Riesz–Thorin theorem explained

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 spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.

Motivation

First we need the following definition:

Definition. Let be two numbers such that . Then for define by: .

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

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

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

In particular, the above result implies that is included in, the sumset of and in the space of all measurable functions. Therefore, we have the following chain of inclusions:

In practice, we often encounter operators defined on the sumset . For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps boundedly into, and Plancherel's theorem shows that the Fourier transform maps boundedly into itself, hence the Fourier transform

extends to by setting$$\backslash mathcal(f\_1+f\_2)\; =\; \backslash mathcal\_(f\_1)\; +\; \backslash mathcal\_(f\_2)$$for all and . It is therefore natural to investigate the behavior of such operators on the intermediate subspaces .

To this end, we go back to our example and note that the Fourier transform on the sumset was obtained by taking the sum of two instantiations of the same operator, namely$$\backslash mathcal\_:L^1(\backslash mathbf^d)\; \backslash to\; L^\backslash infty(\backslash mathbf^d),$$$$\backslash mathcal\_:L^2(\backslash mathbf^d)\; \backslash to\; L^2(\backslash mathbf^d).$$

These really are the same operator, in the sense that they agree on the subspace . Since the intersection contains simple functions, it is dense in both and . Densely defined continuous operators admit unique extensions, and so we are justified in considering

and to be the same.

Therefore, the problem of studying operators on the sumset essentially reduces to the study of operators that map two natural domain spaces, and, boundedly to two target spaces: and, respectively. Since such operators map the sumset space to, it is natural to expect that these operators map the intermediate space to the corresponding intermediate space .

Statement of the theorem

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

In other words, if is simultaneously of type and of type, then is of type for all . In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of is the collection of all points in the unit square such that is of type . The interpolation theorem states that the Riesz diagram of is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram.

The interpolation theorem was originally stated and proved by Marcel Riesz in 1927.^[2] The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that and . Olof Thorin extended the interpolation theorem to the entire square, removing the lower-triangle restriction. The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis.^[3]

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 (the case $p\_0\; =\; p\_1$ trivially follows from). Let $f$ be a simple function, that is $$f\; =\; \backslash sum\_^m\; a\_j\; \backslash mathbf\_$$ for some finite $m\backslash in\backslash mathbb$, $a\_j\; =\; \backslash left\backslash vert\; a\_j\backslash right\backslash vert\backslash mathrm^\; \backslash in\; \backslash mathbb$ and $A\_j\backslash in\backslash Sigma\_1$, $j=1,2,\backslash dots,m$. Similarly, let $g$ denote a simple function $\backslash Omega\_2\; \backslash to\; \backslash mathbb$, namely $$g\; =\; \backslash sum\_^n\; b\_k\; \backslash mathbf\_$$ for some finite $n\backslash in\backslash mathbb$, $b\_k\; =\; \backslash left\backslash vert\; b\_k\backslash right\backslash vert\backslash mathrm^\; \backslash in\; \backslash mathbb$ and $B\_k\backslash in\backslash Sigma\_2$, $k=1,2,\backslash dots,n$.

