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 [[Lp space|space Template:Math]] and also on a certain space Template:Math, then it is also continuous on the space Template:Math, for any intermediate Template:Mvar between Template:Mvar and Template:Mvar. In other words, Template:Math is a space which is intermediate between Template:Math and Template:Math.

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

$X_0\cap X_1$

and

$X_0+X_1=\{z\in Z:z=x_0+x_1,x_0\in X_0,x_1\in X_1\}.$

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

One can define norms on Template:Math and Template:Math by

$\Vert x{\Vert }_{X_0\cap X_1}:=\max ({\Vert x\Vert }_{X_0},{\Vert x\Vert }_{X_1}),$

$\Vert x{\Vert }_{X_0+X_1}:=\inf \{{\Vert x_0\Vert }_{X_0}+{\Vert x_1\Vert }_{X_1}:x=x_0+x_1,x_0\in X_0,x_1\in X_1\}.$

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 Template:Mvar that are intermediate spaces between Template:Math and Template:Math 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 Template:Math, where the two Banach spaces are continuously embedded in the space Template:Mvar of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces Template:Math, for Template:Math are intermediate between Template:Math and Template:Math. More generally,

$L^{p_0}(𝐑)\cap L^{p_1}(𝐑)\subset L^p(𝐑)\subset L^{p_0}(𝐑)+L^{p_1}(𝐑),\text{when}1\le p_0\le p\le p_1\le \mathrm{\infty },$

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

Definition. Given two compatible couples Template:Math and Template:Math, an interpolation pair is a couple Template:Math of Banach spaces with the two following properties:

• The space X is intermediate between Template:Math and Template:Math, and Y is intermediate between Template:Math and Template:Math.
• If Template:Math is any linear operator from Template:Math to Template:Math, which maps continuously Template:Math to Template:Math and Template:Math to Template:Math, then it also maps continuously Template:Math to Template:Math.

The interpolation pair Template:Math is said to be of exponent Template:Mvar (with Template:Math) if there exists a constant Template:Math such that

$\Vert L{\Vert }_{X,Y}\le C\Vert L{\Vert }_{X_0,Y_0}^{1-\theta }\Vert L{\Vert }_{X_1,Y_1}^{\theta }$

for all operators Template:Mvar as above. The notation Template:Math is for the norm of Template:Math as a map from Template:Math to Template:Math. If Template:Math, we say that Template:Math is an exact interpolation pair of exponent Template:Mvar.

Complex interpolation

If the scalars are complex numbers, properties of complex analytic functions are used to define an interpolation space. Given a compatible couple (X₀, X₁) of Banach spaces, the linear space $ℱ(X_0,X_1)$ consists of all functions Template:Math, that are analytic on Template:Math continuous on Template:Math and for which all the following subsets are bounded:

Template:Math,

Template:Math,

Template:Math.

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

$\Vert f{\Vert }_{ℱ(X_0,X_1)}=\max \{\underset{t\in 𝐑}{\sup }\Vert f(it){\Vert }_{X_0},\underset{t\in 𝐑}{\sup }\Vert f(1+it){\Vert }_{X_1}\}.$

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

$(X_0,X_1{)}_{\theta }=\{x\in X_0+X_1:x=f(\theta ),f\in ℱ(X_0,X_1)\}.$

The norm on the complex interpolation space Template:Math is defined by

$\Vert x{\Vert }_{\theta }=\inf \{\Vert f{\Vert }_{ℱ(X_0,X_1)}:f(\theta )=x,f\in ℱ(X_0,X_1)\}.$

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

Theorem. Given two compatible couples of Banach spaces Template:Math and Template:Math, the pair Template:Math is an exact interpolation pair of exponent Template:Mvar, i.e., if Template:Math, is a linear operator bounded from Template:Math to Template:Math, then Template:Mvar is bounded from Template:Math to Template:Math and $$\Vert T{\Vert }_{\theta }\le \Vert T{\Vert }_0^{1-\theta }\Vert T{\Vert }_1^{\theta }.$$

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

${(L^{p_0}(R,\mathrm{\Sigma },\mu ),L^{p_1}(R,\mathrm{\Sigma },\mu ))}_{\theta }=L^p(R,\mathrm{\Sigma },\mu ),\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1},$

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 Template:Mvar is in Template:Math. 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 from 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 Template:Math of real numbers.

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

$K(x,t;X_0,X_1)=\inf \{{\Vert x_0\Vert }_{X_0}+t{\Vert x_1\Vert }_{X_1}:x=x_0+x_1,x_0\in X_0,x_1\in X_1\}.$

Changing the order of the two spaces results in:

$K(x,t;X_0,X_1)=tK(x,t^{-1};X_1,X_0).$

Let

$\begin{array}{cccc}\Vert x{\Vert }_{\theta ,q;K}& ={\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{(t^{-\theta }K(x,t;X_0,X_1))}^q\frac{dt}{t}\right)}^{\frac{1}{q}},& & 0<\theta <1,1\le q<\mathrm{\infty },\\ \Vert x{\Vert }_{\theta ,\mathrm{\infty };K}& =\underset{t>0}{\sup }{t}^{-\theta }K(x,t;X_0,X_1),& & 0\le \theta \le 1.\end{array}$

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

Example

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

$f_1(x)=\{\begin{array}{cc}f(x)& |f(x)|<s,\\ \frac{sf(x)}{|f(x)|}& \text{otherwise}\end{array}$

The optimal choice of Template:Mvar leads to the formula

$K(f,t;L^1,L^{\mathrm{\infty }})=\underset{0}{\overset{t}{\int }}{f}^{*}(u)du,$

where Template:Math is the decreasing rearrangement of Template:Math.

