Riesz–Thorin theorem

Template:Short description Template:About 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.

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 $ℱ$ extends to Template:Math by setting $$ℱ(f_1+f_2)={ℱ}_{L^1}(f_1)+{ℱ}_{L^2}(f_2)$$ 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 $${ℱ}_{L^1}:L^1({𝐑}^d)\to L^{\mathrm{\infty }}({𝐑}^d),$$ $${ℱ}_{L^2}:L^2({𝐑}^d)\to L^2({𝐑}^d).$$

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 ${ℱ}_{L^1}$ and ${ℱ}_{L^2}$ 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; 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 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. 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.

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 (Template:EquationNote)). Let $f$ be a simple function, that is $$f=\sum _{j=1}^m{a}_j{\mathrm{𝟏}}_{A_j}$$ for some finite $m\in \mathbb{N}$, $a_j=\left|a_j\right|{\mathrm{e}}^{\mathrm{i}{\alpha }_j}\in ℂ$ and $A_j\in {\mathrm{\Sigma }}_1$, $j=1,2,\dots ,m$. Similarly, let $g$ denote a simple function ${\mathrm{\Omega }}_2\to ℂ$, namely $$g=\sum _{k=1}^n{b}_k{\mathrm{𝟏}}_{B_k}$$ for some finite $n\in \mathbb{N}$, $b_k=\left|b_k\right|{\mathrm{e}}^{\mathrm{i}{\beta }_k}\in ℂ$ 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 $$\begin{array}{cccc}u:ℂ& \to ℂ& v:ℂ& \to ℂ\\ z& \mapsto u(z)=\frac{1-z}{p_0}+\frac{z}{p_1}& z& \mapsto v(z)=\frac{1-z}{q_0}+\frac{z}{q_1}.\end{array}$$ 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: $$\begin{array}{cc}{f}_z& =\sum _{j=1}^m{\left|a_j\right|}^{\frac{u(z)}{u(\theta )}}{\mathrm{e}}^{\mathrm{i}{\alpha }_j}{\mathrm{𝟏}}_{A_j}\\ g_z& =\sum _{k=1}^n{\left|b_k\right|}^{\frac{1-v(z)}{1-v(\theta )}}{\mathrm{e}}^{\mathrm{i}{\beta }_k}{\mathrm{𝟏}}_{B_k}\end{array}$$ 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 ${\gamma }_{j,k}={\mathrm{e}}^{\mathrm{i}({\alpha }_j+{\beta }_k)}\underset{{\mathrm{\Omega }}_2}{\int }(T{\mathrm{𝟏}}_{A_j}){\mathrm{𝟏}}_{B_k}\mathrm{d}{\mu }_2$ are constants independent of $z$. We readily see that $\mathrm{\Phi }(z)$ is an entire function, bounded on the strip $0\le ℝ\mathrm{e}z\le 1$. Then, in order to prove (Template:EquationNote), we only need to show that Template:NumBlk for all $f_z$ and $g_z$ as constructed above. Indeed, if (Template:EquationNote) 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 (Template:EquationNote)

Let us now prove that our claim (Template:EquationNote) 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 $z=\mathrm{i}y$, $$\begin{array}{cc}|f_{\mathrm{i}y}(\xi )|& =\left|{\left|a_{\hat{ȷ}}\right|}^{\frac{u(\mathrm{i}y)}{u(\theta )}}\right|\\ & =|\exp (\log \left|a_{\hat{ȷ}}\right|\frac{p_{\theta }}{p_0})\exp (-\mathrm{i}y\log \left|a_{\hat{ȷ}}\right|p_{\theta }(\frac{1}{p_0}-\frac{1}{p_1})\left)\right|\\ & ={\left|a_{\hat{ȷ}}\right|}^{\frac{p_{\theta }}{p_0}}\\ & ={|f(\xi )|}^{\frac{p_{\theta }}{p_0}}\end{array}$$ 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 $$\begin{array}{cc}|\mathrm{\Phi }(\mathrm{i}y)|& \le \Vert T{f}_{\mathrm{i}y}{\Vert }_{q_0}\Vert g_{\mathrm{i}y}{\Vert }_{{q_0}^{\prime }}\\ & \le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert f_{\mathrm{i}y}{\Vert }_{p_0}\Vert g_{\mathrm{i}y}{\Vert }_{{q_0}^{\prime }}\\ & =\Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert f\underset{p_{\theta }}{\overset{\frac{p_{\theta }}{p_0}}{\Vert }}\Vert g\underset{{q_{\theta }}^{\prime }}{\overset{\frac{{q_{\theta }}^{\prime }}{{q_0}^{\prime }}}{\Vert }}\\ & =\Vert T{\Vert }_{L^{p_0}\to L^{q_0}}.\end{array}$$