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

Next, we define the two complex functions Note that, for $z=\backslash theta$, $u(\backslash theta)\; =\; p\_\backslash theta^$ and $v(\backslash theta)\; =\; q\_\backslash theta^$. We then extend $f$ and $g$ to depend on a complex parameter $z$ as follows: so that $f\_\backslash theta\; =\; f$ and $g\_\backslash theta\; =\; g$. Here, we are implicitly excluding the case $q\_0\; =\; q\_1\; =\; 1$, which yields $v\backslash equiv\; 1$: 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 $$\backslash Phi(z)\; =\; \backslash int\_\; (T\; f\_z)\; g\_z\; \backslash ,\backslash mathrm\backslash mu\_2=\; \backslash sum\_^m\; \backslash sum\_^n\; \backslash left\backslash vert\; a\_j\backslash right\backslash vert^\; \backslash left\backslash vert\; b\_k\backslash right\backslash vert^\; \backslash gamma\_$$ where $\backslash gamma\_\; =\; \backslash mathrm^\; \backslash int\_\; (T\; \backslash mathbf\_)\backslash mathbf\_\; \backslash ,\backslash mathrm\backslash mu\_2$ are constants independent of $z$. We readily see that $\backslash Phi(z)$ is an entire function, bounded on the strip $0\; \backslash le\; \backslash operatornamez\; \backslash le\; 1$. Then, in order to prove, we only need to show that for all $f\_z$ and $g\_z$ as constructed above. Indeed, if holds true, by Hadamard three-lines theorem, $$\backslash left\backslash vert\backslash Phi(\backslash theta\; +\; \backslash mathrm0)\backslash right\backslash vert\; =\; \backslash biggl\backslash vert\backslash int\_\; (Tf)\; g\; \backslash ,\backslash mathrm\backslash mu\_2\backslash biggr\backslash vert\; \backslash le\; \backslash |T\backslash |\_^\backslash |T\backslash |\_^\backslash theta$$ for all $f$ and $g$. This means, by fixing $f$, that $$\backslash sup\_g\; \backslash biggl\backslash vert\backslash int\_\; (Tf)\; g\; \backslash ,\backslash mathrm\backslash mu\_2\backslash biggr\backslash vert\; \backslash le\; \backslash |T\backslash |\_^\; \backslash |T\backslash |\_^\backslash theta$$ where the supremum is taken with respect to all $g$ simple functions with $\backslash lVert\; g\backslash rVert\_\; =\; 1$. The left-hand side can be rewritten by means of the following lemma.^[4]

In our case, the lemma above implies $$\backslash lVert\; Tf\backslash rVert\_\; \backslash le\; \backslash |T\backslash |\_^\; \backslash |T\backslash |\_^\backslash theta$$ for all simple function $f$ with $\backslash lVert\; f\backslash rVert\_\; =\; 1$. Equivalently, for a generic simple function, $$\backslash lVert\; Tf\backslash rVert\_\; \backslash le\; \backslash |T\backslash |\_^\; \backslash |T\backslash |\_^\backslash theta\; \backslash lVert\; f\backslash rVert\_.$$

Proof of

Let us now prove that our claim is indeed certain. The sequence $(A\_j)\_^m$ consists of disjoint subsets in $\backslash Sigma\_1$ and, thus, each $\backslash xi\backslash in\; \backslash Omega\_1$ belongs to (at most) one of them, say $A\_$. Then, for $z=\backslash mathrmy$, which implies that $\backslash lVert\; f\_\backslash rVert\_\; \backslash le\backslash lVert\; f\backslash rVert\_^$. With a parallel argument, each $\backslash zeta\; \backslash in\; \backslash Omega\_2$ belongs to (at most) one of the sets supporting $g$, say $B\_$, and $$\backslash left\backslash vert\; g\_(\backslash zeta)\backslash right\backslash vert\; =\; \backslash left\backslash vert\; b\_\backslash right\backslash vert^=\; \backslash left\backslash vert\; g(\backslash zeta)\backslash right\backslash vert^=\; \backslash left\backslash vert\; g(\backslash zeta)\backslash right\backslash vert^\backslash implies\; \backslash lVert\; g\_\backslash rVert\_\; \backslash le\backslash lVert\; g\backslash rVert\_^.$$

We can now bound $\backslash Phi(\backslash mathrmy)$: By applying Hölder’s inequality with conjugate exponents $q\_0$ and $q\_0\text{'}$, we have

We can repeat the same process for $z=1+\backslash mathrmy$ to obtain $\backslash left\backslash vert\; f\_(\backslash xi)\backslash right\backslash vert\; =\; \backslash left\backslash vert\; f(\backslash xi)\backslash right\backslash vert^$, $\backslash left\backslash vert\; g\_(\backslash zeta)\backslash right\backslash vert\; =\backslash left\backslash vert\; g(\backslash zeta)\backslash right\backslash vert^$ and, finally, $$\backslash left\backslash vert\backslash Phi(1+\backslash mathrmy)\backslash right\backslash vert\; \backslash le\; \backslash |T\backslash |\_\; \backslash lVert\; f\_\backslash rVert\_\; \backslash lVert\; g\_\backslash rVert\_\; =\backslash |T\backslash |\_.$$

Extension to all measurable functions in L^p_θ

So far, we have proven that when $f$ is a simple function. As already mentioned, the inequality holds true for all $f\backslash in\; L^(\backslash Omega\_1)$ by the density of simple functions in $L^(\backslash Omega\_1)$.

