Interpolation space#Complex interpolation

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 and also on a certain space {{math|L^q}}, then it is also continuous on the space {{math|L^r}}, for any intermediate {{mvar|r}} between {{mvar|p}} and {{mvar|q}}. In other words, {{math|L^r}} is a space which is intermediate between {{math|L^p}} and {{math|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,The seminal papers in this direction are {{citation

| last = Lions |first = Jacques-Louis

| title = Une construction d'espaces d'interpolation

| language = French

| journal = C. R. Acad. Sci. Paris

| volume = 251

| year = 1960

| pages = 1853–1855}} and {{harvtxt|Calderón|1964}}.

real interpolation,first defined in {{citation

| last1 = Lions | first1 = Jacques-Louis

| last2 = Peetre | first2 = Jaak

| title = Propriétés d'espaces d'interpolation

| language = French

| journal = C. R. Acad. Sci. Paris

| volume = 253

| year = 1961

|pages = 1747–1749}}, developed in {{harvtxt|Lions|Peetre|1964}}, with notation slightly different (and more complicated, with four parameters instead of two) from today's notation. It was put later in today's form in {{citation

| last = Peetre | first = Jaak

| title = Nouvelles propriétés d'espaces d'interpolation

| language = French

| journal = C. R. Acad. Sci. Paris

| volume = 256

| year = 1963

| pages = 1424–1426}}, and

{{citation

| last = Peetre | first = Jaak

| title = A theory of interpolation of normed spaces

| series = Notas de Matemática

| volume = 39

| publisher = Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas

| location = Rio de Janeiro

| year = 1968

| pages = iii+86}}.

as well as other tools (see e.g. fractional derivative).

A Banach space {{mvar|X}} is said to be continuously embedded in a Hausdorff topological vector space {{mvar|Z}} when {{mvar|X}} is a linear subspace of {{mvar|Z}} such that the inclusion map from {{mvar|X}} into {{mvar|Z}} is continuous. A compatible couple {{math|(X₀, X₁)}} of Banach spaces consists of two Banach spaces {{math|X₀}} and {{math|X₁}} that are continuously embedded in the same Hausdorff topological vector space {{mvar|Z}}.see {{harvtxt|Bennett|Sharpley|1988}}, pp. 96–105. The embedding in a linear space {{mvar|Z}} allows to consider the two linear subspaces

:$X\_0\; \backslash cap\; X\_1$

and

:$X\_0\; +\; X\_1\; =\; \backslash left\; \backslash \{\; z\; \backslash in\; Z\; :\; z\; =\; x\_0\; +\; x\_1,\; \backslash \; x\_0\; \backslash in\; X\_0,\; \backslash ,\; x\_1\; \backslash in\; X\_1\; \backslash right\; \backslash \}.$

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of {{math|X₀}} and {{math|X₁}}. It depends in an essential way from the specific relative position that {{math|X₀}} and {{math|X₁}} occupy in a larger space {{mvar|Z}}.

One can define norms on {{math|X₀ ∩ X₁}} and {{math|X₀ + X₁}} by

:$\backslash |x\backslash |\_\{X\_0\; \backslash cap\; X\_1\}\; :=\; \backslash max\; \backslash left\; (\; \backslash left\; \backslash |x\; \backslash right\; \backslash |\_\{X\_0\},\; \backslash left\; \backslash |x\; \backslash right\; \backslash |\_\{X\_1\}\; \backslash right\; ),$

:$\backslash |x\backslash |\_\{X\_0\; +\; X\_1\}\; :=\; \backslash inf\; \backslash left\; \backslash \{\; \backslash left\; \backslash |x\_0\; \backslash right\; \backslash |\_\{X\_0\}\; +\; \backslash left\; \backslash |x\_1\; \backslash right\; \backslash |\_\{X\_1\}\; \backslash \; :\; \backslash \; x\; =\; x\_0\; +\; x\_1,\; \backslash ;\; x\_0\; \backslash in\; X\_0,\; \backslash ;\; x\_1\; \backslash in\; X\_1\; \backslash right\; \backslash \}.$

