# Riesz–Thorin 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 Lp 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

Let $0< p_0 < p_1 \leq \infty$ and if $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$. By splitting up the function $f \in L^{p_\theta}$ as the product $|f| = |f|^{1-\theta}|f|^{\theta}$ and applying Hölder's inequality to its $p_\theta$'s power, we obtain the following result, foundational in the study of $L^p$-spaces:

Proposition (log-convexity of $L^p$-norms). If $0< p_0 < p_1 \leq \infty$ and if $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ for some $0 < \theta < 1$, then $\|f\|_{p_\theta} \leq \|f\|_{p_0}^{1-\theta}\|f\|_{p_1}^\theta$ for all $f \in L^{p_0} \cap L^{p_1}$.

This result, whose name derives from the convexity of the map $p \mapsto \log \|f\|_p$ on $[0,\infty]$, implies that $L^{p_0} \cap L^{p_1}$ is included in $L^{p_\theta}$.

On the other hand, if we take the layer-cake decomposition $f = f\chi_{\{|f|>1\}} + f\chi_{\{|f| \leq 1\}}$, then we see that $f\chi_{\{|f|>1\}} \in L^{p_0}$ and $f\chi_{\{|f| \leq 1\}} \in L^{p_1}$, whence we obtain the following result:

Proposition. If $0< p_0 < p_1 \leq \infty$ and if $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ for some $0 < \theta < 1$, then each $f \in L^{p_\theta}$ can be written as a sum of $g \in L^{p_0}$ and $h \in L^{p_1}$.

In particular, the above result implies that $L^{p_\theta}$ is included in $L^{p_0} + L^{p_1}$, the sumset of $L^{p_0}$ and $L^{p_1}$ in the space of all measurable functions. Therefore, we have the following chain of inclusions:

Corollary. $L^{p_0} \cap L^{p_1} \subset L^{p_\theta} \subset L^{p_0} + L^{p_1}$.

In practice, we often encounter operators defined on the sumset $L^{p_0}+L^{p_1}$. For example, the Riemann-Lebesgue lemma shows that the Fourier transform maps $L^1(\mathbb{R}^d)$ boundedly into $L^\infty(\mathbb{R}^d)$, and Plancherel's theorem shows that the Fourier transform maps $L^2(\mathbb{R}^d)$ boundedly into itself, whence the Fourier transform $\mathcal{F}$ extends to $(L^1+L^2)(\mathbb{R}^d)$ by setting $\mathcal{F}(f_1+f_2) = \mathcal{F}_{L^1}(f_1) + \mathcal{F}_{L^2}(f_2)$ for all $f_1 \in L^1(\mathbb{R}^d)$ and $f_2 \in L^2(\mathbb{R}^d)$. It is therefore natural to investigate the behavior of such operators on the intermediate subspaces $L^{p_\theta}$.

To this end, we go back to our example and note that the Fourier transform on the sumset $L^1 + L^2$ was obtained by taking the sum of two instantiations of the same operator, namely $\mathcal{F}_{L^1}:L^1(\mathbb{R}^d) \to L^\infty(\mathbb{R}^d)$ and $\mathcal{F}_{L^2}:L^2(\mathbb{R}^d) \to L^2(\mathbb{R}^d)$. These really are the same operator, in the sense that they agree on the subspace $(L^1 \cap L^2)(\mathbb{R}^d)$. Since the intersection contains simple functions, it is dense in both $L^1(\mathbb{R}^d)$ and $L^2(\mathbb{R}^d)$. Densely defined continuous functions admit unique extensions, and so we are justified in considering $\mathcal{F}_{L^1}$ and $\mathcal{F}_{L^2}$ to be the same.

Therefore, the problem of studying operators on the sumset $L^{p_0} + L^{p_1}$ essentially reduces to the study of operators that map two natural domain spaces, $L^{p_0}$ and $L^{p_1}$, boundedly to two target spaces: $L^{q_0}$ and $L^{q_1}$, respectively. Since such operators map the sumset space $L^{p_0} + L^{p_1}$ to $L^{q_0} + L^{q_1}$, it is natural to expect that these operators map the intermediate space $L^{p_\theta}$ to the corresponding intermediate space$L^{q_\theta}$.

## Statement of the Theorem

There are several ways to state the Riesz-Thorin interpolation theorem;[1] to be consistent with the notations in the previous section, we shall use the sumset formulation.

