Interpolation Space

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach 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 and also on a certain space , then it is also continuous on the space , for any intermediate between and . In other words, is a space which is intermediate between and .

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).

The setting of interpolation

A Banach space is said to be ''continuously embedded'' in a Hausdorff topological vector space when is a linear subspace of such that the inclusion map from into is continuous. A compatible couple of Banach spaces consists of two Banach spaces and that are continuously embedded in the same Hausdorff topological vector space . The embedding in a linear space allows to consider the two linear subspaces

:$X_0 \cap X_1$

and

:$X_0 + X_1 = \left \.$

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of and . It depends in an essential way from the specific ''relative position'' that and occupy in a larger space .

One can define norms on and by

:$\, x\, _ := \max \left ( \left \, x \right \, _, \left \, x \right \, _ \right ),$
:$\, x\, _ := \inf \left \.$

Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:

:$X_0 \cap X_1 \subset X_0, \ X_1 \subset X_0 + X_1.$

Interpolation studies the family of spaces that are intermediate spaces between and in the sense that

:$X_0 \cap X_1 \subset X \subset X_0 + X_1,$

where the two inclusions maps are continuous.

An example of this situation is the pair , where the two Banach spaces are continuously embedded in the space of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces , for are intermediate between and . More generally,

:$L^(\mathbf) \cap L^(\mathbf) \subset L^p(\mathbf) \subset L^(\mathbf) + L^(\mathbf), \ \ \text \ \ 1 \le p_0 \le p \le p_1 \le \infty,$

with continuous injections, so that, under the given condition, is intermediate between and .

:Definition. Given two compatible couples and , an interpolation pair is a couple of Banach spaces with the two following properties:
:*The space ''X'' is intermediate between and , and ''Y'' is intermediate between and .
:*If is any linear operator from to , which maps continuously to and to , then it also maps continuously to .

The interpolation pair is said to be of exponent (with ) if there exists a constant such that
:$\, L\, _ \leq C \, L\, _^ \; \, L\, _^$
for all operators as above. The notation is for the norm of as a map from to . If , we say that is an exact interpolation pair of exponent .

Complex interpolation

If the scalars are complex numbers, properties of complex analytic functions are used to define an interpolation space. Given a compatible couple (''X''₀, ''X''₁) of Banach spaces, the linear space $\mathcal(X_0, X_1)$ consists of all functions , that are analytic on continuous on and for which all the following subsets are bounded:

:,
:,
:.

$\mathcal(X_0, X_1)$ is a Banach space under the norm

:$\, f\, _ = \max \left\.$

Definition. For , the complex interpolation space is the linear subspace of consisting of all values ''f''(''θ'') when ''f'' varies in the preceding space of functions,

:$(X_0, X_1)_\theta = \left \.$

The norm on the complex interpolation space is defined by

:$\ \, x\, _\theta = \inf \left \.$

Equipped with this norm, the complex interpolation space is a Banach space.

:Theorem. Given two compatible couples of Banach spaces and , the pair is an exact interpolation pair of exponent , i.e., if , is a linear operator bounded from to , then is bounded from to and $\, T\, _\theta \le \, T\, _0^ \, T\, _1^\theta.$

The family of spaces (consisting of complex valued functions) behaves well under complex interpolation. If is an arbitrary measure space, if and , then

:$\left( L^(R, \Sigma, \mu), L^(R, \Sigma, \mu) \right)_\theta = L^p(R, \Sigma, \mu), \qquad \frac = \frac + \frac,$

with equality of norms. This fact is closely related to the Riesz–Thorin theorem.

Real interpolation

There are two ways for introducing the real interpolation method. The first and most commonly used when actually identifying examples of interpolation spaces is the K-method. The second method, the J-method, gives the same interpolation spaces as the K-method when the parameter is in . That the J- and K-methods agree is important for the study of duals of interpolation spaces: basically, the dual of an interpolation space constructed by the K-method appears to be a space constructed form the dual couple by the J-method; see below.

K-method

The K-method of real interpolation can be used for Banach spaces over the field of real numbers.

Definition. Let be a compatible couple of Banach spaces. For and every , let

:$K(x, t; X_0, X_1) = \inf \left \.$

Changing the order of the two spaces results in:

:$K(x, t; X_0, X_1) = t K \left (x, t^; X_1, X_0 \right).$

Let

:$\begin \, x\, _ &= \left( \int_0^\infty \left( t^ K(x, t; X_0, X_1) \right)^q \, \tfrac \right)^, && 0 < \theta < 1, 1 \leq q < \infty, \\ \, x\, _ &= \sup_ \; t^ K(x, t; X_0, X_1), && 0 \le \theta \le 1. \end$

The K-method of real interpolation consists in taking to be the linear subspace of consisting of all such that .

