Riesz–Thorin theorem

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 spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and . 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 be two numbers such that . Then for define by: .

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

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

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

In particular, the above result implies that is included in , the sumset of and 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 . For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps boundedly into , and Plancherel's theorem shows that the Fourier transform maps boundedly into itself, hence the Fourier transform <math>\mathcal{F}</math> extends to 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 and . It is therefore natural to investigate the behavior of such operators on the intermediate subspaces .

To this end, we go back to our example and note that the Fourier transform on the sumset 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 . Since the intersection contains simple functions, it is dense in both and . 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 essentially reduces to the study of operators that map two natural domain spaces, and , boundedly to two target spaces: and , respectively. Since such operators map the sumset space to , it is natural to expect that these operators map the intermediate space to the corresponding intermediate space .

Statement of the theorem

There are several ways to state the Riesz–Thorin interpolation theorem; to be consistent with the notations in the previous section, we shall use the sumset formulation.

\|T\|^{\theta}_{L^{p_1} \to L^{q_1.</math>|

In other words, if is simultaneously of type and of type , then is of type for all . In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of is the collection of all points in the unit square such that is of type . The interpolation theorem states that the Riesz diagram of 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. The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that and . 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.

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 ()). 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 (), we only need to show that &&\text{and} & \left\vert\Phi(1 + \mathrm{i}y)\right\vert &\le \|T\|_{L^{p_1} \to L^{q_1\end{aligned}</math>| for all <math display="inline">f_z</math> and <math display="inline">g_z</math> as constructed above. Indeed, if () 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.

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 ()

Let us now prove that our claim () 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 Lpθ

So far, we have proven that \lVert f\rVert_{p_\theta}</math>| 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 () 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 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:

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 and the space of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.

Applications

Hausdorff–Young inequality

It has been shown in the first section that the Fourier transform <math>\mathcal{F}</math> maps boundedly into and into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions into functions <math>\hat{f}:\mathbf{Z} \to \mathbf{C}</math> whose values are the Fourier coefficients

maps boundedly into and into . 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 and . 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

Let be a fixed integrable function and let be the operator of convolution with , i.e., for each function we have .

It follows from Fubini's theorem that is bounded from to and it is trivial that it is bounded from to (both bounds are by ). Therefore the Riesz–Thorin theorem gives

We take this inequality and switch the role of the operator and the operand, or in other words, we think of as the operator of convolution with , and get that is bounded from to Lp. Further, since is in we get, in view of Hölder's inequality, that is bounded from to , where again . So interpolating we get

where the connection between p, r and s is

The Hilbert transform

The Hilbert transform of 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 boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on or , 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 and . 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 , and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps boundedly into itself for all . Interpolation now establishes the bound

<math display="block"> \|\mathcal{H}f\|_p \leq A_p \|f\|_p</math>

for all , and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the 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 . 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. 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 -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

for any unconditional Banach space of sequences , 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>.