Riesz-Thorin interpolation theorem. Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be $\sigma$-finite measure spaces, $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 \leq q_1 \leq \infty$, and $T:L^{p_0}(\mu_1) + L^{p_1}(\mu_1) \to L^{q_0}(\mu_2) + L^{q_1}(\mu_2)$ be a linear operator that maps $L^{p_0}(\mu_1)$ and $L^{p_1}(\mu_1)$ boundedly into $L^{q_0}(\mu_2)$ and $L^{q_1}(\mu_2)$, respectively. If $0 < \theta < 1$, $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$, and $\frac{1}{q_\theta} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$, then $T$ maps $L^{p_\theta}(\mu_1)$ boundedly into $L^{q_\theta}(\mu_2)$ and satisfies the operator norm estimate $\|T\|_{L^{p_\theta} \to L^{q_\theta}} \leq \|T\|^{1-\theta}_{L^{p_0} \to L^{q_0}} \|T\|^{\theta}_{L^{p_1} \to L^{q_1}}$.

In other words, if $T$ is simultaneously of type $(p_0,q_0)$ and of type $(p_1,q_1)$, then $T$ is of type $(p_\theta,q_\theta)$ for all $0 < \theta < 1$. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of $T$ is the collection of all points $\left(\frac{1}{p},\frac{1}{q}\right)$ in the unit square $[0,1] \times [0,1]$ such that $T$ is of type $(p,q)$. The interpolation theorem states that the Riesz diagram of $T$ 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.[2] The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that $p_0 \leq q_0$ and $p_1 \leq q_1$. 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.[3]

## Sketch of Proof

The proof of the Riesz-Thorin interpolation theorem relies crucially on the Hadamard three-lines theorem to establish the requisite bounds. By the characterization of the dual spaces of $L^p$-spaces, we see that

$\|Tf\|_{q_\theta} = \sup_{\|g\|_{p_\theta} \leq 1} \left| \int (Tf)g \, d\mu_2\right|$.

By suitably defining variants $f_z$ and $g_z$ of $f$ and $g$ for each $z \in \mathbb{C}$, we obtain the entire function

$\phi(z) = \int (Tf_z)g_z \, d\mu_2$

whose value at $z=\theta$ is $\int (Tf)g$. We can then use the hypotheses to establish upper bounds of $\Phi$ on the lines $\mathrm{Re} z = 0$ and $\mathrm{Re} z = 1$, whence the Hadamard three-lines theorem establishes the interpolated bound of $\Phi$ on the line $\mathrm{Re} z = \theta$. It now suffices to check that the bound at $z = \theta$ is what we wanted.

## Interpolation of Analytic Families of Operators

The proof outline presented in the above section readily generalizes to the case in which the operator $T$ is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function

$\varphi(z) = \int (T_z f_z)g_z \, d\mu_2$,

from which we obtain the following theorem of Elias Stein, published in his 1956 thesis:[4]

Stein interpolation theorem. Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be $\sigma$-finite measure spaces, $1 \leq p_0 \leq p_1 \leq \infty$, and $1 \leq q_0 \leq q_1 \leq \infty$. We take a collection of linear operators $\{T_z\}_{0 \leq \mathrm{Re} z \leq 1}$ on the space of simple functions in $L^1(\mu_1)$ into the space of all $\mu_2$-measurable functions on $\Omega_2$ such that the mapping

$z \mapsto \int (T_zf)g \, d\mu_2$

is continuous on the strip $S = \{z \in \mathbb{C} : 0 \leq \mathrm{Re} z \leq 1\}$ and holomorphic in the interior of $S$ for all simple functions $f$ and $g$. We assume also that the operators satisfy the uniform bound

$\sup_{z \in S} e^{-k|\mathrm{Im} z|} \left| \int (T_zf)g \, \mu_2 \right| < \infty$

for some constant $k < \pi$.

Assume that $T_z$ maps $L^{p_0}(\mu_1)$ boundedly to $L^{q_0}(\mu_2)$ whenever $\mathrm{Re} z = 0$, and that $T_z$ maps $L^{p_1}(\mu_1)$ boundedly to $L^{q_1}(\mu_2)$ whenever $\mathrm{Re} z = 1$. If the operator norms satisfy the uniform bound

$\sup_{\mathrm{Re} z = 0 \mbox{ or } 1} e^{-k|\mathrm{Im}z|} \log\|T_z\| < \infty$,

then, for each $0 < \theta < 1$, the operator $T_\theta$ maps $L^{p_\theta}(\mu_1)$ boundedly into $L^{q_\theta}(\mu_2)$.

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 $H^1(\mathbb{R}^d)$ and the space $BMO$ of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.[5]

## Applications

### Hausdorff−Young inequality

We have seen in the first section that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^d)$ boundedly into $L^\infty(\mathbb{R}^d)$ and $L^2(\mathbb{R}^d)$ into itself. A similar argument shows that the Fourier series operator that transforms periodic functions $f:\mathbb{T} \to \mathbb{C}$ into functions $\hat{f}:\mathbb{Z} \to \mathbb{C}$ whose values are the Fourier coefficients

$\hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx$

maps $L^1(\mathbb{T})$ boundedly into $l^\infty(\mathbb{Z})$ and $L^2(\mathbb{T})$ into $l^2(\mathbb{Z})$. The Riesz-Thorin interpolation theorem now implies that

$\|\mathcal{F}f\|_{L^{p'}(\mathbb{R}^d)} \leq \|f\|_{L^p(\mathbb{R}^d)}$ and $\|\hat{f}\|_{l^{p'}(\mathbb{Z})} \leq \|f\|_{L^p(\mathbb{T})}$,

where $1 \leq p \leq 2$ and $\frac{1}{p} + \frac{1}{p'} = 1$. This is the Hausdorff–Young inequality.

