Interpolation of Banach Spaces — Mathematics Knowledge System

Overview

Interpolation theory creates new Banach spaces 'between' two given spaces (X_0, X_1). The two main methods are the complex interpolation method [X_0, X_1]_theta (Calderon) and the real interpolation method (X_0, X_1)_{theta,q} (Lions–Peetre). If a linear operator T is bounded from X_0 to Y_0 and from X_1 to Y_1, then T is bounded on the interpolation spaces with an interpolated norm bound. This is the Riesz–Thorin theorem and Marcinkiewicz theorem in abstract form. Interpolation generates scales of spaces like the Sobolev and Besov families.

Intuition

Suppose you know that an operator T maps L^1 to L^1 and L^inf to L^inf. Can you conclude T maps L^2 to L^2? Interpolation theory says yes: L^2 = [L^1, L^inf]_{1/2} lies 'halfway between' L^1 and L^inf, and T's bound on the interpolation space is controlled by its bounds on the endpoints. More generally, interpolation lets you deduce boundedness on a whole 'scale' of spaces from just the endpoint cases.

Formal Definition

Definition

An interpolation couple is a pair (X_0, X_1) of Banach spaces continuously embedded in a Hausdorff topological vector space Z. The sum X_0 + X_1 and intersection X_0 cap X_1 are Banach spaces. Complex interpolation: [X_0, X_1]_theta = {x in X_0+X_1 : x = f(theta) for some analytic f: strip -> X_0+X_1 with f(it) in X_0, f(1+it) in X_1}. Real interpolation: the K-functional K(t,x;X_0,X_1) = inf{||x_0||_{X_0} + t||x_1||_{X_1} : x = x_0+x_1}, and (X_0,X_1)_{theta,q} = {x : ||x||_{theta,q} = (integral_0^inf (t^{-theta}K(t,x))^q dt/t)^{1/q} < inf}.

K(t,x; X_0,X_1) = \inf_{x=x_0+x_1}\left(\|x_0\|_{X_0} + t\|x_1\|_{X_1}\right)

K-functional (real interpolation)

\|x\|_{(X_0,X_1)_{\theta,q}} = \left(\int_0^\infty \left(t^{-\theta}K(t,x)\right)^q \frac{dt}{t}\right)^{1/q}

Real interpolation norm

\|T\|_{[X_0,X_1]_\theta \to [Y_0,Y_1]_\theta} \le \|T\|_{X_0\to Y_0}^{1-\theta}\|T\|_{X_1\to Y_1}^\theta

Interpolation bound (complex method)

Notation

Notation	Meaning
$[X_0,X_1]_\theta$	Complex interpolation space (theta in (0,1))
$(X_0,X_1)_{\theta,q}$	Real interpolation space
$K(t,x)$	Peetre K-functional
$(X_0,X_1)_{\theta,2}$	Real interpolation with q=2 (Lions–Peetre)

Theorems

Theorem 13.1: Theorem 13.1 (Riesz–Thorin interpolation)

If T: L^{p_0}(mu) -> L^{q_0}(nu) with norm M_0 and T: L^{p_1}(mu) -> L^{q_1}(nu) with norm M_1, then for theta in (0,1), with 1/p = (1-theta)/p_0 + theta/p_1 and 1/q = (1-theta)/q_0 + theta/q_1, T: L^p -> L^q with norm at most M_0^{1-theta} M_1^theta.

Theorem 13.2: Theorem 13.2 (Complex interpolation of L^p)

[L^{p_0}, L^{p_1}]_theta = L^p where 1/p = (1-theta)/p_0 + theta/p_1. Similarly for real interpolation: (L^{p_0}, L^{p_1})_{theta,p} = L^p with equivalent norms.

Theorem 13.3: Theorem 13.3

The Sobolev spaces satisfy [H^{s_0}, H^{s_1}]_theta = H^s where s = (1-theta)s_0 + theta s_1, and the Besov spaces satisfy (H^{s_0}, H^{s_1})_{theta,q} = B^s_{2,q}.

Worked Examples

1
The Fourier transform F: f |-> (f_hat(n)) acts as F: L^1(T) -> l^inf with norm ||F||_{1->inf} = 1 (trivially: |f_hat(n)| <= ||f||_1).
2
Parseval's theorem: F: L^2(T) -> l^2 is an isometry, so ||F||_{2->2} = 1.
3
By Riesz–Thorin with p_0 = 1, q_0 = inf, p_1 = 2, q_1 = 2, theta = 2/p' - 1 = 1 - 2/p' = (p-2)/(p-2p)...
$\frac{1}{p} = \frac{1-\theta}{1} + \frac{\theta}{2} \implies \theta = 2 - \frac{2}{p} = \frac{2(p-1)}{p}$
4
So F: L^p -> l^{p'} with norm at most 1^{1-theta} * 1^theta = 1. This is the Hausdorff–Young inequality.

✓ Answer

Riesz–Thorin interpolation between L^1 -> l^inf (trivial) and L^2 -> l^2 (Parseval) gives the Hausdorff–Young inequality ||f_hat||_{l^{p'}} <= ||f||_{L^p} for 1 <= p <= 2.

Practice Problems

Mediumfree response

State the Riesz–Thorin interpolation theorem and give one application.

MediumMultiple choice

The complex interpolation space [L^1(R), L^inf(R)]_theta is:

L^theta(R)L^{1/theta}(R)L^{1/(1-theta)}(R)L^p(R) where 1/p = 1 - theta

Quiz

Summary

Interpolation creates new Banach spaces between two spaces X_0 and X_1.
Complex interpolation [X_0,X_1]_theta and real interpolation (X_0,X_1)_{theta,q} are the two main methods.
Riesz–Thorin: bounded at two L^p endpoints implies bounded on all intermediate L^p, with interpolated norm bound.
K-functional K(t,x) = inf_{x=x_0+x_1}(||x_0||_{X_0} + t||x_1||_{X_1}) is the basic tool of real interpolation.
[L^{p_0}, L^{p_1}]_theta = L^p and [H^{s_0}, H^{s_1}]_theta = H^s (interpolation of scales).

References

BookBergh, J. & Lofstrom, J. — Interpolation Spaces: An Introduction, Springer, 1976
BookLunardi, A. — Interpolation Theory (3rd ed.), Edizioni della Normale, 2018
Websiteen.wikipedia.org