Example

An important example is that of the couple , where the functional can be computed explicitly. The measure is supposed -finite. In this context, the best way of cutting the function as sum of two functions and is, for some to be chosen as function of , to let be given for all by

:$f_1(x) = \begin f(x) & , f(x), < s, \\ \frac & \text \end$

The optimal choice of leads to the formula

:$K \left (f, t; L^1, L^\infty \right ) = \int_0^t f^*(u) \, d u,$

where is the decreasing rearrangement of .

J-method

As with the K-method, the J-method can be used for real Banach spaces.

Definition. Let be a compatible couple of Banach spaces. For and for every vector , let $J(x, t; X_0, X_1) = \max \left ( \, x\, _, t \, x\, _ \right ).$

A vector in belongs to the interpolation space if and only if it can be written as

:$x = \int_0^\infty v(t) \, \frac,$

where is measurable with values in and such that

:$\Phi(v) = \left( \int_0^\infty \left( t^ J(v(t), t; X_0, X_1) \right)^q \, \tfrac \right)^ < \infty.$

The norm of in is given by the formula

:$\, x\, _ := \inf_v \left\.$

Relations between the interpolation methods

The two real interpolation methods are equivalent when .

:Theorem. Let be a compatible couple of Banach spaces. If and , then $J_(X_0, X_1) = K_(X_0, X_1),$ with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if and form a direct sum, then the intersection and the J-spaces are the null space, and a simple computation shows that the K-spaces are also null.

When , one can speak, up to an equivalent renorming, about ''the'' Banach space obtained by the real interpolation method with parameters and . The notation for this real interpolation space is . One has that

:$(X_0, X_1)_ = (X_1, X_0)_, \qquad 0 < \theta < 1, 1 \le q \le \infty.$

For a given value of , the real interpolation spaces increase with : if and , the following continuous inclusion holds true:

:$(X_0, X_1)_ \subset (X_0, X_1)_.$

:Theorem. Given , and two compatible couples and , the pair is an exact interpolation pair of exponent .

A complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method. However, there is a general relationship.

:Theorem. Let be a compatible couple of Banach spaces. If , then $(X_0, X_1)_ \subset (X_0, X_1)_\theta \subset (X_0, X_1)_.$

Examples

When and , the space of continuously differentiable functions on , the interpolation method, for , gives the Hölder space of exponent . This is because the K-functional of this couple is equivalent to

:$\sup \left\.$

Only values are interesting here.

Real interpolation between spaces gives the family of Lorentz spaces. Assuming and , one has:

:$\left ( L^1(\mathbf, \Sigma, \mu), L^\infty(\mathbf, \Sigma, \mu) \right)_ = L^(\mathbf, \Sigma, \mu), \qquad \text \tfrac = 1 - \theta,$

with equivalent norms. This follows from an inequality of Hardy and from the value given above of the K-functional for this compatible couple. When , the Lorentz space is equal to , up to renorming. When , the Lorentz space is equal to weak-.

The reiteration theorem

An intermediate space of the compatible couple is said to be of class ''θ'' if

:$(X_0, X_1)_ \subset X \subset (X_0, X_1)_,$

with continuous injections. Beside all real interpolation spaces with parameter and , the complex interpolation space is an intermediate space of class of the compatible couple .

The reiteration theorems says, in essence, that interpolating with a parameter behaves, in some way, like forming a convex combination : taking a further convex combination of two convex combinations gives another convex combination.

:Theorem. Let be intermediate spaces of the compatible couple , of class and respectively, with . When and , one has $(A_0, A_1)_ = (X_0, X_1)_, \qquad \eta = (1 - \theta) \theta_0 + \theta \theta_1.$

It is notable that when interpolating with the real method between and , only the values of and matter. Also, and can be complex interpolation spaces between and , with parameters and respectively.

There is also a reiteration theorem for the complex method.

:Theorem. Let be a compatible couple of complex Banach spaces, and assume that is dense in and in . Let and , where . Assume further that is dense in . Then, for every , $\left( \left (X_0, X_1 \right )_, \left (X_0, X_1 \right )_ \right)_\theta = (X_0, X_1)_\eta, \qquad \eta = (1 - \theta) \theta_0 + \theta \theta_1.$

The density condition is always satisfied when or .

Duality

Let be a compatible couple, and assume that is dense in ''X''₀ and in ''X''₁. In this case, the restriction map from the (continuous) dual $X'_j$ of , to the dual of is one-to-one. It follows that the pair of duals $\left (X'_0, X'_1 \right )$ is a compatible couple continuously embedded in the dual .

