Riesz–Thorin theorem

Motivation

First we need the following definition:

Definition. Let p₀, p₁ be two numbers such that 0 < p₀ < p₁ ≤ ∞. Then for 0 < θ < 1 define p_θ by:  1/p_θ = 1 − θ/p₀ + θ/p₁.

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

Proposition (log-convexity of L^p-norms). Each  f  ∈ L^p₀ ∩ L^p₁ satisfies: ∥ f ∥ p θ ≤ ∥ f ∥ p 0 1 − θ ∥ f ∥ p 1 θ .

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

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

Proposition. Each  f  in L^p_θ can be written as a sum:  f  = g + h, where g ∈ L^p₀ and h ∈ L^p₁.

In particular, the above result implies that L^p_θ is included in L^p₀ + L^p₁, the sumset of L^p₀ and L^p₁ in the space of all measurable functions. Therefore, we have the following chain of inclusions:

Corollary. L^p₀ ∩ L^p₁ ⊂ L^p_θ ⊂ L^p₀ + L^p₁.

In practice, we often encounter operators defined on the sumset L^p₀ + L^p₁. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps L¹(R^d) boundedly into L^∞(R^d), and Plancherel's theorem shows that the Fourier transform maps L²(R^d) boundedly into itself, whence the Fourier transform F extends to (L¹ + L²) (R^d) by setting

F ( f 1 + f 2 ) = F L 1 ( f 1 ) + F L 2 ( f 2 )

for all  f₁  ∈ L¹(R^d) and  f₂  ∈ L²(R^d). It is therefore natural to investigate the behavior of such operators on the intermediate subspaces L^p_θ.

To this end, we go back to our example and note that the Fourier transform on the sumset L¹ + L² was obtained by taking the sum of two instantiations of the same operator, namely

F L 1 : L 1 ( R d ) → L ∞ ( R d ) , F L 2 : L 2 ( R d ) → L 2 ( R d ) .

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

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

Statement of the theorem

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

Riesz–Thorin interpolation theorem. Let (Ω₁, Σ₁, μ₁) and (Ω₂, Σ₂, μ₂) be σ-finite measure spaces. Suppose 1 ≤ p₀ ≤ p₁ ≤ ∞, 1 ≤ q₀ ≤ q₁ ≤ ∞, and let T : L^p₀(μ₁) + L^p₁(μ₁) → L^q₀(μ₂) + L^q₁(μ₂) be a linear operator that maps L^p₀(μ₁) (resp. L^p₁(μ₁)) boundedly into L^q₀(μ₂) (resp. L^q₁(μ₂)). For 0 < θ < 1, let p_θ, q_θ be defined as above. Then T maps L^p_θ(μ₁) boundedly into L^q_θ(μ₂) and satisfies the operator norm estimate ∥ T ∥ L p θ → L q θ ≤ ∥ T ∥ L p 0 → L q 0 1 − θ ∥ T ∥ L p 1 → L q 1 θ .

In other words, if T is simultaneously of type (p₀, q₀) and of type (p₁, q₁), then T is of type (p_θ, q_θ) for all 0 < θ < 1. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of T is the collection of all points (1/p, 1/q) in the unit square [0, 1] × [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. The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that p₀ ≤ q₀ and p₁ ≤ q₁. Olof Thorin extended the interpolation theorem to the entire square, removing the lower-triangle restriction. The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis.

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

∥ T f ∥ q θ = sup ∥ g ∥ p θ ≤ 1 | ∫ ( T f ) g d μ 2 | .

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

ϕ ( z ) = ∫ ( T f z ) g z d μ 2

whose value at z = θ is

∫ ( T f ) g .

We can then use the hypotheses to establish upper bounds of Φ on the lines Re(z) = 0 and Re(z) = 1, whence the Hadamard three-lines theorem establishes the interpolated bound of Φ on the line Re(z) = θ. It now suffices to check that the bound at z = θ 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

φ ( z ) = ∫ ( T z f z ) g z d μ 2 ,

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

Stein interpolation theorem. Let (Ω₁, Σ₁, μ₁) and (Ω₂, Σ₂, μ₂) be σ-finite measure spaces. Suppose 1 ≤ p₀ , p₁ ≤ ∞, 1 ≤ q₀ , q₁ ≤ ∞, and define: S = {z ∈ C : 0 < Re(z) < 1} , S = {z ∈ C : 0 ≤ Re(z) ≤ 1} . We take a collection of linear operators {T_z : z ∈ S} on the space of simple functions in L¹(μ₁) into the space of all μ₂-measurable functions on Ω₂. We assume the following further properties on this collection of linear operators:

The mapping

is continuous on S and holomorphic on S for all simple functions  f  and g.

