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{{Short description|Theorem on operator interpolation}}
{{About||more theorems that are called Riesz's theorem|Riesz theorem (disambiguation){{!}}Riesz theorem}}
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 {{math|[[lp space|''L<sup>p</sup>'']]}} spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to {{math|''L''<sup>2</sup>}} which is a [[Hilbert space]], or to {{math|''L''<sup>1</sup>}} and {{math|''L''<sup>∞</sup>}}. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The [[Marcinkiewicz theorem]] is similar but applies also to a class of non-linear maps.

==Motivation==
First we need the following definition:

:'''Definition.''' Let {{math|''p''<sub>0</sub>, ''p''<sub>1</sub>}} be two numbers such that {{math|1 ≤ ''p''<sub>0</sub> < ''p''<sub>1</sub> ≤ ∞}}. Then for {{math|0 < ''θ'' < 1}} define {{math|''p<sub>θ</sub>''}} by: {{math|1=&thinsp;{{sfrac|1|''p<sub>θ</sub>''}} = {{sfrac|1 − ''θ''|''p''<sub>0</sub>}} + {{sfrac|''θ''|''p''<sub>1</sub>}}}}.

By splitting up the function {{math|&thinsp;''f''&thinsp;}} in {{math|''L<sup>p<sub>θ</sub></sup>''}} as the product {{math|1={{!}}&thinsp;''f''&thinsp;{{!}} = {{!}}&thinsp;''f''&thinsp;{{!}}<sup>1−''θ''</sup> {{!}}&thinsp;''f''&thinsp;{{!}}<sup>''θ''</sup>}} and applying [[Hölder's inequality]] to its {{math|''p<sub>θ</sub>''}} power, we obtain the following result, foundational in the study of {{math|''L<sup>p</sup>''}}-spaces:

{{math theorem|name=Proposition (log-convexity of {{math|''L<sup>p</sup>''}}-norms)
|math_statement=Each {{math|&thinsp;''f''&thinsp; ∈ ''L''<sup>''p''<sub>0</sub></sup> ∩ ''L''<sup>''p''<sub>1</sub></sup>}} satisfies: 
{{NumBlk|:|<math display="block">\|f\|_{p_\theta} \leq \|f\|_{p_0}^{1-\theta}\|f\|_{p_1}^\theta.</math>|{{EquationRef|1}}}}
}}

This result, whose name derives from the convexity of the map {{math|{{frac|1|''p''}} ↦ log {{!!}}&thinsp;''f''&thinsp;{{!!}}<sub>''p''</sub>}} on {{math|[0, ∞]}}, implies that {{math|''L''<sup>''p''<sub>0</sub></sup> ∩ ''L''<sup>''p''<sub>1</sub></sup> ⊂ ''L<sup>p<sub>θ</sub></sup>''}}.

On the other hand, if we take the ''layer-cake decomposition'' {{math|1=&thinsp;''f''&thinsp; = &thinsp;''f''&thinsp;'''1'''<sub>{{mset|{{abs|&thinsp;''f''&thinsp;}}>1}}</sub> + &thinsp;''f''&thinsp;'''1'''<sub>{{mset|{{abs|&thinsp;''f''&thinsp;}}≤1}}</sub>}}, then we see that {{math|&thinsp;''f''&thinsp;'''1'''<sub>{{mset|{{abs|&thinsp;''f''&thinsp;}}>1}}</sub> ∈ ''L''<sup>''p''<sub>0</sub></sup>}} and {{math|&thinsp;''f''&thinsp;'''1'''<sub>{{mset|{{abs|&thinsp;''f''&thinsp;}}≤1}}</sub> ∈ ''L''<sup>''p''<sub>1</sub></sup>}}, whence we obtain the following result:

{{math theorem|name=Proposition
|math_statement=Each {{math|&thinsp;''f''&thinsp;}} in {{math|''L<sup>p<sub>θ</sub></sup>''}} can be written as a sum: {{math|1=&thinsp;''f''&thinsp; = ''g'' + ''h''}}, where {{math|''g'' ∈ ''L''<sup>''p''<sub>0</sub></sup>}} and {{math|''h'' ∈ ''L''<sup>''p''<sub>1</sub></sup>}}.
}}

