Riesz–Thorin theorem

Template:Short description 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 L^p spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L² which is a Hilbert space, or to L¹ and L^∞. 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 p₀, p₁ be two numbers such that 1 ≤ p₀ < p₁ ≤ ∞. Then for 0 < θ < 1 define p_θ by:  Template:Sfrac = Template:Sfrac + Template:Sfrac.

By splitting up the function  f  in L^p_θ as the product | f | = | f |^1−θ | f |^θ and applying Hölder's inequality to its p_θ power, we obtain the following result, foundational in the study of L^p-spaces:

This result, whose name derives from the convexity of the map 1⁄p ↦ log || f ||_p on [0, ∞], implies that L^p₀ ∩ L^p₁ ⊂ L^p_θ.

On the other hand, if we take the layer-cake decomposition  f  =  f 1_{Template:Mset} +  f 1_{Template:Mset}, then we see that  f 1_{Template:Mset} ∈ L^p₀ and  f 1_{Template:Mset} ∈ L^p₁, whence we obtain the following result:

In particular, the above result implies that L^p_θ is included in L^p₀ + L^p₁, the sumset of L^p₀ and L^p₁ 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 L^p₀ + L^p₁. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps L¹(R^d) boundedly into L^∞(R^d), and Plancherel's theorem shows that the Fourier transform maps L²(R^d) boundedly into itself, hence the Fourier transform ${ℱ}$ extends to (L¹ + L²) (R^d) by setting $${ℱ}(f_1+f_2)={{ℱ}}_{L^1}(f_1)+{{ℱ}}_{L^2}(f_2)$$ for all  f₁  ∈ L¹(R^d) and  f₂  ∈ L²(R^d). It is therefore natural to investigate the behavior of such operators on the intermediate subspaces L^p_θ.

To this end, we go back to our example and note that the Fourier transform on the sumset L¹ + L² was obtained by taking the sum of two instantiations of the same operator, namely $${{ℱ}}_{L^1}:L^1({\mathbf{𝐑}}^d)\to L^{\mathrm{\infty }}({\mathbf{𝐑}}^d),$$ $${{ℱ}}_{L^2}:L^2({\mathbf{𝐑}}^d)\to L^2({\mathbf{𝐑}}^d).$$

These really are the same operator, in the sense that they agree on the subspace (L¹ ∩ L²) (R^d). Since the intersection contains simple functions, it is dense in both L¹(R^d) and L²(R^d). Densely defined continuous operators admit unique extensions, and so we are justified in considering ${{ℱ}}_{L^1}$ and ${{ℱ}}_{L^2}$ to be the same.

Therefore, the problem of studying operators on the sumset L^p₀ + L^p₁ essentially reduces to the study of operators that map two natural domain spaces, L^p₀ and L^p₁, boundedly to two target spaces: L^q₀ and L^q₁, respectively. Since such operators map the sumset space L^p₀ + L^p₁ to L^q₀ + L^q₁, it is natural to expect that these operators map the intermediate space L^p_θ to the corresponding intermediate space L^q_θ.

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.

In other words, if Template:Mvar is simultaneously of type (p₀, q₀) and of type (p₁, q₁), then Template:Mvar is of type (p_θ, q_θ) for all 0 < θ < 1. 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:Sfrac, Template:Sfrac) in the unit square [0, 1] × [0, 1] such that Template:Mvar is of type (p, q). 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. The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that p₀ ≤ q₀ and p₁ ≤ q₁. 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 $p_0<p_1$ (the case $p_0=p_1$ trivially follows from (1)). Let $f$ be a simple function, that is $$f=\sum _{j=1}^m{a}_j{\mathbf{𝟏}}_{A_j}$$ for some finite $m\in \mathbb{\mathbb{N}}$, $a_j=\left|a_j\right|{\mathrm{e}}^{\mathrm{i}{\alpha }_j}\in \mathbb{ℂ}$ and $A_j\in {\mathrm{\Sigma }}_1$, $j=1,2,\dots ,m$. Similarly, let $g$ denote a simple function ${\mathrm{\Omega }}_2\to \mathbb{ℂ}$, namely $$g=\sum _{k=1}^n{b}_k{\mathbf{𝟏}}_{B_k}$$ for some finite $n\in \mathbb{\mathbb{N}}$, $b_k=\left|b_k\right|{\mathrm{e}}^{\mathrm{i}{\beta }_k}\in \mathbb{ℂ}$ and $B_k\in {\mathrm{\Sigma }}_2$, $k=1,2,\dots ,n$.