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

:$X\_0\; \backslash cap\; X\_1\; \backslash subset\; X\_0,\; \backslash \; X\_1\; \backslash subset\; X\_0\; +\; X\_1.$

Interpolation studies the family of spaces {{mvar|X}} that are intermediate spaces between {{math|X₀}} and {{math|X₁}} in the sense that

:$X\_0\; \backslash cap\; X\_1\; \backslash subset\; X\; \backslash subset\; X\_0\; +\; X\_1,$

where the two inclusions maps are continuous.

An example of this situation is the pair {{math|(L¹(R), L^∞(R))}}, where the two Banach spaces are continuously embedded in the space {{mvar|Z}} of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces {{math|L^p(R)}}, for {{math|1 ≤ p ≤ ∞}} are intermediate between {{math|L¹(R)}} and {{math|L^∞(R)}}. More generally,

:$L^\{p\_0\}(\backslash mathbf\{R\})\; \backslash cap\; L^\{p\_1\}(\backslash mathbf\{R\})\; \backslash subset\; L^p(\backslash mathbf\{R\})\; \backslash subset\; L^\{p\_0\}(\backslash mathbf\{R\})\; +\; L^\{p\_1\}(\backslash mathbf\{R\}),\; \backslash \; \backslash \; \backslash text\{when\}\; \backslash \; \backslash \; 1\; \backslash le\; p\_0\; \backslash le\; p\; \backslash le\; p\_1\; \backslash le\; \backslash infty,$

with continuous injections, so that, under the given condition, {{math|L^p(R)}} is intermediate between {{math|L^p₀(R)}} and {{math|L^p₁(R)}}.

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

:*The space X is intermediate between {{math|X₀}} and {{math|X₁}}, and Y is intermediate between {{math|Y₀}} and {{math|Y₁}}.

:*If {{math|L}} is any linear operator from {{math|X₀ + X₁}} to {{math|Y₀ + Y₁}}, which maps continuously {{math|X₀}} to {{math|Y₀}} and {{math|X₁}} to {{math|Y₁}}, then it also maps continuously {{math|X}} to {{math|Y}}.

The interpolation pair {{math|(X, Y)}} is said to be of exponent {{mvar|θ}} (with {{math|0 < θ < 1}}) if there exists a constant {{math|C}} such that

:$\backslash |L\backslash |\_\{X,Y\}\; \backslash leq\; C\; \backslash |L\backslash |\_\{X\_0,Y\_0\}^\{1-\backslash theta\}\; \backslash ;\; \backslash |L\backslash |\_\{X\_1,Y\_1\}^\{\backslash theta\}$

for all operators {{mvar|L}} as above. The notation {{math|{{!!}}L{{!!}}_X,Y}} is for the norm of {{math|L}} as a map from {{math|X}} to {{math|Y}}. If {{math|C {{=}} 1}}, we say that {{math|(X, Y)}} is an exact interpolation pair of exponent {{mvar|θ}}.

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 $\backslash mathcal\{F\}(X\_0,\; X\_1)$ consists of all functions {{math| f  : C → X₀ + X₁}}, that are analytic on {{math|S {{=}} {z : 0 < Re(z) < 1},}} continuous on {{math|{{overline|S}} {{=}} {z : 0 ≤ Re(z) ≤ 1},}} and for which all the following subsets are bounded:

:{{math|{ f (z) : z ∈ S} ⊂ X₀ + X₁}},

:{{math|{ f (it) : t ∈ R} ⊂ X₀}},

:{{math|{ f (1 + it) : t ∈ R} ⊂ X₁}}.