In particular, the above result implies that {{math|''L<sup>p<sub>θ</sub></sup>''}} is included in {{math|''L''<sup>''p''<sub>0</sub></sup> + ''L''<sup>''p''<sub>1</sub></sup>}}, the [[Minkowski addition|sumset]] of {{math|''L''<sup>''p''<sub>0</sub></sup>}} and {{math|''L''<sup>''p''<sub>1</sub></sup>}} in the space of all measurable functions. Therefore, we have the following chain of inclusions:

{{math theorem|name=Corollary
|math_statement= {{math|''L''<sup>''p''<sub>0</sub></sup> ∩ ''L''<sup>''p''<sub>1</sub></sup> ⊂ ''L<sup>p<sub>θ</sub></sup>'' ⊂ ''L''<sup>''p''<sub>0</sub></sup> + ''L''<sup>''p''<sub>1</sub></sup>}}.
}}

In practice, we often encounter [[Operator (mathematics)|operators]] defined on the [[sumset]] {{math|''L''<sup>''p''<sub>0</sub></sup> + ''L''<sup>''p''<sub>1</sub></sup>}}. For example, the [[Riemann–Lebesgue lemma]] shows that the [[Fourier transform]] maps {{math|''L''<sup>1</sup>('''R'''<sup>''d''</sup>)}} [[Bounded operator|boundedly]] into {{math|''L''<sup>∞</sup>('''R'''<sup>''d''</sup>)}}, and [[Plancherel theorem|Plancherel's theorem]] shows that the Fourier transform maps {{math|''L''<sup>2</sup>('''R'''<sup>''d''</sup>)}} boundedly into itself, hence the Fourier transform <math>\mathcal{F}</math> extends to {{math|(''L''<sup>1</sup> + ''L''<sup>2</sup>) ('''R'''<sup>''d''</sup>)}} 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 {{math|&thinsp;''f''<sub>1</sub>&thinsp; ∈ ''L''<sup>1</sup>('''R'''<sup>''d''</sup>)}} and {{math|&thinsp;''f''<sub>2</sub>&thinsp; ∈ ''L''<sup>2</sup>('''R'''<sup>''d''</sup>)}}. It is therefore natural to investigate the behavior of such operators on the ''intermediate subspaces'' {{math|''L<sup>p<sub>θ</sub></sup>''}}.

To this end, we go back to our example and note that the Fourier transform on the sumset {{math|''L''<sup>1</sup> + ''L''<sup>2</sup>}} 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 {{math|(''L''<sup>1</sup> ∩ ''L''<sup>2</sup>) ('''R'''<sup>''d''</sup>)}}. Since the intersection contains [[simple function]]s, it is dense in both {{math|''L''<sup>1</sup>('''R'''<sup>''d''</sup>)}} and {{math|''L''<sup>2</sup>('''R'''<sup>''d''</sup>)}}. 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 {{math|''L''<sup>''p''<sub>0</sub></sup> + ''L''<sup>''p''<sub>1</sub></sup>}} essentially reduces to the study of operators that map two natural domain spaces, {{math|''L''<sup>''p''<sub>0</sub></sup>}} and {{math|''L''<sup>''p''<sub>1</sub></sup>}}, boundedly to two target spaces: {{math|''L''<sup>''q''<sub>0</sub></sup>}} and {{math|''L''<sup>''q''<sub>1</sub></sup>}}, respectively. Since such operators map the sumset space {{math|''L''<sup>''p''<sub>0</sub></sup> + ''L''<sup>''p''<sub>1</sub></sup>}} to {{math|''L''<sup>''q''<sub>0</sub></sup> + ''L''<sup>''q''<sub>1</sub></sup>}}, it is natural to expect that these operators map the intermediate space {{math|''L<sup>p<sub>θ</sub></sup>''}} to the corresponding intermediate space {{math|''L<sup>q<sub>θ</sub></sup>''}}.

==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 {{math|''L''<sup>''p''<sub>0</sub></sup> ∩ ''L''<sup>''p''<sub>1</sub></sup>}}. 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.