Formally, let $f\backslash in\; L^(\backslash Omega\_1)$ and let $(f\_n)\_n$ be a sequence of simple functions such that $\backslash left\backslash vert\; f\_n\backslash right\backslash vert\; \backslash le\; \backslash left\backslash vert\; f\backslash right\backslash vert$, for all $n$, and $f\_n\; \backslash to\; f$ pointwise. Let $E=\backslash$ and define $g\; =\; f\; \backslash mathbf\_E$, $g\_n=\; f\_n\; \backslash mathbf\_E$, $h\; =\; f\; -\; g\; =\; f\; \backslash mathbf\_$ and $h\_n\; =\; f\_n\; -\; g\_n$. Note that, since we are assuming $p\_0\; \backslash le\; p\_\backslash theta\; \backslash le\; p\_1$, and, equivalently, $g\backslash in\; L^(\backslash Omega\_1)$ and $h\backslash in\; L^(\backslash Omega\_1)$.

Let us see what happens in the limit for $n\backslash to\backslash infty$. Since $\backslash left\backslash vert\; f\_n\backslash right\backslash vert\; \backslash le\backslash left\backslash vert\; f\backslash right\backslash vert$, $\backslash left\backslash vert\; g\_n\backslash right\backslash vert\; \backslash le\; \backslash left\backslash vert\; g\backslash right\backslash vert$ and $\backslash left\backslash vert\; h\_n\backslash right\backslash vert\; \backslash le\; \backslash left\backslash vert\; h\backslash right\backslash vert$, by the dominated convergence theorem one readily has Similarly, $\backslash left\backslash vert\; f\; -\; f\_n\backslash right\backslash vert\; \backslash le\; 2\backslash left\backslash vert\; f\backslash right\backslash vert$, $\backslash left\backslash vert\; g-g\_n\backslash right\backslash vert\; \backslash le\; 2\backslash left\backslash vert\; g\backslash right\backslash vert$ and $\backslash left\backslash vert\; h-\; h\_n\backslash right\backslash vert\; \backslash le\; 2\backslash left\backslash vert\; h\backslash right\backslash vert$ imply 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 $(p\_\backslash theta,\; q\_\backslash theta)$ for a generic $f$)

It is now easy to prove that $Tg\_n\; \backslash to\; Tg$ and $Th\_n\; \backslash to\; Th$ in measure: For any $\backslash epsilon\; >\; 0$, Chebyshev’s inequality yields $$\backslash mu\_2(y\backslash in\; \backslash Omega\_2:\; \backslash left\backslash vert\; Tg\; -\; Tg\_n\backslash right\backslash vert\; >\; \backslash epsilon)\; \backslash le\; \backslash frac$$ and similarly for $Th\; -\; Th\_n$. Then, $Tg\_n\; \backslash to\; Tg$ and $Th\_n\; \backslash to\; Th$ a.e. for some subsequence and, in turn, $Tf\_n\; \backslash to\; Tf$ a.e. Then, by Fatou’s lemma and recalling that holds true for simple functions, $$\backslash lVert\; Tf\backslash rVert\_\; \backslash le\; \backslash liminf\_\; \backslash lVert\; T\; f\_n\backslash rVert\_\; \backslash le\backslash |T\backslash |\_\; \backslash liminf\_\; \backslash lVert\; f\_n\backslash rVert\_\; =\; \backslash |T\backslash |\_\backslash lVert\; f\backslash rVert\_.$$

Interpolation of analytic families of operators

The proof outline presented in the above section readily generalizes to the case in which the operator is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function$$\backslash varphi(z)\; =\; \backslash int\; (T\_z\; f\_z)g\_z\; \backslash ,\; d\backslash mu\_2,$$from which we obtain the following theorem of Elias Stein, published in his 1956 thesis:^[5]

The theory of real Hardy spaces and the space of bounded mean oscillations permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space and the space of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.^[6]

Applications

Hausdorff–Young inequality

maps boundedly into and into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions into functions whose values are the Fourier coefficients$$\backslash hat(n)\; =\; \backslash frac\; \backslash int\_^\; f(x)\; e^\; \backslash ,\; dx,$$maps boundedly into and into . The Riesz–Thorin interpolation theorem now implies the following:where and . This is the Hausdorff–Young inequality.

