="block">\begin{aligned} \lVert Tg - Tg_n\rVert_{p_0} & \le \|T\|_{L^{p_0} \to L^{q_0}} \lVert g - g_n\rVert_{p_0} \to 0 & \lVert Th - Th_n\rVert_{p_1} & \le \|T\|_{L^{p_1} \to L^{q_1}} \lVert h - h_n\rVert_{p_1} \to 0.\end{aligned}</math>

It is now easy to prove that <math display="inline">Tg_n \to Tg</math> and <math display="inline">Th_n \to Th</math> in measure: For any <math display="inline">\epsilon > 0</math>, Chebyshev’s inequality yields <math display="block">\mu_2(y\in \Omega_2: \left\vert Tg - Tg_n\right\vert > \epsilon) \le \frac{\lVert Tg - Tg_n\rVert_{q_0}^{q_0}} {\epsilon^{q_0}}</math> and similarly for <math display="inline">Th - Th_n</math>. Then, <math display="inline">Tg_n \to Tg</math> and <math display="inline">Th_n \to Th</math> a.e. for some subsequence and, in turn, <math display="inline">Tf_n \to Tf</math> a.e. Then, by Fatou’s lemma and recalling that (Template:EquationNote) holds true for simple functions, <math display="block">\lVert Tf\rVert_{q_\theta} \le \liminf_{n\to\infty} \lVert T f_n\rVert_{q_\theta} \le \|T\|_{L^{p_\theta} \to L^{q_\theta}} \liminf_{n\to\infty} \lVert f_n\rVert_{p_\theta} = \|T\|_{L^{p_\theta} \to L^{q_\theta}} \lVert f\rVert_{p_\theta}.</math>

Interpolation of analytic families of operators

The proof outline presented in the above section readily generalizes to the case in which the operator Template:Mvar is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function <math display="block">\varphi(z) = \int (T_z f_z)g_z \, d\mu_2,</math> 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 Template:Mvar 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 a blog post of Tao for a high-level exposition of the theorem.</ref>

The theory of real Hardy spaces and the space of bounded mean oscillations permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space Template:Math and the space Template:Math of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.<ref>Fefferman and Stein (1972)</ref>

Applications

Hausdorff–Young inequality

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} It has been shown in the first section that the Fourier transform <math>\mathcal{F}</math> maps Template:Math boundedly into Template:Math and Template:Math into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions Template:Math into functions <math>\hat{f}:\mathbf{Z} \to \mathbf{C}</math> whose values are the Fourier coefficients <math display="block">\hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx ,</math> maps Template:Math boundedly into Template:Math and Template:Math into Template:Math. The Riesz–Thorin interpolation theorem now implies the following: <math display="block">\begin{align} \left \|\mathcal{F}f \right \|_{L^{q}(\mathbf{R}^d)} &\leq \|f\|_{L^p(\mathbf{R}^d)} \\ \left \|\hat{f} \right \|_{\ell^{q}(\mathbf{Z})} &\leq \|f\|_{L^p(\mathbf{T})} \end{align}</math> where Template:Math and Template:Math. This is the Hausdorff–Young inequality.

The Hausdorff–Young inequality can also be established for the Fourier transform on locally compact Abelian groups. The norm estimate of 1 is not optimal. See the main article for references.

Convolution operators

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let Template:Math be a fixed integrable function and let Template:Mvar be the operator of convolution with Template:Math, i.e., for each function Template:Mvar we have Template:Math.

It follows from Fubini's theorem that Template:Mvar is bounded from Template:Math to Template:Math and it is trivial that it is bounded from Template:Math to Template:Math (both bounds are by Template:Math). Therefore the Riesz–Thorin theorem gives <math display="block">\| f * g \|_p \leq \|f\|_1 \|g\|_p.</math>

We take this inequality and switch the role of the operator and the operand, or in other words, we think of Template:Mvar as the operator of convolution with Template:Mvar, and get that Template:Mvar is bounded from Template:Math to L^p. Further, since Template:Mvar is in Template:Math we get, in view of Hölder's inequality, that Template:Mvar is bounded from Template:Math to Template:Math, where again Template:Math. So interpolating we get <math display="block">\|f*g\|_s\leq \|f\|_r\|g\|_p</math> where the connection between p, r and s is <math display="block">\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.</math>

The Hilbert transform

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The Hilbert transform of Template:Math is given by <math display="block"> \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> where p.v. indicates the Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier: <math display="block"> \widehat{\mathcal{H}f}(\xi) = -i \, \sgn(\xi) \hat{f}(\xi).</math>

It follows from the Plancherel theorem that the Hilbert transform maps Template:Math boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on Template:Math or Template:Math, and so we cannot use the Riesz–Thorin interpolation theorem directly. To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions Template:Math and Template:Math. We can show, however, that <math display="block">(\mathcal{H}f)^2 = f^2 + 2\mathcal{H}(f\mathcal{H}f)</math> for all Schwartz functions Template:Math, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps Template:Math boundedly into itself for all Template:Math. Interpolation now establishes the bound <math display="block"> \|\mathcal{H}f\|_p \leq A_p \|f\|_p</math> for all Template:Math, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the Template:Math case.

Comparison with the real interpolation method

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 Template: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 operators, do not have good endpoint estimates.<ref>Elias Stein is quoted for saying that interesting operators in harmonic analysis are rarely bounded on Template:Math and Template: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 <math display="block"> \mu \left( \{x : Tf(x) > \alpha \} \right) \leq \left( \frac{C_{p,q} \|f\|_p}{\alpha} \right)^q,</math> 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. 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 spaces and do not necessarily produce norm estimates on the Template:Math-spaces.

Mityagin's theorem

B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of spaces of sequences with unconditional bases (cf. below).

Assume: <math display="block">\|A\|_{\ell_1 \to \ell_1}, \|A\|_{\ell_\infty \to \ell_\infty} \leq M.</math>

Then <math display="block">\|A\|_{X \to X} \leq M</math>

for any unconditional Banach space of sequences Template:Mvar, that is, for any <math>(x_i) \in X</math> and any <math>(\varepsilon_i) \in \{-1, 1 \}^\infty</math>, <math>\| (\varepsilon_i x_i) \|_X = \| (x_i) \|_X </math>.

The proof is based on the Krein–Milman theorem.

See also

• Marcinkiewicz interpolation theorem
• Interpolation space

References

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• Template:Citation. Translated from the Russian and edited by G. P. Barker and G. Kuerti.
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External links

• Template:Springer