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 $g=f{\mathrm{𝟏}}_E$, $g_n=f_n{\mathrm{𝟏}}_E$, $h=f-g=f{\mathrm{𝟏}}_{E^{\mathrm{c}}}$ and $h_n=f_n-g_n$. Note that, since we are assuming $p_0\le p_{\theta }\le p_1$, $$\begin{array}{cc}\Vert f\underset{p_{\theta }}{\overset{p_{\theta }}{\Vert }}& =\underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}{\mathrm{𝟏}}_E\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f{\mathrm{𝟏}}_E\right|}^{p_0}\mathrm{d}{\mu }_1=\underset{{\mathrm{\Omega }}_1}{\int }{\left|g\right|}^{p_0}\mathrm{d}{\mu }_1=\Vert g\underset{p_0}{\overset{p_0}{\Vert }}\\ \Vert f\underset{p_{\theta }}{\overset{p_{\theta }}{\Vert }}& =\underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f\right|}^{p_{\theta }}{\mathrm{𝟏}}_{E^{\mathrm{c}}}\mathrm{d}{\mu }_1\ge \underset{{\mathrm{\Omega }}_1}{\int }{\left|f{\mathrm{𝟏}}_{E^{\mathrm{c}}}\right|}^{p_1}\mathrm{d}{\mu }_1=\underset{{\mathrm{\Omega }}_1}{\int }{\left|h\right|}^{p_1}\mathrm{d}{\mu }_1=\Vert h\underset{p_1}{\overset{p_1}{\Vert }}\end{array}$$ 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 $$\begin{array}{cccccc}\Vert f_n{\Vert }_{p_{\theta }}& \to \Vert f{\Vert }_{p_{\theta }}& \Vert g_n{\Vert }_{p_0}& \to \Vert g{\Vert }_{p_0}& \Vert h_n{\Vert }_{p_1}& \to \Vert h{\Vert }_{p_1}.\end{array}$$ 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 $$\begin{array}{cccccc}\Vert f-f_n{\Vert }_{p_{\theta }}& \to 0& \Vert g-g_n{\Vert }_{p_0}& \to 0& \Vert h-h_n{\Vert }_{p_1}& \to 0\end{array}$$ 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 $f$) $$\begin{array}{cccc}\Vert Tg-T{g}_n{\Vert }_{p_0}& \le \Vert T{\Vert }_{L^{p_0}\to L^{q_0}}\Vert g-g_n{\Vert }_{p_0}\to 0& \Vert Th-T{h}_n{\Vert }_{p_1}& \le \Vert T{\Vert }_{L^{p_1}\to L^{q_1}}\Vert h-h_n{\Vert }_{p_1}\to 0.\end{array}$$

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 (Template:EquationNote) 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 Template:Math and the space Template:Math of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.

Applications

Hausdorff–Young inequality

Template:Main It has been shown in the first section that the Fourier transform $ℱ$ 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 $\hat{f}:𝐙\to 𝐂$ whose values are the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)e^{-inx}dx,$$ maps Template:Math boundedly into Template:Math and Template:Math into Template:Math. The Riesz–Thorin interpolation theorem now implies the following: $$\begin{array}{cc}{\Vert ℱf\Vert }_{L^q({𝐑}^d)}& \le \Vert f{\Vert }_{L^p({𝐑}^d)}\\ {\Vert \hat{f}\Vert }_{{\ell }^q(𝐙)}& \le \Vert f{\Vert }_{L^p(𝐓)}\end{array}$$ 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

Template:Main 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 $$\Vert f*g{\Vert }_p\le \Vert f{\Vert }_1\Vert g{\Vert }_p.$$

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 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 $$\Vert f*g{\Vert }_s\le \Vert f{\Vert }_r\Vert g{\Vert }_p$$ 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 Template:Math 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 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 $$(\mathscr{H}f{)}^2=f^2+2\mathscr{H}(f\mathscr{H}f)$$ 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 $$\Vert \mathscr{H}f{\Vert }_p\le A_p\Vert f{\Vert }_p$$ 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. 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 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: $$\Vert A{\Vert }_{{\ell }_1\to {\ell }_1},\Vert A{\Vert }_{{\ell }_{\mathrm{\infty }}\to {\ell }_{\mathrm{\infty }}}\le M.$$

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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External links

• Template:Springer