J-method

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

Definition. Let Template:Math be a compatible couple of Banach spaces. For Template:Math and for every vector Template:Math, let $$J(x,t;X_0,X_1)=\max (\Vert x{\Vert }_{X_0},t\Vert x{\Vert }_{X_1}).$$

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

$x=\underset{0}{\overset{\mathrm{\infty }}{\int }}v(t)\frac{dt}{t},$

where Template:Math is measurable with values in Template:Math and such that

$\mathrm{\Phi }(v)={\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{(t^{-\theta }J(v(t),t;X_0,X_1))}^q\frac{dt}{t}\right)}^{\frac{1}{q}}<\mathrm{\infty }.$

The norm of Template:Mvar in Template:Math is given by the formula

$\Vert x{\Vert }_{\theta ,q;J}:=\underset{v}{\inf }\{\mathrm{\Phi }(v):x=\underset{0}{\overset{\mathrm{\infty }}{\int }}v(t)\frac{dt}{t}\}.$

Relations between the interpolation methods

The two real interpolation methods are equivalent when Template:Math.

Theorem. Let Template:Math be a compatible couple of Banach spaces. If Template:Math and Template:Math, then $$J_{\theta ,q}(X_0,X_1)=K_{\theta ,q}(X_0,X_1),$$ with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if Template:Math and Template:Math 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 Template:Math, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters Template:Mvar and Template:Mvar. The notation for this real interpolation space is Template:Math. One has that

$(X_0,X_1{)}_{\theta ,q}=(X_1,X_0{)}_{1-\theta ,q},0<\theta <1,1\le q\le \mathrm{\infty }.$

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

$(X_0,X_1{)}_{\theta ,q}\subset (X_0,X_1{)}_{\theta ,r}.$

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

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 Template:Math be a compatible couple of Banach spaces. If Template:Math, then $$(X_0,X_1{)}_{\theta ,1}\subset (X_0,X_1{)}_{\theta }\subset (X_0,X_1{)}_{\theta ,\mathrm{\infty }}.$$

Examples

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

$\sup \{|f(u)|,\frac{|f(u)-f(v)|}{1+t^{-1}|u-v|}:u,v\in [0,1]\}.$

Only values Template:Math are interesting here.

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

${(L^1(𝐑,\mathrm{\Sigma },\mu ),L^{\mathrm{\infty }}(𝐑,\mathrm{\Sigma },\mu ))}_{\theta ,q}=L^{p,q}(𝐑,\mathrm{\Sigma },\mu ),\text{where }\frac{1}{p}=1-\theta ,$

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 Template:Math, the Lorentz space Template:Math is equal to Template:Math, up to renorming. When Template:Math, the Lorentz space Template:Math is equal to [[Lp space#Weak Lp|weak-Template:Math]].

The reiteration theorem

An intermediate space Template:Mvar of the compatible couple Template:Math is said to be of class θ if

$(X_0,X_1{)}_{\theta ,1}\subset X\subset (X_0,X_1{)}_{\theta ,\mathrm{\infty }},$

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

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

Theorem. Let Template:Math be intermediate spaces of the compatible couple Template:Math, of class Template:Math and Template:Math respectively, with Template:Math. When Template:Math and Template:Math, one has $$(A_0,A_1{)}_{\theta ,q}=(X_0,X_1{)}_{\eta ,q},\eta =(1-\theta ){\theta }_0+\theta {\theta }_1.$$

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

There is also a reiteration theorem for the complex method.

Theorem. Let Template:Math be a compatible couple of complex Banach spaces, and assume that Template:Math is dense in Template:Math and in Template:Math. Let Template:Math and Template:Math, where Template:Math. Assume further that Template:Math is dense in Template:Math. Then, for every Template:Math, $${({(X_0,X_1)}_{{\theta }_0},{(X_0,X_1)}_{{\theta }_1})}_{\theta }=(X_0,X_1{)}_{\eta },\eta =(1-\theta ){\theta }_0+\theta {\theta }_1.$$

The density condition is always satisfied when Template:Math or Template:Math.

Duality

Let Template:Math be a compatible couple, and assume that Template:Math is dense in X₀ and in X₁. In this case, the restriction map from the (continuous) dual $X{\text{'}}_j$ of Template:Math, Template:Math to the dual of Template:Math is one-to-one. It follows that the pair of duals $(X{\text{'}}_0,X{\text{'}}_1)$ is a compatible couple continuously embedded in the dual Template:Math.