Note that, since we are assuming ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ to be $\sigma$-finite metric spaces, $f\in L^r({\mu }_1)$ and $g\in L^r({\mu }_2)$ for all $r\in [1,\mathrm{\infty }]$. Then, by proper normalization, we can assume $\Vert f{\Vert }_{p_{\theta }}=1$ and $\Vert g{\Vert }_{{q_{\theta }}^{\prime }}=1$, with ${q_{\theta }}^{\prime }=q_{\theta }(q_{\theta }-1{)}^{-1}$ and with $p_{\theta }$, $q_{\theta }$ as defined by the theorem statement.

Next, we define the two complex functions $$\begin{array}{rlrl}u:\mathbb{ℂ}& \to \mathbb{ℂ}& v:\mathbb{ℂ}& \to \mathbb{ℂ}\\ z& \mapsto u(z)=\frac{1-z}{p_0}+\frac{z}{p_1}& z& \mapsto v(z)=\frac{1-z}{q_0}+\frac{z}{q_1}.\end{array}$$ Note that, for $z=\theta$, $u(\theta )=p_{\theta }^{-1}$ and $v(\theta )=q_{\theta }^{-1}$. We then extend $f$ and $g$ to depend on a complex parameter $z$ as follows: $$\begin{array}{rl}{f}_z& =\sum _{j=1}^m{\left|a_j\right|}^{\frac{u(z)}{u(\theta )}}{\mathrm{e}}^{\mathrm{i}{\alpha }_j}{\mathbf{𝟏}}_{A_j}\\ g_z& =\sum _{k=1}^n{\left|b_k\right|}^{\frac{1-v(z)}{1-v(\theta )}}{\mathrm{e}}^{\mathrm{i}{\beta }_k}{\mathbf{𝟏}}_{B_k}\end{array}$$ so that $f_{\theta }=f$ and $g_{\theta }=g$. Here, we are implicitly excluding the case $q_0=q_1=1$, which yields $v\equiv 1$: In that case, one can simply take $g_z=g$, independently of $z$, and the following argument will only require minor adaptations.

Let us now introduce the function $$\mathrm{\Phi }(z)=\underset{{\mathrm{\Omega }}_2}{\int }(T{f}_z)g_z\mathrm{d}{\mu }_2=\sum _{j=1}^m\sum _{k=1}^n{\left|a_j\right|}^{\frac{u(z)}{u(\theta )}}{\left|b_k\right|}^{\frac{1-v(z)}{1-v(\theta )}}{\gamma }_{j,k}$$ where ${\gamma }_{j,k}={\mathrm{e}}^{\mathrm{i}({\alpha }_j+{\beta }_k)}\underset{{\mathrm{\Omega }}_2}{\int }(T{\mathbf{𝟏}}_{A_j}){\mathbf{𝟏}}_{B_k}\mathrm{d}{\mu }_2$ are constants independent of $z$. We readily see that $\mathrm{\Phi }(z)$ is an entire function, bounded on the strip $0\le \mathbb{ℝ}\mathrm{e}z\le 1$. Then, in order to prove (2), we only need to show that Template:NumBlk for all $f_z$ and $g_z$ as constructed above. Indeed, if (3) holds true, by Hadamard three-lines theorem, $$|\mathrm{\Phi }(\theta +\mathrm{i}0)|=|\underset{{\mathrm{\Omega }}_2}{\int }(Tf)g\mathrm{d}{\mu }_2|\le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}^{1-\theta }\Vert T{\Vert }_{L^{p_1}\to L^{q_1}}^{\theta }$$ for all $f$ and $g$. This means, by fixing $f$, that $$\underset{g}{\sup }|\underset{{\mathrm{\Omega }}_2}{\int }(Tf)g\mathrm{d}{\mu }_2|\le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}^{1-\theta }\Vert T{\Vert }_{L^{p_1}\to L^{q_1}}^{\theta }$$ where the supremum is taken with respect to all $g$ simple functions with $\Vert g{\Vert }_{{q_{\theta }}^{\prime }}=1$. The left-hand side can be rewritten by means of the following lemma.

