Interpolation space

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives.

History

The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem. In simple terms, if a linear function is continuous on a certain space L^p and also on a certain space L^q, then it is also continuous on the space L^r, for any intermediate r between p and q. In other words, L^r is a space which is intermediate, or between L^p and L^q.

In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability.

Many methods were designed to generate such spaces of functions, including the Fourier transform, complex interpolation, real interpolation, as well as other tools (see e.g. fractional derivative).

Technical discussion

In order to discuss some of the main results of the theory, it is necessary for the reader to have some familiarity with the theory of Banach spaces. In this article, we are interested in the following situation. X and Z are Banach spaces, and X is a subset of Z, but the norm of X is not the same as the one of Z. An example of this can be obtained by taking X to be the Sobolev space $H^1(\mathbb{R})$ and by taking Z to be $L^2(\mathbb{R})$. We say that X is continuously included in Z when we have there is a constant $C<\infty$ such that

$\|u\|_Z \leq C \|u \|_X \quad \text{ for all } u \in X.$

This is the case of the example X = H¹ and Z = L².

Assume that we are given Banach spaces X and Y, and that they are both subsets of Z. Further define norms on $X \cap Y$ and X + Y by

$\|u\|_{X \cap Y} := \max ( \|u\|_X, \|u\|_Y )$

$\|u\|_{X + Y} := \inf \{ \|u_1\|_X + \|u_2\|_Y | u = u_1 + u_2, \; u_1 \in X, \; u_2 \in Y \}$

The following inclusions are all continuous:

$X \cap Y \subset X,Y \subset X+Y.$

(The space Z plays no further role, it was merely a tool that allows us to make sense of X+Y.) Our interest now is to come up with "intermediate spaces", between X and Y in the following sense:

Definition: With X and Y as above, an interpolation space is a Banach space W with the following property:

If L is a linear operator from X+Y into itself, which is continuous from X into itself and from Y into itself, then it is also continuous from W into itself.

The space W is further said to be of exponent θ (with 0<θ<1) if there exists a constant C such that

$\|L\|_{W;W} \leq C \|L\|_{X;X}^{1-\theta} \|L\|_{Y;Y}^{\theta} \quad \text{ for all such operators }L.$

We have used the notation $\|L\|_{A;B}$ to denote the norm of the operator L as a map from A to B. If C=1 (which is the smallest possible), we further say that W is an exact interpolation space.

There are many ways of obtaining interpolation spaces (and the Riesz-Thorin theorem is an example of this for L^p spaces). A method for arbitrary Banach spaces is the complex interpolation method.

Complex interpolation

If the field of scalars is the complex numbers, then we may use properties of complex analytic functions to define an interpolation space.

Definition: For two Banach spaces X and Y, the complex interpolation method consists in looking at the space of analytic functions f with values in X+Y, defined on the open strip $0 < \Re z < 1$, and continuous on the closed strip $0 \leq \Re z \leq 1$, and such that

f(iy) is bounded in X, f(1+iy) is bounded in Y.

We define the norm

$\|f\| = \max\{\sup_y\|f(iy)\|_X,\sup_y\|f(1+iy)\|_Y\}.$

For 0 < θ < 1, one defines

$[X,Y]_\theta = \{ u \in X+Y \}, \quad \text{ with the norm } \|u\| = \inf_{f(\theta)=u}\|f\|.$

It is then easy to show that we have the

Theorem: W = [X, Y]_θ is an exact interpolation space of exponent θ.

Real interpolation (the K-method)

The K-method of real interpolation can be used even when the field of scalars is the real numbers.

Definition: For any $u \in X+Y$, let $K(t,u)=\min_{u=u_1+u_2} \|u_1\|_X + t \|u_2\|_Y$ and let

$\|u\|_{\theta,q;K} = \left( \int_0^\infty t^{-\theta} (K(t,u))^q \, {dt \over t} \right)^{1/q}, \quad 1 \leq q \leq \infty.$

Then, the K-method of real interpolation consists in taking K_θ,q(X,Y) to be the set of all u in X+Y such that $\|u\|_{\theta,q;K}<\infty$.

Then, K_θ,q(X,Y) is an exact interpolation space of power θ.

Real interpolation (the J-method)

As with the K-method, the J-method can also be used for vector spaces over the real numbers.

Definition: For any $u \in X \cap Y$, let $J(t,u)=\max(\|u\|_X,t \|u\|_Y)$. Then, u is in J_θ,q(X,Y) if and only if it can be written as $u = \int_0^\infty v(t) \, dt/t$, where v(t) is measurable with values in $X \cap Y$, and such that

$\Phi(v) = \left( \int_0^\infty t^{-\theta} (J(t,v(t)))^q \, {dt \over t} \right)^{1/q} < \infty.$

The norm of u is $\|u\|_{\theta,q;J} := \inf_v \Phi(v)$.

Again, J_θ,q(X,Y) is an exact interpolation space of power θ.

Relations between the interpolation methods

The two real interpolation methods are often equivalent.

Theorem: If 0<θ<1 and 1 ≤ q ≤ ∞, then J_θ,q(X,Y) = K_θ,q(X,Y) with equivalence of norms.

When the two methods are equivalent, we write [X,Y]_θ,q for the real interpolation method. By contrast, the complex interpolation method is usually not equivalent to the real interpolation method. However, there is still a relationship.

Theorem: If 0 < θ < 1, then

$[X,Y]_{\theta,1} \subset [X,Y]_\theta \subset [X,Y]_{\theta,\infty}.\,$

References

• Bergh, Jöran; Löfström, Jörgen (1976), Interpolation Spaces: An Introduction, Springer-Verlag, ISBN 3-540-07875-4 
• Tartar, Luc (2007), An Introduction to Sobolev Spaces and Interpolation, Springer, ISBN 978-3-540-71482-8