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

Theorem. Let Template:Math be a compatible couple of complex Banach spaces, and assume that Template:Math is dense in Template:Math and in Template:Math. If Template:Math and Template:Math are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals, $$((X_0,X_1{)}_{\theta }{)}^{\prime }={(X{\text{'}}_0,X{\text{'}}_1)}_{\theta },0<\theta <1.$$

In general, the dual of the space Template:Math is equal to ${(X{\text{'}}_0,X{\text{'}}_1)}^{\theta },$ 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 X₀, X₁ is a reflexive space.

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

Theorem. Let Template:Math and Template:Math a compatible couple of real Banach spaces. Assume that Template:Math is dense in Template:Math and in Template:Math. Then $$\left({(X_0,X_1)}_{\theta ,q}\right)={(X{\text{'}}_0,X{\text{'}}_1)}_{\theta ,q^{\prime }},$$ where $\frac{1}{q^{\prime }}=1-\frac{1}{q}.$

Discrete definitions

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

$\Vert x{\Vert }_{\theta ,q;K}\simeq {\left(\sum _{n\in 𝐙}{\left(2^{-\theta n}K(x,2^n;X_0,X_1)\right)}^q\right)}^{\frac{1}{q}}.$

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

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

$\Vert x^{\prime }{\Vert }_{\theta ,p;J}\simeq \inf \{{\left(\sum _{n\in 𝐙}{(2^{\theta n}\max ({\Vert x{\text{'}}_n\Vert }_{X{\text{'}}_0},2^{-n}{\Vert x{\text{'}}_n\Vert }_{X{\text{'}}_1}))}^p\right)}^{\frac{1}{p}}:x^{\prime }=\sum _{n\in 𝐙}x{\text{'}}_n\}.$

The usual formula for the discrete Template:Math-norm is obtained by changing Template:Mvar to Template:Math.

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 Template:Mvar is compact from Template:Math to a Banach space Template:Mvar and bounded from Template:Math to Template:Mvar, then Template:Mvar is compact from Template:Math to Template:Mvar when Template:Math, Template:Math.

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

$a_n,b_n>0,\sum _{n=1}^{\mathrm{\infty }}\min (a_n,b_n)<\mathrm{\infty }.$

The interpolation space Template:Math consists of the vectors Template:Mvar in Template:Math such that

$\Vert x{\Vert }_{K(X_0,X_1)}=\underset{m\ge 1}{\sup }{\Vert \sum _{n=1}^m{a}_{n}K(x,\frac{b_n}{a_n};X_0,X_1)y_n\Vert }_Y<\mathrm{\infty },$

where {y_n} is the unconditional basis of Template:Mvar. 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 Rⁿ,

$\begin{array}{cccc}& H_p^s& & s\in 𝐑,1\le p\le \mathrm{\infty }\\ & B_{p,q}^s& & s\in 𝐑,1\le p,q\le \mathrm{\infty }\end{array}$

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

$\begin{array}{cc}0& <\theta <1,\\ 1& \le p,p_0,p_1,q,q_0,q_1\le \mathrm{\infty },\\ s,& s_0,s_1\in 𝐑,\\ s_{\theta }& =(1-\theta )s_0+\theta s_1,\\ \frac{1}{p_{\theta }}& =\frac{1-\theta }{p_0}+\frac{\theta }{p_1},\\ \frac{1}{q_{\theta }}& =\frac{1-\theta }{q_0}+\frac{\theta }{q_1}.\end{array}$

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

$\begin{array}{cccc}{(H_{p_0}^{s_0},H_{p_1}^{s_1})}_{\theta }& =H_{p_{\theta }}^{s_{\theta }},& & s_0\ne s_1,1<p_0,p_1<\mathrm{\infty }.\\ {(B_{p_0,q_0}^{s_0},B_{p_1,q_1}^{s_1})}_{\theta }& =B_{p_{\theta },q_{\theta }}^{s_{\theta }},& & s_0\ne s_1.\end{array}$

Real interpolation between Sobolev spaces may give Besov spaces, except when Template:Math,

${(H_{p_0}^s,H_{p_1}^s)}_{\theta ,p_{\theta }}=H_{p_{\theta }}^s.$

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

${(H_p^{s_0},H_p^{s_1})}_{\theta ,q}=B_{p,q}^{s_{\theta }},s_0\ne s_1.$

Also,

$\begin{array}{cccc}{(B_{p,q_0}^{s_0},B_{p,q_1}^{s_1})}_{\theta ,q}& =B_{p,q}^{s_{\theta }},& & s_0\ne s_1.\\ {(B_{p,q_0}^s,B_{p,q_1}^s)}_{\theta ,q}& =B_{p,q_{\theta }}^s.\\ {(B_{p_0,q_0}^{s_0},B_{p_1,q_1}^{s_1})}_{\theta ,q_{\theta }}& =B_{p_{\theta },q_{\theta }}^{s_{\theta }},& & s_0\ne s_1,p_{\theta }=q_{\theta }.\end{array}$

See also

• Fundamental lemma of interpolation theory
• Riesz–Thorin theorem
• Marcinkiewicz interpolation theorem

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. Template:ISBN.
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