In our case, the lemma above implies $$\Vert Tf{\Vert }_{q_{\theta }}\le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}^{1-\theta }\Vert T{\Vert }_{L^{p_1}\to L^{q_1}}^{\theta }$$ for all simple function $f$ with $\Vert f{\Vert }_{p_{\theta }}=1$. Equivalently, for a generic simple function, $$\Vert Tf{\Vert }_{q_{\theta }}\le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}^{1-\theta }\Vert T{\Vert }_{L^{p_1}\to L^{q_1}}^{\theta }\Vert f{\Vert }_{p_{\theta }}.$$

Proof of (3)

Let us now prove that our claim (3) is indeed certain. The sequence $(A_j{)}_{j=1}^m$ consists of disjoint subsets in ${\mathrm{\Sigma }}_1$ and, thus, each $\xi \in {\mathrm{\Omega }}_1$ belongs to (at most) one of them, say $A_{\hat{ȷ}}$. Then, for $z=\mathrm{i}y$, $$\begin{array}{rl}|f_{\mathrm{i}y}(\xi )|& =\left|{\left|a_{\hat{ȷ}}\right|}^{\frac{u(\mathrm{i}y)}{u(\theta )}}\right|\\ & =|\exp (\log \left|a_{\hat{ȷ}}\right|\frac{p_{\theta }}{p_0})\exp (-\mathrm{i}y\log \left|a_{\hat{ȷ}}\right|p_{\theta }(\frac{1}{p_0}-\frac{1}{p_1})\left)\right|\\ & ={\left|a_{\hat{ȷ}}\right|}^{\frac{p_{\theta }}{p_0}}\\ & ={|f(\xi )|}^{\frac{p_{\theta }}{p_0}}\end{array}$$ which implies that $\Vert f_{\mathrm{i}y}{\Vert }_{p_0}\le \Vert f{\Vert }_{p_{\theta }}^{\frac{p_{\theta }}{p_0}}$. With a parallel argument, each $\zeta \in {\mathrm{\Omega }}_2$ belongs to (at most) one of the sets supporting $g$, say $B_{\hat{k}}$, and $$|g_{\mathrm{i}y}(\zeta )|={\left|b_{\hat{k}}\right|}^{\frac{1-1/q_0}{1-1/q_{\theta }}}={|g(\zeta )|}^{\frac{1-1/q_0}{1-1/q_{\theta }}}={|g(\zeta )|}^{\frac{{q_{\theta }}^{\prime }}{{q_0}^{\prime }}}⟹\Vert g_{\mathrm{i}y}{\Vert }_{{q_0}^{\prime }}\le \Vert g\underset{{q_{\theta }}^{\prime }}{\overset{\frac{{q_{\theta }}^{\prime }}{{q_0}^{\prime }}}{\Vert }}.$$

We can now bound $\mathrm{\Phi }(\mathrm{i}y)$: By applying Hölder’s inequality with conjugate exponents $q_0$ and ${q_0}^{\prime }$, we have $$\begin{array}{rl}|\mathrm{\Phi }(\mathrm{i}y)|& \le \Vert T{f}_{\mathrm{i}y}{\Vert }_{q_0}\Vert g_{\mathrm{i}y}{\Vert }_{{q_0}^{\prime }}\\ & \le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert f_{\mathrm{i}y}{\Vert }_{p_0}\Vert g_{\mathrm{i}y}{\Vert }_{{q_0}^{\prime }}\\ & =\Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert f\underset{p_{\theta }}{\overset{\frac{p_{\theta }}{p_0}}{\Vert }}\Vert g\underset{{q_{\theta }}^{\prime }}{\overset{\frac{{q_{\theta }}^{\prime }}{{q_0}^{\prime }}}{\Vert }}\\ & =\Vert T{\Vert }_{L^{p_0}\to L^{q_0}}.\end{array}$$

We can repeat the same process for $z=1+\mathrm{i}y$ to obtain $|f_{1+\mathrm{i}y}(\xi )|={|f(\xi )|}^{p_{\theta }/p_1}$, $|g_{1+\mathrm{i}y}(\zeta )|={|g(\zeta )|}^{{q_{\theta }}^{\prime }/{q_1}^{\prime }}$ and, finally, $$|\mathrm{\Phi }(1+\mathrm{i}y)|\le \Vert T{\Vert }_{L^{p_1}\to L^{q_1}}\Vert f_{1+\mathrm{i}y}{\Vert }_{p_1}\Vert g_{1+\mathrm{i}y}{\Vert }_{{q_1}^{\prime }}=\Vert T{\Vert }_{L^{p_1}\to L^{q_1}}.$$

Extension to all measurable functions in L^p_θ

So far, we have proven that Template:NumBlk when $f$ is a simple function. As already mentioned, the inequality holds true for all $f\in L^{p_{\theta }}({\mathrm{\Omega }}_1)$ by the density of simple functions in $L^{p_{\theta }}({\mathrm{\Omega }}_1)$.