$\backslash mathcal\{F\}(X\_0,\; X\_1)$ is a Banach space under the norm

:$\backslash |f\backslash |\_\{\backslash mathcal\{F\}(X\_0,\; X\_1)\}\; =\; \backslash max\; \backslash left\backslash \{\; \backslash sup\_\{t\; \backslash in\; \backslash mathbf\{R\}\}\; \backslash |f(it)\backslash |\_\{X\_0\},\; \backslash ;\; \backslash sup\_\{t\; \backslash in\; \backslash mathbf\{R\}\}\backslash |f(1\; +\; it)\backslash |\_\{X\_1\}\; \backslash right\backslash \}.$

Definition.see p. 88 in {{harvtxt|Bergh|Löfström|1976}}. For {{math|0 < θ < 1}}, the complex interpolation space {{math|(X₀, X₁)_θ}} is the linear subspace of {{math|X₀ + X₁}} consisting of all values f(θ) when f varies in the preceding space of functions,

:$(X\_0,\; X\_1)\_\backslash theta\; =\; \backslash left\; \backslash \{\; x\; \backslash in\; X\_0\; +\; X\_1\; :\; x\; =\; f(\backslash theta),\; \backslash ;\; f\; \backslash in\; \backslash mathcal\{F\}(X\_0,\; X\_1)\; \backslash right\; \backslash \}.$

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

:$\backslash \; \backslash |x\backslash |\_\backslash theta\; =\; \backslash inf\; \backslash left\; \backslash \{\; \backslash |f\backslash |\_\{\backslash mathcal\{F\}(X\_0,\; X\_1)\}\; \backslash \; :\backslash \; f(\backslash theta)\; =\; x,\; \backslash ;\; f\; \backslash in\; \backslash mathcal\{F\}(X\_0,\; X\_1)\; \backslash right\; \backslash \}.$

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

:Theorem.see Theorem 4.1.2, p. 88 in {{harvtxt|Bergh|Löfström|1976}}. Given two compatible couples of Banach spaces {{math|(X₀, X₁)}} and {{math|(Y₀, Y₁)}}, the pair {{math|((X₀, X₁)_θ, (Y₀, Y₁)_θ)}} is an exact interpolation pair of exponent {{mvar|θ}}, i.e., if {{math|T : X₀ + X₁ → Y₀ + Y₁}}, is a linear operator bounded from {{math|X_j}} to {{math|Y_j, j {{=}} 0, 1}}, then {{mvar|T}} is bounded from {{math|(X₀, X₁)_θ}} to {{math|(Y₀, Y₁)_θ}} and $$\backslash |T\backslash |\_\backslash theta\; \backslash le\; \backslash |T\backslash |\_0^\{1\; -\; \backslash theta\}\; \backslash |T\backslash |\_1^\backslash theta.$$

The family of {{math|L^p}} spaces (consisting of complex valued functions) behaves well under complex interpolation.see Chapter 5, p. 106 in {{harvtxt|Bergh|Löfström|1976}}. If {{math|(R, Σ, μ)}} is an arbitrary measure space, if {{math|1 ≤ p₀, p₁ ≤ ∞}} and {{math|0 < θ < 1}}, then

:$\backslash left(\; L^\{p\_0\}(R,\; \backslash Sigma,\; \backslash mu),\; L^\{p\_1\}(R,\; \backslash Sigma,\; \backslash mu)\; \backslash right)\_\backslash theta\; =\; L^p(R,\; \backslash Sigma,\; \backslash mu),\; \backslash qquad\; \backslash frac\{1\}\{p\}\; =\; \backslash frac\{1\; -\; \backslash theta\}\{p\_0\}\; +\; \backslash frac\{\backslash theta\}\{p\_1\},$

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

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 {{mvar|θ}} is in {{math|(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 interpolationsee pp. 293–302 in {{harvtxt|Bennett|Sharpley|1988}}. can be used for Banach spaces over the field {{math|R}} of real numbers.