For some constant k < π, the operators satisfy the uniform bound:

T_z maps L^p₀(μ₁) boundedly to L^q₀(μ₂) whenever Re(z) = 0.

T_z maps L^p₁(μ₁) boundedly to L^q₁(μ₂) whenever Re(z) = 1.

The operator norms satisfy the uniform bound

Then, for each 0 < θ < 1, the operator T_θ maps L^p_θ(μ₁) boundedly into L^q_θ(μ₂).

The theory of real Hardy spaces and the space of bounded mean oscillations permits us to wield the Stein interpolation theorem argument in dealing with operators on the Hardy space H¹(R^d) and the space BMO of bounded mean oscillations; this is a result of Charles Fefferman and Elias Stein.

Hausdorff–Young inequality

It has been shown in the first section that the Fourier transform F maps L¹(R^d) boundedly into L^∞(R^d) and L²(R^d) into itself. A similar argument shows that the Fourier series operator that transforms periodic functions  f  : T → C into functions f ^ : Z → C whose values are the Fourier coefficients

f ^ ( n ) = 1 2 π ∫ − π π f ( x ) e − i n x d x

maps L¹(T) boundedly into ℓ^∞(Z) and L²(T) into ℓ²(Z). The Riesz–Thorin interpolation theorem now implies the following:

∥ F f ∥ L q ( R d ) ≤ ∥ f ∥ L p ( R d ) ∥ f ^ ∥ ℓ q ( Z ) ≤ ∥ f ∥ L p ( T )

where 1 ≤ p ≤ 2 and 1/p + 1/q = 1. This is the Hausdorff–Young inequality.

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

Convolution operators

Let  f  be a fixed integrable function and let 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 ≤ ∥ 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 L^p. Further, since g is in L^p we get, in view of Hölder's inequality, that S is bounded from L^q to L^∞, where again 1/p + 1/q = 1. So interpolating we get

∥ f ∗ g ∥ s ≤ ∥ f ∥ r ∥ g ∥ p

where the connection between p, r and s is

1 r + 1 p = 1 + 1 s .

The Hilbert transform

The Hilbert transform of  f  : R → C is given by

H f ( x ) = 1 π p . v . ∫ − ∞ ∞ f ( x − t ) t d t = ( 1 π p . v . 1 t ∗ 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:

H f ^ ( ξ ) = − i s g n ( ξ ) f ^ ( ξ ) .

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

Nevertheless, the Hilbert transform is not bounded on L¹(R) or L^∞(R), and so we cannot use the Riesz–Thorin interpolation theorem directly. To see why we do not have these endpoint bounds, it suffices to compute the Hilbert transform of the simple functions 1_(−1,1)(x) and 1_(0,1)(x) − 1_(0,1)(−x). We can show, however, that

( H f ) 2 = f 2 + 2 H ( f H f )

for all Schwartz functions  f  : R → C, and this identity can be used in conjunction with the Cauchy–Schwarz inequality to show that the Hilbert transform maps L²ⁿ(R^d) boundedly into itself for all n ≥ 2. Interpolation now establishes the bound

∥ H f ∥ p ≤ A p ∥ f ∥ p

for all 2 ≤ p < ∞, and the self-adjointness of the Hilbert transform can be used to carry over these bounds to the 1 < p ≤ 2 case.

Comparison with the real interpolation method

While the Riesz–Thorin interpolation theorem and its variants are powerful tools that yield a clean estimate on the interpolated operator norms, they suffer from numerous defects: some minor, some more severe. Note first that the complex-analytic nature of the proof of the Riesz–Thorin interpolation theorem forces the scalar field to be C. For extended-real-valued functions, this restriction can be bypassed by redefining the function to be finite everywhere—possible, as every integrable function must be finite almost everywhere. A more serious disadvantage is that, in practice, many operators, such as the Hardy–Littlewood maximal operator and the Calderón–Zygmund operators, do not have good endpoint estimates. In the case of the Hilbert transform in the previous section, we were able to bypass this problem by explicitly computing the norm estimates at several midway points. This is cumbersome and is often not possible in more general scenarios. Since many such operators satisfy the weak-type estimates

μ ( { x : T f ( x ) > α } ) ≤ ( C p , q ∥ f ∥ p α ) 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 ∥ ℓ 1 → ℓ 1 , ∥ A ∥ ℓ ∞ → ℓ ∞ ≤ M .

Then

∥ A ∥ X → X ≤ M

for any unconditional Banach space of sequences X, that is, for any ( x i ) ∈ X and any ( ε i ) ∈ { − 1 , 1 } ∞ , ∥ ( ε i x i ) ∥ X = ∥ ( x i ) ∥ X .

The proof is based on the Krein–Milman theorem.