{{math theorem
|name=Riesz–Thorin interpolation theorem
|math_statement= Let {{math|(Ω<sub>1</sub>, Σ<sub>1</sub>, ''μ''<sub>1</sub>)}} and {{math|(Ω<sub>2</sub>, Σ<sub>2</sub>, ''μ''<sub>2</sub>)}} be [[Σ-finite measure|{{mvar|σ}}-finite measure]] spaces. Suppose {{math|1 ≤ ''p''<sub>0</sub> , ''q''<sub>0</sub> , ''p''<sub>1</sub> , ''q''<sub>1</sub> ≤ ∞}}, and let {{math|''T'' : ''L''<sup>''p''<sub>0</sub></sup>(''μ''<sub>1</sub>) + ''L''<sup>''p''<sub>1</sub></sup>(''μ''<sub>1</sub>) → ''L''<sup>''q''<sub>0</sub></sup>(''μ''<sub>2</sub>) + ''L''<sup>''q''<sub>1</sub></sup>(''μ''<sub>2</sub>)}} be a [[linear operator]] that [[Bounded operator|boundedly]] maps {{math|''L''<sup>''p''<sub>0</sub></sup>(''μ''<sub>1</sub>)}} into {{math|''L''<sup>''q''<sub>0</sub></sup>(''μ''<sub>2</sub>)}} and {{math|''L''<sup>''p''<sub>1</sub></sup>(''μ''<sub>1</sub>)}} into {{math|''L''<sup>''q''<sub>1</sub></sup>(''μ''<sub>2</sub>)}}. For {{math|0 < ''θ'' < 1}}, let {{math|''p<sub>θ</sub>'', ''q<sub>θ</sub>''}} be defined as above. Then {{mvar|T}} boundedly maps {{math|''L<sup>p<sub>θ</sub></sup>''(''μ''<sub>1</sub>)}} into {{math|''L<sup>q<sub>θ</sub></sup>''(''μ''<sub>2</sub>)}} and satisfies the [[operator norm]] estimate 
{{NumBlk||<math display="block">\|T\|_{L^{p_\theta} \to L^{q_\theta}} \leq \|T\|^{1-\theta}_{L^{p_0} \to L^{q_0}} \|T\|^{\theta}_{L^{p_1} \to L^{q_1}}.</math>|{{EquationRef|2}}}}
}}

In other words, if {{mvar|T}} is simultaneously of '''type''' {{math|(''p''<sub>0</sub>, ''q''<sub>0</sub>)}} and of type {{math|(''p''<sub>1</sub>, ''q''<sub>1</sub>)}}, then {{mvar|T}} is of type {{math|(''p<sub>θ</sub>'', ''q<sub>θ</sub>'')}} for all {{math|0 < ''θ'' < 1}}. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, '''the Riesz diagram''' of {{mvar|T}} is the collection of all points {{math|({{sfrac|1|''p''}}, {{sfrac|1|''q''}})}} in the unit square {{math|[0, 1] × [0, 1]}} such that {{mvar|T}} is of type {{math|(''p'', ''q'')}}. The interpolation theorem states that the Riesz diagram of {{mvar|T}} 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 {{math|''p''<sub>0</sub> ≤ ''q''<sub>0</sub>}} and {{math|''p''<sub>1</sub> ≤ ''q''<sub>1</sub>}}. [[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 ({{EquationNote|1}})). 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 ({{EquationNote|2}}), we only need to show that {{NumBlk|:|<math display="block">\begin{aligned}
\left\vert\Phi(\mathrm{i}y)\right\vert &\le \|T\|_{L^{p_0} \to L^{q_0}} &&\text{and} & \left\vert\Phi(1 + \mathrm{i}y)\right\vert &\le \|T\|_{L^{p_1} \to L^{q_1}}\end{aligned}</math>|{{EquationRef|3}}}} for all <math display="inline">f_z</math> and <math display="inline">g_z</math> as constructed above. Indeed, if ({{EquationNote|3}}) 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>{{cite web |last1=Bernard |first1=Calista |title=Interpolation theorems and applications |url=http://math.uchicago.edu/~may/REU2013/REUPapers/Bernard.pdf}}</ref>