Definition. Let {{math|(X₀, X₁)}} be a compatible couple of Banach spaces. For {{math|t > 0}} and every {{math|x ∈ X₀ + X₁}}, let

:$K(x,\; t;\; X\_0,\; X\_1)\; =\; \backslash inf\; \backslash left\; \backslash \{\; \backslash left\; \backslash |x\_0\; \backslash right\; \backslash |\_\{X\_0\}\; +\; t\; \backslash left\; \backslash |x\_1\; \backslash right\; \backslash |\_\{X\_1\}\; \backslash \; :\backslash \; x\; =\; x\_0\; +\; x\_1,\; \backslash ;\; x\_0\; \backslash in\; X\_0,\; \backslash ,\; x\_1\; \backslash in\; X\_1\; \backslash right\; \backslash \}.$

Changing the order of the two spaces results in:see Proposition 1.2, p. 294 in {{harvtxt|Bennett|Sharpley|1988}}.

:$K(x,\; t;\; X\_0,\; X\_1)\; =\; t\; K\; \backslash left\; (x,\; t^\{-1\};\; X\_1,\; X\_0\; \backslash right).$

Let

:$\backslash begin\{align\}$

\|x\|_{\theta,q; K} &= \left( \int_0^\infty \left( t^{-\theta} K(x, t; X_0, X_1) \right)^q \, \tfrac{dt}{t} \right)^{\frac{1}{q}}, && 0 < \theta < 1, 1 \leq q < \infty, \\

\|x\|_{\theta,\infty; K} &= \sup_{t > 0} \; t^{-\theta} K(x, t; X_0, X_1), && 0 \le \theta \le 1.

\end{align}

The K-method of real interpolation consists in taking {{math|K_θ,q(X₀, X₁) }} to be the linear subspace of {{math|X₀ + X₁}} consisting of all {{mvar|x}} such that {{math|{{!!}}x{{!!}}_θ,q;K < ∞}}.

An important example is that of the couple {{math|(L¹(R, Σ, μ), L^∞(R, Σ, μ))}}, where the functional {{math|K(t, f ; L¹, L^∞)}} can be computed explicitly. The measure {{mvar|μ}} is supposed σ-finite measure. In this context, the best way of cutting the function {{math| f  ∈ L¹ + L^∞}} as sum of two functions {{math| f₀ ∈ L¹ }} and {{math| f₁ ∈ L^∞ }} is, for some {{math|s > 0}} to be chosen as function of {{mvar|t}}, to let {{math| f₁(x)}} be given for all {{math|x ∈ R}} by

:$f\_1(x)\; =\; \backslash begin\{cases\}$

f(x) & |f(x)| < s, \\

\frac{s f(x)}

\end{cases}

The optimal choice of {{mvar|s}} leads to the formulasee p. 298 in {{harvtxt|Bennett|Sharpley|1988}}.

:$K\; \backslash left\; (f,\; t;\; L^1,\; L^\backslash infty\; \backslash right\; )\; =\; \backslash int\_0^t\; f^*(u)\; \backslash ,\; d\; u,$

where {{math| f ^∗}} is the decreasing rearrangement of {{math| f }}.

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

Definition. Let {{math|(X₀, X₁)}} be a compatible couple of Banach spaces. For {{math|t > 0}} and for every vector {{math|x ∈ X₀ ∩ X₁}}, let $$J(x,\; t;\; X\_0,\; X\_1)\; =\; \backslash max\; \backslash left\; (\; \backslash |x\backslash |\_\{X\_0\},\; t\; \backslash |x\backslash |\_\{X\_1\}\; \backslash right\; ).$$

A vector {{mvar|x}} in {{math|X₀ + X₁}} belongs to the interpolation space {{math|J_θ,q(X₀, X₁)}} if and only if it can be written as

:$x\; =\; \backslash int\_0^\backslash infty\; v(t)\; \backslash ,\; \backslash frac\{dt\}\{t\},$

where {{math|v(t)}} is measurable with values in {{math|X₀ ∩ X₁}} and such that

:

