In the field ofmathematical analysis, aninterpolation space is a space which lies "in between" two otherBanach spaces. The main applications are inSobolev spaces, where spaces of functions that have a noninteger number ofderivatives are interpolated from the spaces of functions with integer number of derivatives.

The theory of interpolation of vector spaces began by an observation ofJózef Marcinkiewicz, later generalized and now known as theRiesz-Thorin theorem. In simple terms, if a linear function is continuous on a certainspaceL^p and also on a certain spaceL^q, then it is also continuous on the spaceL^r, for any intermediater betweenp andq. In other words,L^r is a space which is intermediate betweenL^p andL^q.

In the development of Sobolev spaces, it became clear that thetrace spaces were not any of the usual function spaces (with integer number of derivatives), andJacques-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 theFourier transform, complex interpolation,^[1] real interpolation,^[2] as well as other tools (see e.g.fractional derivative).

ABanach spaceX is said to becontinuously embedded in aHausdorfftopological vector spaceZ whenX is a linear subspace ofZ such that the inclusion map fromX intoZ is continuous. Acompatible couple(X₀,X₁) of Banach spaces consists of two Banach spacesX₀ andX₁ that are continuously embedded in the same Hausdorff topological vector spaceZ.^[3] The embedding in a linear spaceZ allows to consider the two linear subspaces

${\displaystyle X_0\cap X_1}$

and

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

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes ofX₀ andX₁. It depends in an essential way from the specificrelative position thatX₀ andX₁ occupy in a larger spaceZ.

One can define norms onX₀ ∩X₁ andX₀ +X₁ by

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

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

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

${\displaystyle X_0\cap X_1\subset X_0,X_1\subset X_0+X_1.}$

Interpolation studies the family of spacesX that areintermediate spaces betweenX₀ andX₁ in the sense that

${\displaystyle X_0\cap X_1\subset X\subset X_0+X_1,}$

where the two inclusions maps are continuous.

An example of this situation is the pair(L¹(R),L^∞(R)), where the two Banach spaces are continuously embedded in the spaceZ of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spacesL^p(R), for1 ≤p ≤ ∞ are intermediate betweenL¹(R) andL^∞(R). More generally,

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

with continuous injections, so that, under the given condition,L^p(R) is intermediate betweenL^p₀(R) andL^p₁(R).

Definition. Given two compatible couples(X₀,X₁) and(Y₀,Y₁), aninterpolation pair is a couple(X,Y) of Banach spaces with the two following properties:

• The spaceX is intermediate betweenX₀ andX₁, andY is intermediate betweenY₀ andY₁.
• IfL is any linear operator fromX₀ +X₁ toY₀ +Y₁, which maps continuouslyX₀ toY₀ andX₁ toY₁, then it also maps continuouslyX toY.

The interpolation pair(X,Y) is said to be ofexponentθ (with0 <θ < 1) if there exists a constantC such that

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

for all operatorsL as above. The notation||L||_X,Y is for the norm ofL as a map fromX toY. IfC = 1, we say that(X,Y) is anexact interpolation pair of exponentθ.

If the scalars arecomplex numbers, properties of complexanalytic functions are used to define an interpolation space. Given a compatible couple (X₀,X₁) of Banach spaces, the linear space${\displaystyle \mathcal{F}(X_0,X_1)}$ consists of all functions f  :C →X₀ +X₁, that are analytic onS = {z : 0 < Re(z) < 1}, continuous onS = {z : 0 ≤ Re(z) ≤ 1}, and for which all the following subsets are bounded:

{ f (z) :z ∈S} ⊂X₀ +X₁,

{ f (it) :t ∈R} ⊂X₀,

{ f (1 +it) :t ∈R} ⊂X₁.

${\displaystyle \mathcal{F}(X_0,X_1)}$ is a Banach space under the norm

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

Definition.^[4] For0 <θ < 1, thecomplex interpolation space(X₀,X₁)_θ is the linear subspace ofX₀ +X₁ consisting of all valuesf(θ) whenf varies in the preceding space of functions,

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

The norm on the complex interpolation space(X₀,X₁)_θ is defined by

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

Equipped with this norm, the complex interpolation space(X₀,X₁)_θ is a Banach space.