Formally, let $f\in L^{p_{\theta }}({\mathrm{\Omega }}_1)$ and let $(f_n{)}_n$ be a sequence of simple functions such that $\left|f_n\right|\le \left|f\right|$, for all $n$, and $f_n\to f$ pointwise. Let $E=\{x\in {\mathrm{\Omega }}_1:|f(x)|>1\}$ and define $g=f{\mathbf{𝟏}}_E$, $g_n=f_n{\mathbf{𝟏}}_E$, $h=f-g=f{\mathbf{𝟏}}_{E^{\mathrm{c}}}$ and $h_n=f_n-g_n$. Note that, since we are assuming $p_0\le p_{\theta }\le p_1$, $$\begin{array}{rl}\Vert f\underset{p_{\theta }}{\overset{p_{\theta }}{\Vert }}& =\underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}{\mathbf{𝟏}}_E\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f{\mathbf{𝟏}}_E\right|}^{p_0}\mathrm{d}{\mu }_1=\underset{{\mathrm{\Omega }}_1}{\int }{\left|g\right|}^{p_0}\mathrm{d}{\mu }_1=\Vert g\underset{p_0}{\overset{p_0}{\Vert }}\\ \Vert f\underset{p_{\theta }}{\overset{p_{\theta }}{\Vert }}& =\underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}{\mathbf{𝟏}}_{E^{\mathrm{c}}}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f{\mathbf{𝟏}}_{E^{\mathrm{c}}}\right|}^{p_1}\mathrm{d}{\mu }_1=\underset{{\mathrm{\Omega }}_1}{\int }{\left|h\right|}^{p_1}\mathrm{d}{\mu }_1=\Vert h\underset{p_1}{\overset{p_1}{\Vert }}\end{array}$$ and, equivalently, $g\in L^{p_0}({\mathrm{\Omega }}_1)$ and $h\in L^{p_1}({\mathrm{\Omega }}_1)$.

Let us see what happens in the limit for $n\to \mathrm{\infty }$. Since $\left|f_n\right|\le \left|f\right|$, $\left|g_n\right|\le \left|g\right|$ and $\left|h_n\right|\le \left|h\right|$, by the dominated convergence theorem one readily has $$\begin{array}{rlrlrl}\Vert f_n{\Vert }_{p_{\theta }}& \to \Vert f{\Vert }_{p_{\theta }}& \Vert g_n{\Vert }_{p_0}& \to \Vert g{\Vert }_{p_0}& \Vert h_n{\Vert }_{p_1}& \to \Vert h{\Vert }_{p_1}.\end{array}$$ Similarly, $|f-f_n|\le 2\left|f\right|$, $|g-g_n|\le 2\left|g\right|$ and $|h-h_n|\le 2\left|h\right|$ imply $$\begin{array}{rlrlrl}\Vert f-f_n{\Vert }_{p_{\theta }}& \to 0& \Vert g-g_n{\Vert }_{p_0}& \to 0& \Vert h-h_n{\Vert }_{p_1}& \to 0\end{array}$$ and, by the linearity of $T$ as an operator of types $(p_0,q_0)$ and $(p_1,q_1)$ (we have not proven yet that it is of type $(p_{\theta },q_{\theta })$ for a generic $f$) $$\begin{array}{rlrl}\Vert Tg-T{g}_n{\Vert }_{p_0}& \le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert g-g_n{\Vert }_{p_0}\to 0& \Vert Th-T{h}_n{\Vert }_{p_1}& \le \Vert T{\Vert }_{L^{p_1}\to L^{q_1}}\Vert h-h_n{\Vert }_{p_1}\to 0.\end{array}$$

It is now easy to prove that $T{g}_n\to Tg$ and $T{h}_n\to Th$ in measure: For any $ϵ>0$, Chebyshev’s inequality yields $${\mu }_2(y\in {\mathrm{\Omega }}_2:|Tg-T{g}_n|>ϵ)\le \frac{\Vert Tg-T{g}_n\underset{q_0}{\overset{q_0}{\Vert }}}{{ϵ}^{q_0}}$$ and similarly for $Th-T{h}_n$. Then, $T{g}_n\to Tg$ and $T{h}_n\to Th$ a.e. for some subsequence and, in turn, $T{f}_n\to Tf$ a.e. Then, by Fatou’s lemma and recalling that (4) holds true for simple functions, $$\Vert Tf{\Vert }_{q_{\theta }}\le \underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}\Vert T{f}_n{\Vert }_{q_{\theta }}\le \Vert T{\Vert }_{L^{p_{\theta }}\to L^{q_{\theta }}}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}\Vert f_n{\Vert }_{p_{\theta }}=\Vert T{\Vert }_{L^{p_{\theta }}\to L^{q_{\theta }}}\Vert f{\Vert }_{p_{\theta }}.$$