For the complex interpolation method, the following duality result holds:

:Theorem.see 12.1 and 12.2, p. 121 in . Let be a compatible couple of complex Banach spaces, and assume that is dense in and in . If and are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals, $( (X_0, X_1)_\theta )' = \left(X'_0, X'_1 \right )_\theta, \qquad 0 < \theta < 1.$

In general, the dual of the space is equal to $\left (X'_0, X'_1 \right )^,$ a space defined by a variant of the complex method. The upper-θ and lower-θ methods do not coincide in general, but they do if at least one of ''X''₀, ''X''₁ is a reflexive space.

For the real interpolation method, the duality holds provided that the parameter ''q'' is finite:

:Theorem. Let and a compatible couple of real Banach spaces. Assume that is dense in and in . Then $\left ( \left (X_0, X_1 \right )_ \right )' = \left (X'_0, X'_1 \right )_,$ where $\tfrac = 1 - \tfrac.$

Discrete definitions

Since the function varies regularly (it is increasing, but is decreasing), the definition of the -norm of a vector , previously given by an integral, is equivalent to a definition given by a series. This series is obtained by breaking into pieces of equal mass for the measure ,

:$\, x\, _ \simeq \left( \sum_ \left( 2^ K \left (x, 2^n; X_0, X_1 \right ) \right)^q \right)^.$

In the special case where is continuously embedded in , one can omit the part of the series with negative indices . In this case, each of the functions defines an equivalent norm on .

The interpolation space is a "diagonal subspace" of an -sum of a sequence of Banach spaces (each one being isomorphic to ). Therefore, when is finite, the dual of is a quotient of the -sum of the duals, , which leads to the following formula for the discrete -norm of a functional ''x in the dual of :

:$\, x'\, _ \simeq \inf \left\.$

The usual formula for the discrete -norm is obtained by changing to .

The discrete definition makes several questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators. Lions and Peetre have proved that:

:Theorem. If the linear operator is compact from to a Banach space and bounded from to , then is compact from to when , .

Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:

:Theorem. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.

A general interpolation method

The space used for the discrete definition can be replaced by an arbitrary sequence space ''Y'' with unconditional basis, and the weights , , that are used for the -norm, can be replaced by general weights

:$a_n, b_n > 0, \ \ \sum_^\infty \min(a_n, b_n) < \infty.$

The interpolation space consists of the vectors in such that

:$\, x\, _ = \sup_ \left \, \sum_^m a_n K \left (x, \tfrac; X_0, X_1 \right) \, y_n \right\, _Y < \infty,$

where is the unconditional basis of . This abstract method can be used, for example, for the proof of the following result:

Theorem. A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with symmetric basis.

Interpolation of Sobolev and Besov spaces

Several interpolation results are available for Sobolev spaces and Besov spaces on R^''n'',Theorem 6.4.5, p. 152 in .

:$\begin &H^s_p && s \in \mathbf, 1 \le p \le \infty \\ &B^s_ && s \in \mathbf, 1 \le p, q \le \infty \end$

These spaces are spaces of measurable functions on when , and of tempered distributions on when . For the rest of the section, the following setting and notation will be used:

:\begin
0 &< \theta < 1, \\
1 &\le p, p_0, p_1, q, q_0, q_1 \le \infty, \\
s, &s_0, s_1 \in \mathbf, \\
s_\theta &= (1 - \theta) s_0 + \theta s_1, \\ pt\frac 1 &= \frac + \frac, \\ pt\frac 1 &= \frac + \frac.
\end

Complex interpolation works well on the class of Sobolev spaces $H^_$ (the Bessel potential spaces) as well as Besov spaces:

:$\begin \left (H^_, H^_ \right )_\theta &= H^_, && s_0 \ne s_1, 1 < p_0, p_1 < \infty. \\ \left (B^_, B^_ \right)_\theta &= B^_, && s_0 \ne s_1. \end$

Real interpolation between Sobolev spaces may give Besov spaces, except when ,

:$\left (H^_, H^_ \right)_ = H^_.$

When but , real interpolation between Sobolev spaces gives a Besov space:

:$\left (H^_p, H^_p \right)_ = B^_, \qquad s_0 \ne s_1.$

Also,

:$\begin \left (B^_, B^_ \right)_ &= B^_, && s_0 \ne s_1. \\ \left (B^s_, B^s_ \right )_ &= B^_. \\ \left (B^_, B^_ \right )_ &= B^_, && s_0 \ne s_1, p_\theta =q_\theta. \end$

See also

* Riesz–Thorin theorem
* Marcinkiewicz interpolation theorem

Notes

References

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*Leoni, Giovanni (2017).
A First Course in Sobolev Spaces: Second Edition
'. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. .
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{{Topological vector spaces

Banach spaces
Fourier analysis
Sobolev spaces