The Hausdorff-Young inequality can also be established for the Fourier transform on locally compact abelian groups. We also note that 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 T be the operator of convolution with f, i.e., for each function g we have

$Tg = f * g.$

It is well known that T 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

$\|f*g\|_p\leq \|f\|_1\|g\|_p.$

We take this inequality and switch the role of the operator and the operand, or in other words, we think of S as the operator of convolution with g, and get that S is bounded from L1 to Lp. Further, since g is in Lp we get, in view of Hölder's inequality, that S is bounded from Lq to L, where again 1/p + 1/q = 1. So interpolating we get

$\|f*g\|_s\leq \|f\|_r\|g\|_p$

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 $f:\mathbb{R} \to \mathbb{C}$ is given by

$\mathcal{H}f(x) = \frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^\infty \frac{f(x-t)}{t} \, dt = \left(\frac{1}{\pi} \, \mathrm{p.v.} \frac{1}{t} \ast f\right)(x)$,

where p.v. indicates the Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier:

$\widehat{\mathcal{H}f}(\xi) = -i \, \mathrm{sgn}(\xi) \hat{f}(\xi)$.

It follows from the Plancherel theorem that the Hilbert transform maps $L^2(\mathbb{R})$ boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on $L^1(\mathbb{R})$ or $L^\infty(\mathbb{R})$, 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 $\chi_{(-1,1)}(x)$ and $\chi_{(0,1)}(x)- \chi_{(0,1)}(-x)$. We can show, however, that

$(\mathcal{H}f)^2 = f^2 + 2\mathcal{H}(f\mathcal{H}f)$

for all Schwartz functions $f:\mathbb{R} \to \mathbb{C}$, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps $L^{2^n}(\mathbb{R}^d)$ boundedly into itself for all $n \geq 2$. Interpolation now establishes the bound

$\|\mathcal{H}f\|_p \leq A_p \|f\|_p$

for all $2 \leq p < \infty$, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the $1 < p \leq 2$ 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 $\mathbb{C}$. 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.[6] 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

$\mu\left(\{x : Tf(x) > \alpha\}\right) \leq \left( \frac{C_{p,q} \|f\|_p}{\alpha} \right)^q$,

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 $\|A\|_{\ell_1 \to \ell_1} \leq M$, $\|A\|_{\ell_\infty \to \ell_\infty} \leq M$. Then $\|A\|_{X \to X} \leq M$ for any unconditional Banach space of sequences X (that is, for any $(x_i) \in X$ and any $(\epsilon_i) \in \{ -1, +1 \}^\infty$, $\| (\epsilon_i x_i) \|_X = \| (x_i) \|_X$).

The proof is based on the Krein–Milman theorem.

## Notes

1. ^ 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 $L^{p_0} \cap L^{p_1}$. In contrast, Duoanddikoetxea (2001), Tao (2010), and Stein and Shakarchi (2011) use the sumset formulation, which we adopt in this section.
2. ^ 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.
3. ^ Thorin (1948)
4. ^ 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 $z$ 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 a blog post of Tao for a high-level exposition of the theorem.
5. ^ Fefferman and Stein (1972)
6. ^ Elias Stein is quoted for saying that interesting operators in harmonic analysis are rarely bounded on $L^1$ and $L^\infty$.

## References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience.
• Fefferman, Charles; Stein, Elias M. (1972), "$H^p$ Spaces of Several variables", Acta Mathematica 129: 137–193
• Glazman, I.M.; Lyubich, Yu.I. (1974), Finite-dimensional linear analysis: a systematic presentation in problem form, Cambridge, Mass.: The M.I.T. Press. Translated from the Russian and edited by G. P. Barker and G. Kuerti.
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft. 256, Springer, ISBN 3-540-12104-8, MR 0717035.
• Mitjagin [Mityagin], B.S. (1965), "An interpolation theorem for modular spaces (Russian)", Mat. Sb. (N.S.) 66 (108): 473–482.
• Thorin, G. O. (1948), "Convexity theorems generalizing those of M. Riesz and Hadamard with some applications", Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 9: 1–58, MR 0025529
• Riesz, Marcel (1927), "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires", Acta Mathematica 49: 465–497
• Stein, Elias M. (1956), "Interpolation of Linear Operators", Trans. Amer. Math. Soc. 83: 482–492
• Stein, Elias M.; Shakarchi, Rami (2011), Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press
• Stein, Elias M.; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press