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

Convolution operators

See main article: Young's convolution inequality. Let be a fixed integrable function and let be the operator of convolution with, i.e., for each function we have .

It is well known that is bounded from to and it is trivial that it is bounded from to (both bounds are by). Therefore the Riesz–Thorin theorem gives$$\backslash |\; f\; *\; g\; \backslash |\_p\; \backslash leq\; \backslash |f\backslash |\_1\; \backslash |g\backslash |\_p.$$

We take this inequality and switch the role of the operator and the operand, or in other words, we think of as the operator of convolution with, and get that is bounded from to L^p. Further, since is in we get, in view of Hölder's inequality, that is bounded from to, where again . So interpolating we get$$\backslash |f*g\backslash |\_s\backslash leq\; \backslash |f\backslash |\_r\backslash |g\backslash |\_p$$where the connection between p, r and s is$$\backslash frac+\backslash frac=1+\backslash frac.$$

The Hilbert transform

See main article: Hilbert transform.

The Hilbert transform of is given by$$\backslash mathcalf(x)\; =\; \backslash frac\; \backslash ,\; \backslash mathrm\; \backslash int\_^\backslash infty\; \backslash frac\; \backslash ,\; dt\; =\; \backslash left(\backslash frac\; \backslash ,\; \backslash mathrm\; \backslash frac\; \backslash ast\; f\backslash right)(x),$$where p.v. indicates the Cauchy principal value of the integral. The Hilbert transform is a Fourier multiplier operator with a particularly simple multiplier:$$\backslash widehat(\backslash xi)\; =\; -i\; \backslash ,\; \backslash sgn(\backslash xi)\; \backslash hat(\backslash xi).$$

It follows from the Plancherel theorem that the Hilbert transform maps boundedly into itself.

Nevertheless, the Hilbert transform is not bounded on or, and so we cannot use the Riesz–Thorin interpolation theorem directly. To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions and . We can show, however, that$$(\backslash mathcalf)^2\; =\; f^2\; +\; 2\backslash mathcal(f\backslash mathcalf)$$for all Schwartz functions, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps boundedly into itself for all . Interpolation now establishes the bound$$\backslash |\backslash mathcalf\backslash |\_p\; \backslash leq\; A\_p\; \backslash |f\backslash |\_p$$for all, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the case.

Comparison with the real interpolation method

While the Riesz–Thorin interpolation theorem and its variants are powerful tools that yield a clean estimate on the interpolated operator norms, they suffer from numerous defects: some minor, some more severe. Note first that the complex-analytic nature of the proof of the Riesz–Thorin interpolation theorem forces the scalar field to be . For extended-real-valued functions, this restriction can be bypassed by redefining the function to be finite everywhere—possible, as every integrable function must be finite almost everywhere. A more serious disadvantage is that, in practice, many operators, such as the Hardy–Littlewood maximal operator and the Calderón–Zygmund operators, do not have good endpoint estimates.^[7] 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$$\backslash mu\; \backslash left(\backslash \; \backslash right)\; \backslash leq\; \backslash left(\backslash frac\; \backslash right)^q,$$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 -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:$$\backslash |A\backslash |\_,\; \backslash |A\backslash |\_\; \backslash leq\; M.$$

Then$$\backslash |A\backslash |\_\; \backslash leq\; M$$

for any unconditional Banach space of sequences, that is, for any

and any , .

The proof is based on the Krein–Milman theorem.

See also

• Marcinkiewicz interpolation theorem
• Interpolation space

References

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

Notes and References

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 . 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. Web site: Bernard . Calista . Interpolation theorems and applications .
5. 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 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.
6. Fefferman and Stein (1972)
7. [Elias Stein]