The norm of {{mvar|x}} in {{math|J_θ,q(X₀, X₁)}} is given by the formula

:$\backslash |x\backslash |\_\{\backslash theta,q;J\}\; :=\; \backslash inf\_v\; \backslash left\backslash \{\; \backslash Phi(v)\; \backslash \; :\backslash \; x\; =\; \backslash int\_0^\backslash infty\; v(t)\; \backslash ,\; \backslash tfrac\{dt\}\{t\}\; \backslash right\backslash \}.$

The two real interpolation methods are equivalent when {{math|0 < θ < 1}}.see Theorem 2.8, p. 314 in {{harvtxt|Bennett|Sharpley|1988}}.

:Theorem. Let {{math|(X₀, X₁)}} be a compatible couple of Banach spaces. If {{math|0 < θ < 1}} and {{math|1 ≤ q ≤ ∞}}, then $$J\_\{\backslash theta,q\}(X\_0,\; X\_1)\; =\; K\_\{\backslash theta,q\}(X\_0,\; X\_1),$$ with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if {{math|X₀}} and {{math|X₁}} 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 {{math|0 < θ < 1}}, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters {{mvar|θ}} and {{mvar|q}}. The notation for this real interpolation space is {{math|(X₀, X₁)_θ,q}}. One has that

:

For a given value of {{mvar|θ}}, the real interpolation spaces increase with {{mvar|q}}:see Proposition 1.10, p. 301 in {{harvtxt|Bennett|Sharpley|1988}} if {{math|0 < θ < 1}} and {{math| 1 ≤ q ≤ r ≤ ∞}}, the following continuous inclusion holds true:

:$(X\_0,\; X\_1)\_\{\backslash theta,\; q\}\; \backslash subset\; (X\_0,\; X\_1)\_\{\backslash theta,\; r\}.$

:Theorem. Given {{math|0 < θ < 1}}, {{math|1 ≤ q ≤ ∞}} and two compatible couples {{math|(X₀, X₁)}} and {{math|(Y₀, Y₁)}}, the pair {{math|((X₀, X₁)_θ,q, (Y₀, Y₁)_θ,q)}} is an exact interpolation pair of exponent {{mvar|θ}}.see Theorem 1.12, pp. 301–302 in {{harvtxt|Bennett|Sharpley|1988}}.

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 {{math|(X₀, X₁)}} be a compatible couple of Banach spaces. If {{math|0 < θ < 1}}, then $$(X\_0,\; X\_1)\_\{\backslash theta,\; 1\}\; \backslash subset\; (X\_0,\; X\_1)\_\backslash theta\; \backslash subset\; (X\_0,\; X\_1)\_\{\backslash theta,\; \backslash infty\}.$$

When {{math|X₀ {{=}} C([0, 1])}} and {{math|X₁ {{=}} C¹([0, 1])}}, the space of continuously differentiable functions on {{math|[0, 1]}}, the {{math|(θ, ∞)}} interpolation method, for {{math|0 < θ < 1}}, gives the Hölder space {{math|C^0,θ}} of exponent {{mvar|θ}}. This is because the K-functional {{math|K(f, t; X₀, X₁)}} of this couple is equivalent to

:$\backslash sup\; \backslash left\backslash \{\; |f(u)|,\; \backslash ,\; \backslash frac$

Only values {{math|0 < t < 1}} are interesting here.

Real interpolation between {{math|L^p}} spaces givessee Theorem 1.9, p. 300 in {{harvtxt|Bennett|Sharpley|1988}}. the family of Lorentz spaces. Assuming {{math|0 < θ < 1}} and {{math|1 ≤ q ≤ ∞}}, one has:

:$\backslash left\; (\; L^1(\backslash mathbf\{R\},\; \backslash Sigma,\; \backslash mu),\; L^\backslash infty(\backslash mathbf\{R\},\; \backslash Sigma,\; \backslash mu)\; \backslash right)\_\{\backslash theta,\; q\}\; =\; L^\{p,\; q\}(\backslash mathbf\{R\},\; \backslash Sigma,\; \backslash mu),\; \backslash qquad\; \backslash text\{where\; \}\; \backslash tfrac\{1\}\{p\}\; =\; 1\; -\; \backslash theta,$

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 {{math|q {{=}} p}}, the Lorentz space {{math|L^p,p}} is equal to {{math|L^p}}, up to renorming. When {{math|q {{=}} ∞}}, the Lorentz space {{math|L^p,∞}} is equal to Lp space#Weak Lp.