Theorem.^[5] Given two compatible couples of Banach spaces(X₀,X₁) and(Y₀,Y₁), the pair((X₀,X₁)_θ, (Y₀,Y₁)_θ) is an exact interpolation pair of exponentθ, i.e., ifT :X₀ +X₁ →Y₀ +Y₁, is a linear operator bounded fromX_j toY_j,j = 0, 1, thenT is bounded from(X₀,X₁)_θ to(Y₀,Y₁)_θ and$${\displaystyle \Vert T{\Vert }_{\theta }\le \Vert T{\Vert }_0^{1-\theta }\Vert T{\Vert }_1^{\theta }.}$$

The family ofL^p spaces (consisting of complex valued functions) behaves well under complex interpolation.^[6] If(R, Σ,μ) is an arbitrarymeasure space, if1 ≤p₀,p₁ ≤ ∞ and0 <θ < 1, then

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

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

There are two ways for introducing thereal 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 parameterθ is in(0, 1). 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.

The K-method of real interpolation^[7] can be used for Banach spaces over the fieldR ofreal numbers.

Definition. Let(X₀,X₁) be a compatible couple of Banach spaces. Fort > 0 and everyx ∈X₀ +X₁, let

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

Changing the order of the two spaces results in:^[8]

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

Let

The K-method of real interpolation consists in takingK_θ,q(X₀,X₁) to be the linear subspace ofX₀ +X₁ consisting of allx such that||x||_θ,q;K < ∞.

An important example is that of the couple(L¹(R, Σ,μ),L^∞(R, Σ,μ)), where the functionalK(t,f ;L¹,L^∞) can be computed explicitly. The measureμ is supposedσ-finite. In this context, the best way of cutting the function f  ∈L¹ +L^∞ as sum of two functions f₀ ∈L¹  and f₁ ∈L^∞  is, for somes > 0 to be chosen as function oft, to let f₁(x) be given for allx ∈R by

The optimal choice ofs leads to the formula^[9]

${\displaystyle K\left(f,t;L^1,L^{\mathrm{\infty }}\right)={\int }_0^t{f}^{\ast }(u)du,}$

where f ^∗ is thedecreasing rearrangement of f .

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

Definition. Let(X₀,X₁) be a compatible couple of Banach spaces. Fort > 0 and for every vectorx ∈X₀ ∩X₁, let$${\displaystyle J(x,t;X_0,X_1)=max\left(\Vert x{\Vert }_{X_0},t\Vert x{\Vert }_{X_1}\right).}$$

A vectorx inX₀ +X₁ belongs to the interpolation spaceJ_θ,q(X₀,X₁) if and only if it can be written as

${\displaystyle x={\int }_0^{\mathrm{\infty }}v(t)\frac{dt}{t},}$

wherev(t) is measurable with values inX₀ ∩X₁ and such that

The norm ofx inJ_θ,q(X₀,X₁) is given by the formula

${\displaystyle \Vert x{\Vert }_{\theta ,q;J}:=\underset{v}{inf}\left\{\mathrm{\Phi }(v):x={\int }_0^{\mathrm{\infty }}v(t)\frac{dt}{t}\right\}.}$

The two real interpolation methods are equivalent when0 <θ < 1.^[10]

Theorem. Let(X₀,X₁) be a compatible couple of Banach spaces. If0 <θ < 1 and1 ≤q ≤ ∞, then$${\displaystyle J_{\theta ,q}(X_0,X_1)=K_{\theta ,q}(X_0,X_1),}$$ withequivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example ifX₀ andX₁ 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.

When0 <θ < 1, one can speak, up to an equivalent renorming, aboutthe Banach space obtained by the real interpolation method with parametersθ andq. The notation for this real interpolation space is(X₀,X₁)_θ,q. One has that

For a given value ofθ, the real interpolation spaces increase withq:^[11] if0 <θ < 1 and 1 ≤q ≤r ≤ ∞, the following continuous inclusion holds true:

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

Theorem. Given0 <θ < 1,1 ≤q ≤ ∞ and two compatible couples(X₀,X₁) and(Y₀,Y₁), the pair((X₀,X₁)_θ,q, (Y₀,Y₁)_θ,q) is an exact interpolation pair of exponentθ.^[12]

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(X₀,X₁) be a compatible couple of Banach spaces. If0 <θ < 1, then$${\displaystyle (X_0,X_1{)}_{\theta ,1}\subset (X_0,X_1{)}_{\theta }\subset (X_0,X_1{)}_{\theta ,\mathrm{\infty }}.}$$

WhenX₀ =C([0, 1]) andX₁ =C¹([0, 1]), the space of continuously differentiable functions on[0, 1], the(θ, ∞) interpolation method, for0 <θ < 1, gives theHölder spaceC^0,θ of exponentθ. This is because the K-functionalK(f,t;X₀,X₁) of this couple is equivalent to

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

Only values0 <t < 1 are interesting here.