{{math theorem|name=Lemma
|math_statement=Let <math display="inline">1\le p, p' \le \infty</math> be conjugate exponents and let <math display="inline">f</math> be a function in <math display="inline">L^p(\mu_1)</math>. Then <math display="block">\lVert f\rVert_p = \sup \biggl|\int_{\Omega_1} fg \,\mathrm{d}\mu_1\biggr|</math> where the supremum is taken over all simple functions <math display="inline">g</math> in <math display="inline">L^{p'}(\mu_1)</math> such that <math display="inline">\lVert g\rVert_{p'} \le 1</math>.
}}

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 ({{EquationNote|3}})====
Let us now prove that our claim ({{EquationNote|3}}) 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<sup>p<sub>θ</sub></sup>''===
So far, we have proven that {{NumBlk|:|<math display="block">
\lVert Tf\rVert_{q_\theta} \le \|T\|_{L^{p_\theta} \to L^{q_\theta}} \lVert f\rVert_{p_\theta}</math>|{{EquationRef|4}}}} 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 ({{EquationNote|4}}) 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 {{mvar|T}} 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 {{mvar|z}} added to the operator. To compensate for this, a stronger version of the [[Hadamard three-lines theorem]], due to [[Isidore Isaac Hirschman, Jr.]], is used to establish the desired bounds. See Stein and Weiss (1971) for a detailed proof, and [http://terrytao.wordpress.com/2011/05/03/steins-interpolation-theorem/ a blog post of Tao] for a high-level exposition of the theorem.</ref>

{{math theorem
|name=Stein interpolation theorem
|math_statement= Let {{math|(Ω<sub>1</sub>, Σ<sub>1</sub>, ''μ''<sub>1</sub>)}} and {{math|(Ω<sub>2</sub>, Σ<sub>2</sub>, ''μ''<sub>2</sub>)}} be [[Σ-finite measure|{{mvar|σ}}-finite measure]] spaces. Suppose {{math|1 ≤ ''p''<sub>0</sub> , ''p''<sub>1</sub> ≤ ∞, 1 ≤ ''q''<sub>0</sub> , ''q''<sub>1</sub> ≤ ∞}}, and define: 
:{{math|1=''S'' = {''z'' ∈ '''C''' : 0 < Re(''z'') < 1}<nowiki/>}},
:{{math|1={{overline|''S''}} = {''z'' ∈ '''C''' : 0 ≤ Re(''z'') ≤ 1}<nowiki/>}}.
We take a collection of linear operators {{math|{{mset|''T<sub>z</sub>'' : ''z'' ∈ {{overline|''S''}}}} }}on the space of simple functions in {{math|''L''<sup>1</sup>(''μ''<sub>1</sub>)}} into the space of all {{math|''μ''<sub>2</sub>}}-measurable functions on {{math|Ω<sub>2</sub>}}. We assume the following further properties on this collection of linear operators: 
* The mapping <math display="block"> z \mapsto \int (T_zf)g \, d\mu_2</math> is continuous on {{math|{{overline|''S''}}}} and holomorphic on {{mvar|S}} for all simple functions {{math|&thinsp;''f''&thinsp;}} and {{mvar|g}}. 
* For some constant {{math|''k'' < ''π''}}, the operators satisfy the uniform bound: <math display="block"> \sup_{z \in S} e^{-k|\text{Im}(z)|} \log \left| \int (T_zf)g \, d\mu_2 \right| < \infty</math>
* {{math|''T<sub>z</sub>''}} maps {{math|''L''<sup>''p''<sub>0</sub></sup>(''μ''<sub>1</sub>)}} [[Bounded operator|boundedly]] to {{math|''L''<sup>''q''<sub>0</sub></sup>(''μ''<sub>2</sub>)}} whenever {{math|1=Re(''z'') = 0}}.
* {{math|''T<sub>z</sub>''}} maps {{math|''L''<sup>''p''<sub>1</sub></sup>(''μ''<sub>1</sub>)}} boundedly to {{math|''L''<sup>''q''<sub>1</sub></sup>(''μ''<sub>2</sub>)}} whenever {{math|1=Re(''z'') = 1}}. 
* The operator norms satisfy the uniform bound <math display="block">\sup_{\text{Re}(z) = 0, 1} e^{-k|\text{Im}(z)|} \log \left \|T_z \right \| < \infty</math> for some constant {{math|''k'' < ''π''}}.