An intermediate space {{mvar|X}} of the compatible couple {{math|(X₀, X₁)}} is said to be of class θ if see Definition 2.2, pp. 309–310 in {{harvtxt|Bennett|Sharpley|1988}}

:$(X\_0,\; X\_1)\_\{\backslash theta,1\}\; \backslash subset\; X\; \backslash subset\; (X\_0,\; X\_1)\_\{\backslash theta,\backslash infty\},$

with continuous injections. Beside all real interpolation spaces {{math|(X₀, X₁)_θ,q}} with parameter {{mvar|θ}} and {{math|1 ≤ q ≤ ∞}}, the complex interpolation space {{math|(X₀, X₁)_θ}} is an intermediate space of class {{mvar|θ}} of the compatible couple {{math|(X₀, X₁)}}.

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

:Theorem.see Theorem 2.4, p. 311 in {{harvtxt|Bennett|Sharpley|1988}} Let {{math|A₀, A₁}} be intermediate spaces of the compatible couple {{math|(X₀, X₁)}}, of class {{math|θ₀}} and {{math|θ₁}} respectively, with {{math|0 < θ₀ ≠ θ₁ < 1}}. When {{math|0 < θ < 1}} and {{math|1 ≤ q ≤ ∞}}, one has $$(A\_0,\; A\_1)\_\{\backslash theta,\; q\}\; =\; (X\_0,\; X\_1)\_\{\backslash eta,\; q\},\; \backslash qquad\; \backslash eta\; =\; (1\; -\; \backslash theta)\; \backslash theta\_0\; +\; \backslash theta\; \backslash theta\_1.$$

It is notable that when interpolating with the real method between {{math|A₀ {{=}} (X₀, X₁)_θ0,q0}} and {{math|A₁ {{=}} (X₀, X₁)_θ1,q1}}, only the values of {{math|θ₀}} and {{math|θ₁}} matter. Also, {{math|A₀}} and {{math|A₁}} can be complex interpolation spaces between {{math|X₀}} and {{math|X₁}}, with parameters {{math|θ₀}} and {{math|θ₁}} respectively.

There is also a reiteration theorem for the complex method.

:Theorem.see 12.3, p. 121 in {{harvtxt|Calderón|1964}}. Let {{math|(X₀, X₁)}} be a compatible couple of complex Banach spaces, and assume that {{math|X₀ ∩ X₁}} is dense in {{math|X₀}} and in {{math|X₁}}. Let {{math|A₀ {{=}} (X₀, X₁)_θ0}} and {{math|A₁ {{=}} (X₀, X₁)_θ1}}, where {{math|0 ≤ θ₀ ≤ θ₁ ≤ 1}}. Assume further that {{math|X₀ ∩ X₁}} is dense in {{math|A₀ ∩ A₁}}. Then, for every {{math|0 ≤ θ ≤ 1}}, $$\backslash left(\; \backslash left\; (X\_0,\; X\_1\; \backslash right\; )\_\{\backslash theta\_0\},\; \backslash left\; (X\_0,\; X\_1\; \backslash right\; )\_\{\backslash theta\_1\}\; \backslash right)\_\backslash theta\; =\; (X\_0,\; X\_1)\_\backslash eta,\; \backslash qquad\; \backslash eta\; =\; (1\; -\; \backslash theta)\; \backslash theta\_0\; +\; \backslash theta\; \backslash theta\_1.$$

The density condition is always satisfied when {{math|X₀ ⊂ X₁}} or {{math|X₁ ⊂ X₀}}.