Real interpolation betweenL^p spaces gives^[13] the family ofLorentz spaces. Assuming0 <θ < 1 and1 ≤q ≤ ∞, one has:

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

with equivalent norms. This follows from aninequality of Hardy and from the value given above of the K-functional for this compatible couple. Whenq =p, the Lorentz spaceL^p,p is equal toL^p, up to renorming. Whenq = ∞, the Lorentz spaceL^p,∞ is equal toweak-L^p.

An intermediate spaceX of the compatible couple(X₀,X₁) is said to be ofclassθ if^[14]

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

with continuous injections. Beside all real interpolation spaces(X₀,X₁)_θ,q with parameterθ and1 ≤q ≤ ∞, the complex interpolation space(X₀,X₁)_θ is an intermediate space of classθ of the compatible couple(X₀,X₁).

The reiteration theorems says, in essence, that interpolating with a parameterθ behaves, in some way, like forming aconvex combinationa = (1 −θ)x₀ +θx₁: taking a further convex combination of two convex combinations gives another convex combination.

Theorem.^[15] LetA₀,A₁ be intermediate spaces of the compatible couple(X₀,X₁), of classθ₀ andθ₁ respectively, with0 <θ₀ ≠θ₁ < 1. When0 <θ < 1 and1 ≤q ≤ ∞, one has$${\displaystyle (A_0,A_1{)}_{\theta ,q}=(X_0,X_1{)}_{\eta ,q},\eta =(1-\theta ){\theta }_0+\theta {\theta }_1.}$$

It is notable that when interpolating with the real method betweenA₀ = (X₀,X₁)_θ0,q0 andA₁ = (X₀,X₁)_θ1,q1, only the values ofθ₀ andθ₁ matter. Also,A₀ andA₁ can be complex interpolation spaces betweenX₀ andX₁, with parametersθ₀ andθ₁ respectively.

There is also a reiteration theorem for the complex method.

Theorem.^[16] Let(X₀,X₁) be a compatible couple of complex Banach spaces, and assume thatX₀ ∩X₁ is dense inX₀ and inX₁. LetA₀ = (X₀,X₁)_θ0 andA₁ = (X₀,X₁)_θ1, where0 ≤θ₀ ≤θ₁ ≤ 1. Assume further thatX₀ ∩X₁ is dense inA₀ ∩A₁. Then, for every0 ≤θ ≤ 1,$${\displaystyle {\left({\left(X_0,X_1\right)}_{{\theta }_0},{\left(X_0,X_1\right)}_{{\theta }_1}\right)}_{\theta }=(X_0,X_1{)}_{\eta },\eta =(1-\theta ){\theta }_0+\theta {\theta }_1.}$$

The density condition is always satisfied whenX₀ ⊂X₁ orX₁ ⊂X₀.

Let(X₀,X₁) be a compatible couple, and assume thatX₀ ∩X₁ is dense inX₀ and inX₁. In this case, the restriction map from the (continuous)dual${\displaystyle X_j^{\prime }}$ ofX_j,j = 0, 1, to the dual ofX₀ ∩X₁ is one-to-one. It follows that the pair of duals${\displaystyle \left(X_0^{\prime },X_1^{\prime }\right)}$ is a compatible couple continuously embedded in the dual(X₀ ∩X₁)′.

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

Theorem.^[17] Let(X₀,X₁) be a compatible couple of complex Banach spaces, and assume thatX₀ ∩X₁ is dense inX₀ and inX₁. IfX₀ andX₁ arereflexive, then the dual of the complex interpolation space is obtained by interpolating the duals,

In general, the dual of the space(X₀,X₁)_θ is equal^[17] to${\displaystyle {\left(X_0^{\prime },X_1^{\prime }\right)}^{\theta },}$ a space defined by a variant of the complex method.^[18] The upper-θ and lower-θ methods do not coincide in general, but they do if at least one ofX₀,X₁ is a reflexive space.^[19]

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

Theorem.^[20] Let0 <θ < 1, 1 ≤q < ∞ and(X₀,X₁) a compatible couple of real Banach spaces. Assume thatX₀ ∩X₁ is dense inX₀ and inX₁. Then$${\displaystyle {\left({\left(X_0,X_1\right)}_{\theta ,q}\right)}^{\prime }={\left(X_0^{\prime },X_1^{\prime }\right)}_{\theta ,q^{\prime }},}$$ where${\displaystyle \frac{1}{q^{\prime }}=1-\frac{1}{q}.}$

Since the functiont →K(x,t) varies regularly (it is increasing, but⁠1/t⁠K(x,t) is decreasing), the definition of theK_θ,q-norm of a vectorn, previously given by an integral, is equivalent to a definition given by a series.^[21] This series is obtained by breaking(0, ∞) into pieces(2ⁿ, 2ⁿ⁺¹) of equal mass for the measure⁠dt/t⁠,

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

In the special case whereX₀ is continuously embedded inX₁, one can omit the part of the series with negative indicesn. In this case, each of the functionsx →K(x, 2ⁿ;X₀,X₁) defines an equivalent norm onX₁.