Then, for each {{math|0 < ''θ'' < 1}}, the operator {{math|''T<sub>θ</sub>''}} maps {{math|''L<sup>p<sub>θ</sub></sup>''(''μ''<sub>1</sub>)}} boundedly into {{math|''L<sup>q<sub>θ</sub></sup>''(''μ''<sub>2</sub>)}}.
}}

The theory of [[Hardy space#Real Hardy spaces for Rn|real Hardy spaces]] and the [[Bounded mean oscillation|space of bounded mean oscillations]] permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space {{math|''H''<sup>1</sup>('''R'''<sup>''d''</sup>)}} and the space {{math|BMO}} of bounded mean oscillations; this is a result of [[Charles Fefferman]] and [[Elias Stein]].<ref>Fefferman and Stein (1972)</ref>

==Applications==

===Hausdorff–Young inequality===
{{main|Hausdorff–Young inequality}}
It has been shown in the [[#Motivation|first section]] that the [[Fourier transform]] <math>\mathcal{F}</math> maps {{math|''L''<sup>1</sup>('''R'''<sup>''d''</sup>)}} boundedly into {{math|''L''<sup>∞</sup>('''R'''<sup>''d''</sup>)}} and {{math|''L''<sup>2</sup>('''R'''<sup>''d''</sup>)}} into itself. A similar argument shows that the [[Fourier series|Fourier series operator]], which transforms periodic functions {{math|&thinsp;''f''&thinsp; : '''T''' → '''C'''}} 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 {{math|''L''<sup>1</sup>('''T''')}} boundedly into {{math|''ℓ''<sup>∞</sup>('''Z''')}} and {{math|''L''<sup>2</sup>('''T''')}} into {{math|''ℓ''<sup>2</sup>('''Z''')}}. 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 {{math|1 ≤ ''p'' ≤ 2}} and {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}. This is the [[Hausdorff–Young inequality]].

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

===Convolution operators===
{{main|Young's convolution inequality}}
Let {{math|&thinsp;''f''&thinsp;}} be a fixed integrable function and let {{mvar|T}} be the operator of convolution with {{math|&thinsp;''f''&thinsp;}}, i.e., for each function {{mvar|g}} we have {{math|1=''Tg'' = &thinsp;''f''&thinsp; ∗ ''g''}}.

It follows from [[Fubini's theorem]] that {{mvar|T}} is bounded from {{math|''L''<sup>1</sup>}} to {{math|''L''<sup>1</sup>}} and it is trivial that it is bounded from {{math|''L''<sup>∞</sup>}} to {{math|''L''<sup>∞</sup>}} (both bounds are by {{math|{{!!}}&thinsp;''f''&thinsp;{{!!}}<sub>1</sub>}}). 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 {{mvar|S}} as the operator of convolution with {{mvar|g}}, and get that {{mvar|S}} is bounded from {{math|''L''<sup>1</sup>}} to ''L<sup>p</sup>''. Further, since {{mvar|g}} is in {{math|''L<sup>p</sup>''}} we get, in view of Hölder's inequality, that {{mvar|S}} is bounded from {{math|''L<sup>q</sup>''}} to {{math|''L''<sup>∞</sup>}}, where again {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}. 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===
{{main|Hilbert transform}}