Let {{math|(X₀, X₁)}} be a compatible couple, and assume that {{math|X₀ ∩ X₁}} is dense in X₀ and in X₁. In this case, the restriction map from the (continuous) dual $X\text{'}\_j$ of {{math|X_j}}, {{math|j {{=}} 0, 1,}} to the dual of {{math|X₀ ∩ X₁}} is one-to-one. It follows that the pair of duals $\backslash left\; (X\text{'}\_0,\; X\text{'}\_1\; \backslash right\; )$ is a compatible couple continuously embedded in the dual {{math|(X₀ ∩ X₁)′}}.

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

:Theorem.see 12.1 and 12.2, p. 121 in {{harvtxt|Calderón|1964}}. Let {{math|(X₀, X₁)}} be a compatible couple of complex Banach spaces, and assume that {{math|X₀ ∩ X₁}} is dense in {{math|X₀}} and in {{math|X₁}}. If {{math|X₀}} and {{math|X₁}} are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals,

In general, the dual of the space {{math|(X₀, X₁)_θ}} is equal to $\backslash left\; (X\text{'}\_0,\; X\text{'}\_1\; \backslash right\; )^\{\backslash theta\},$ a space defined by a variant of the complex method.Theorem 4.1.4, p. 89 in {{harvtxt|Bergh|Löfström|1976}}. 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.Theorem 4.3.1, p. 93 in {{harvtxt|Bergh|Löfström|1976}}.

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

:Theorem.see Théorème 3.1, p. 23 in {{harvtxt|Lions|Peetre|1964}}, or Theorem 3.7.1, p. 54 in {{harvtxt|Bergh|Löfström|1976}}. Let {{math|0 < θ < 1, 1 ≤ q < ∞}} and {{math|(X₀, X₁)}} a compatible couple of real Banach spaces. Assume that {{math|X₀ ∩ X₁}} is dense in {{math|X₀}} and in {{math|X₁}}. Then $$\backslash left\; (\; \backslash left\; (X\_0,\; X\_1\; \backslash right\; )\_\{\backslash theta,\; q\}\; \backslash right\; )\text{'}\; =\; \backslash left\; (X\text{'}\_0,\; X\text{'}\_1\; \backslash right\; )\_\{\backslash theta,\; q\text{'}\},$$ where $\backslash tfrac\{1\}\{q\text{'}\}\; =\; 1\; -\; \backslash tfrac\{1\}\{q\}.$

Since the function {{math|t → K(x, t)}} varies regularly (it is increasing, but {{math|{{sfrac|1|t}}K(x, t)}} is decreasing), the definition of the {{math|K_θ,q}}-norm of a vector {{mvar|n}}, previously given by an integral, is equivalent to a definition given by a series.see chap. II in {{harvtxt|Lions|Peetre|1964}}. This series is obtained by breaking {{math|(0, ∞)}} into pieces {{math|(2ⁿ, 2ⁿ⁺¹)}} of equal mass for the measure {{math|{{sfrac|dt|t}}}},

:$\backslash |x\backslash |\_\{\backslash theta,\; q;\; K\}\; \backslash simeq\; \backslash left(\; \backslash sum\_\{n\; \backslash in\; \backslash mathbf\{Z\}\}\; \backslash left(\; 2^\{-\backslash theta\; n\}\; K\; \backslash left\; (x,\; 2^n;\; X\_0,\; X\_1\; \backslash right\; )\; \backslash right)^q\; \backslash right)^\{\backslash frac\{1\}\{q\}\}.$

In the special case where {{math|X₀}} is continuously embedded in {{math|X₁}}, one can omit the part of the series with negative indices {{mvar|n}}. In this case, each of the functions {{math|x → K(x, 2ⁿ; X₀, X₁)}} defines an equivalent norm on {{math|X₁}}.