The interpolation space(X₀,X₁)_θ,q is a "diagonal subspace" of anℓ^q-sum of a sequence of Banach spaces (each one being isomorphic toX₀ +X₁). Therefore, whenq is finite, the dual of(X₀,X₁)_θ,q is aquotient of theℓ^p-sum of the duals,⁠1/p⁠ +⁠1/q⁠ = 1, which leads to the following formula for the discreteJ_θ,p-norm of a functionalx' in the dual of(X₀,X₁)_θ,q:

${\displaystyle \Vert x^{\prime }{\Vert }_{\theta ,p;J}\simeq inf\left\{{\left(\sum _{n\in \mathbf{Z}}{\left(2^{\theta n}max\left({\Vert x_n^{\prime }\Vert }_{X_0^{\prime }},2^{-n}{\Vert x_n^{\prime }\Vert }_{X_1^{\prime }}\right)\right)}^p\right)}^{\frac{1}{p}}:x^{\prime }=\sum _{n\in \mathbf{Z}}{x}_n^{\prime }\right\}.}$

The usual formula for the discreteJ_θ,p-norm is obtained by changingn to−n.

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.^[22] If the linear operatorT iscompact fromX₀ to a Banach spaceY and bounded fromX₁ toY, thenT is compact from(X₀,X₁)_θ,q toY when0 <θ < 1,1 ≤q ≤ ∞.

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

Theorem.^[23] A bounded linear operator between two Banach spaces isweakly compact if and only if it factors through areflexive space.

The spaceℓ^q used for the discrete definition can be replaced by an arbitrarysequence spaceY withunconditional basis, and the weightsa_n = 2^−θn,b_n = 2^(1−θ)n, that are used for theK_θ,q-norm, can be replaced by general weights

The interpolation spaceK(X₀,X₁,Y, {a_n}, {b_n}) consists of the vectorsx inX₀ +X₁ such that^[24]

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

Theorem.^[25] A Banach space with unconditional basis is isomorphic to a complemented subspace of a space withsymmetric basis.

Several interpolation results are available forSobolev spaces andBesov spaces onRⁿ,^[26]

These spaces are spaces ofmeasurable functions onRⁿ whens ≥ 0, and oftempered distributions onRⁿ whens < 0. For the rest of the section, the following setting and notation will be used:

Complex interpolation works well on the class of Sobolev spaces${\displaystyle H_p^s}$ (theBessel potential spaces) as well as Besov spaces:

Real interpolation between Sobolev spaces may give Besov spaces, except whens₀ =s₁,

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

Whens₀ ≠s₁ butp₀ =p₁, real interpolation between Sobolev spaces gives a Besov space:

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

Also,

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

• Calderón, Alberto P. (1964), "Intermediate spaces and interpolation, the complex method",Studia Math.,24 (2):113–190,doi:10.4064/sm-24-2-113-190.
• Lions, Jacques-Louis.; Peetre, Jaak (1964),"Sur une classe d'espaces d'interpolation",Inst. Hautes Études Sci. Publ. Math. (in French),19:5–68,doi:10.1007/bf02684796,S2CID 124471748.
• Bennett, Colin; Sharpley, Robert (1988),Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, pp. xiv+469,ISBN 978-0-12-088730-9.
• Bergh, Jöran; Löfström, Jörgen (1976),Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Berlin-New York: Springer-Verlag, pp. x+207,ISBN 978-3-540-07875-3.
• Leoni, Giovanni (2017).A First Course in Sobolev Spaces: Second Edition.Graduate Studies in Mathematics.181. American Mathematical Society. pp. 734.ISBN 978-1-4704-2921-8.
• Lindenstrauss, Joram; Tzafriri, Lior (1979),Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243,ISBN 978-3-540-08888-2.
• Tartar, Luc (2007),An Introduction to Sobolev Spaces and Interpolation, Springer,ISBN 978-3-540-71482-8.

• Banach spaces
• Fourier analysis
• Sobolev spaces

• CS1 French-language sources (fr)