The '''Hilbert transform''' of {{math|&thinsp;''f''&thinsp; : '''R''' → '''C'''}} 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 [[Multiplier (Fourier analysis)|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 {{math|''L''<sup>2</sup>('''R''')}} boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on {{math|''L''<sup>1</sup>('''R''')}} or {{math|''L''<sup>∞</sup>('''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 {{math|'''1'''<sub>(−1,1)</sub>(''x'')}} and {{math|'''1'''<sub>(0,1)</sub>(''x'') − '''1'''<sub>(0,1)</sub>(−''x'')}}. 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 space|Schwartz functions]] {{math|&thinsp;''f''&thinsp; : '''R''' → '''C'''}}, and this identity can be used in conjunction with the [[Cauchy–Schwarz inequality]] to show that the Hilbert transform maps {{math|''L''<sup>2<sup>''n''</sup></sup>('''R'''<sup>''d''</sup>)}} boundedly into itself for all {{math|''n'' ≥ 2}}. Interpolation now establishes the bound
<math display="block"> \|\mathcal{H}f\|_p \leq A_p \|f\|_p</math>
for all {{math|2 ≤ ''p'' < ∞}}, and the [[Self-adjoint operator|self-adjointness]] of the Hilbert transform can be used to carry over these bounds to the {{math|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 {{math|'''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 theory|Calderón–Zygmund operators]], do not have good endpoint estimates.<ref>[[Elias Stein]] is quoted for saying that interesting operators in [[harmonic analysis]] are rarely bounded on {{math|''L''<sup>1</sup>}} and {{math|''L''<sup>∞</sup>}}.</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 function|sublinear]]. This is not a hindrance to applying real interpolation methods, but complex interpolation methods are ill-equipped to handle non-linear operators. On the other hand, real interpolation methods, compared to complex interpolation methods, tend to produce worse estimates on the intermediate operator norms and do not behave as well off the diagonal in the Riesz diagram. The off-diagonal versions of the Marcinkiewicz interpolation theorem require the formalism of [[Lorentz space]]s and do not necessarily produce norm estimates on the {{math|''L<sup>p</sup>''}}-spaces.

==Mityagin's theorem==
B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of [[Sequence space|spaces of sequences]] with [[Schauder basis|unconditional bases]] (cf. below).

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 {{mvar|X}}, 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]]

==Notes==
{{Reflist|30em}}

==References==
* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|authorlink2=Jacob T. Schwartz|title=Linear operators, Parts I and II|publisher=Wiley-Interscience|year=1958}}.
* {{Citation | last1=Fefferman |first1=Charles | last2=Stein | first2=Elias M. | title=<math>H^p</math> Spaces of Several variables | year=1972 | volume=129 | journal=Acta Mathematica | pages=137–193 | doi=10.1007/bf02392215| doi-access=free }}
* {{citation|first1=I.M.|last1=Glazman|first2=Yu.I.|last2=Lyubich|title=Finite-dimensional linear analysis: a systematic presentation in problem form|publisher=The M.I.T. Press|location=Cambridge, Mass.|year=1974}}.  Translated from the Russian and edited by G. P. Barker and G. Kuerti.
* {{citation|mr=0717035|first=L.|last= Hörmander|author-link=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer  |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}}.
* {{citation|first=B.S.|last=Mitjagin [Mityagin]|title=An interpolation theorem for modular spaces'' (Russian)|journal=Mat. Sb. |series=New Series|volume=66|issue=108|year=1965|pages=473–482}}.
* {{Citation | last1=Thorin | first1=G. O. | title=Convexity theorems generalizing those of M. Riesz and Hadamard with some applications | mr=0025529 | year=1948 | journal=Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=9 | pages=1–58}}
* {{Citation |last1= Riesz | first1=Marcel | title=Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires | year=1927 | volume=49 | issue=3–4 | journal=Acta Mathematica | pages=465–497 | doi=10.1007/bf02564121| doi-access=free }}
* {{Citation | last1=Stein | first1=Elias M. | title=Interpolation of Linear Operators | year=1956 | volume=83 | issue=2 | journal=Trans. Amer. Math. Soc.  | pages=482–492 | doi=10.1090/s0002-9947-1956-0082586-0| doi-access=free }}
* {{Citation | last1=Stein |first1=Elias M. | last2=Shakarchi | first2=Rami | title=Functional Analysis: Introduction to Further Topics in Analysis | year=2011 | publisher=Princeton University Press}}
* {{Citation | last1=Stein |first1=Elias M. | last2=Weiss | first2=Guido | title=Introduction to Fourier Analysis on Euclidean Spaces | year=1971 | publisher=Princeton University Press}}

==External links==

* {{springer|title=Riesz convexity theorem|id=p/r082230}}

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