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 $$\phi (z)=\int (T_z{f}_z)g_{z}d{\mu }_2,$$ 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 H¹(R^d) and the space BMO 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 ${ℱ}$ maps L¹(R^d) boundedly into L^∞(R^d) and L²(R^d) into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions  f  : T → C into functions $\hat{f}:\mathbf{𝐙}\to \mathbf{𝐂}$ whose values are the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)e^{-inx}dx,$$ maps L¹(T) boundedly into ℓ^∞(Z) and L²(T) into ℓ²(Z). The Riesz–Thorin interpolation theorem now implies the following: $$\begin{array}{rl}{\Vert {ℱ}f\Vert }_{L^q({\mathbf{𝐑}}^d)}& \le \Vert f{\Vert }_{L^p({\mathbf{𝐑}}^d)}\\ {\Vert \hat{f}\Vert }_{{\ell }^q(\mathbf{𝐙})}& \le \Vert f{\Vert }_{L^p(\mathbf{𝐓})}\end{array}$$ where 1 ≤ p ≤ 2 and Template:Sfrac + Template:Sfrac = 1. 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  f  be a fixed integrable function and let Template:Mvar be the operator of convolution with  f , i.e., for each function Template:Mvar we have Tg =  f  ∗ g.

It follows from Fubini's theorem that Template:Mvar is bounded from L¹ to L¹ and it is trivial that it is bounded from L^∞ to L^∞ (both bounds are by || f ||₁). Therefore the Riesz–Thorin theorem gives $$\Vert f*g{\Vert }_p\le \Vert f{\Vert }_1\Vert g{\Vert }_p.$$

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 L¹ to L^p. Further, since Template:Mvar is in L^p we get, in view of Hölder's inequality, that Template:Mvar is bounded from L^q to L^∞, where again Template:Sfrac + Template:Sfrac = 1. So interpolating we get $$\Vert f*g{\Vert }_s\le \Vert f{\Vert }_r\Vert g{\Vert }_p$$ where the connection between p, r and s is $$\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.$$

The Hilbert transform

The Hilbert transform of  f  : R → C is given by $${\mathscr{H}}f(x)=\frac{1}{\pi }\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{f(x-t)}{t}dt=(\frac{1}{\pi }\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}\frac{1}{t}\ast f)(x),$$ where p.v. indicates the Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier: $$\widehat{{\mathscr{H}}f}(\xi )=-i\mathrm{sgn}(\xi )\hat{f}(\xi ).$$

It follows from the Plancherel theorem that the Hilbert transform maps L²(R) boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on L¹(R) or L^∞(R), 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 1_(−1,1)(x) and 1_(0,1)(x) − 1_(0,1)(−x). We can show, however, that $$({\mathscr{H}}f{)}^2=f^2+2{\mathscr{H}}(f{\mathscr{H}}f)$$ for all Schwartz functions  f  : R → C, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps L²ⁿ(R^d) boundedly into itself for all n ≥ 2. Interpolation now establishes the bound $$\Vert {\mathscr{H}}f{\Vert }_p\le A_p\Vert f{\Vert }_p$$ for all 2 ≤ p < ∞, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the 1 < p ≤ 2 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 C. 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 $$\mu (\{x:Tf(x)>\alpha \})\le {\left(\frac{C_{p,q}\Vert f{\Vert }_p}{\alpha }\right)}^q,$$ 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 L^p-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: $$\Vert A{\Vert }_{{\ell }_1\to {\ell }_1},\Vert A{\Vert }_{{\ell }_{\mathrm{\infty }}\to {\ell }_{\mathrm{\infty }}}\le M.$$

Then $$\Vert A{\Vert }_{X\to X}\le M$$

for any unconditional Banach space of sequences Template:Mvar, that is, for any $(x_i)\in X$ and any $({\epsilon }_i)\in \{-1,1{\}}^{\mathrm{\infty }}$, $\Vert ({\epsilon }_i{x}_i){\Vert }_X=\Vert (x_i){\Vert }_X$.

The proof is based on the Krein–Milman theorem.

See also

• Marcinkiewicz interpolation theorem
• Interpolation space

References

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

• Template:Springer