The interpolation space {{math|(X₀, X₁)_θ,q}} is a "diagonal subspace" of an {{math|ℓ^q}}-sum of a sequence of Banach spaces (each one being isomorphic to {{math|X₀ + X₁}}). Therefore, when {{mvar|q}} is finite, the dual of {{math|(X₀, X₁)_θ,q}} is a quotient of the {{math|ℓ^p}}-sum of the duals, {{math|{{sfrac|1|p}} + {{sfrac|1|q}} {{=}} 1}}, which leads to the following formula for the discrete {{math|J_θ,p}}-norm of a functional x' in the dual of {{math|(X₀, X₁)_θ,q}}:

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

The usual formula for the discrete {{math|J_θ,p}}-norm is obtained by changing {{mvar|n}} to {{math|−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.see chap. 5, Théorème 2.2, p. 37 in {{harvtxt|Lions|Peetre|1964}}. If the linear operator {{mvar|T}} is compact from {{math|X₀}} to a Banach space {{mvar|Y}} and bounded from {{math|X₁}} to {{mvar|Y}}, then {{mvar|T}} is compact from {{math|(X₀, X₁)_θ,q}} to {{mvar|Y}} when {{math|0 < θ < 1}}, {{math|1 ≤ q ≤ ∞}}.

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

:Theorem.{{citation|last1 = Davis|first1 = William J.| last2 = Figiel| first2 = Tadeusz | last3 = Johnson | first3 = William B. | author3-link = William B. Johnson (mathematician) | last4 = Pełczyński |first4 = Aleksander| year = 1974 | title = Factoring weakly compact operators | journal = Journal of Functional Analysis | volume = 17|issue = 3| pages = 311–327 | doi = 10.1016/0022-1236(74)90044-5| doi-access = free }}, see also Theorem 2.g.11, p. 224 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.

The space {{math|ℓ^q}} used for the discrete definition can be replaced by an arbitrary sequence space Y with unconditional basis, and the weights {{math|a_n {{=}} 2^−θn}}, {{math|b_n {{=}} 2^(1−θ)n}}, that are used for the {{math|K_θ,q}}-norm, can be replaced by general weights

:

The interpolation space {{math|K(X₀, X₁, Y, {a_n}, {b_n})}} consists of the vectors {{mvar|x}} in {{math|X₀ + X₁}} such that{{citation| last1 = Johnson| first1 = William B.| last2 = Lindenstrauss | first2 = Joram | contribution = Basic concepts in the geometry of Banach spaces | title = Handbook of the geometry of Banach spaces, Vol. I| pages = 1–84 | publisher = North-Holland | location = Amsterdam | year = 2001}}, and section 2.g in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.

:

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

Theorem.see Theorem 3.b.1, p. 123 in {{citation | last1 = Lindenstrauss | first1 = Joram | author1-link = Joram Lindenstrauss | last2 = Tzafriri | first2 = Lior | location = Berlin | publisher = Springer-Verlag | series = Ergebnisse der Mathematik und ihrer Grenzgebiete | title = Classical Banach Spaces I, Sequence Spaces | volume = 92 | pages = xiii+188 | isbn = 978-3-540-08072-5 | year = 1977}}. A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with symmetric basis.

Several interpolation results are available for Sobolev spaces and Besov spaces on Rⁿ,Theorem 6.4.5, p. 152 in {{harvtxt|Bergh|Löfström|1976}}.

:$\backslash begin\{align\}$

&H^s_p && s \in \mathbf{R}, 1 \le p \le \infty \\

&B^s_{p, q} && s \in \mathbf{R}, 1 \le p, q \le \infty

\end{align}

These spaces are spaces of measurable functions on {{math|Rⁿ}} when {{math|s ≥ 0}}, and of tempered distributions on {{math|Rⁿ}} when {{math|s < 0}}. For the rest of the section, the following setting and notation will be used:

:$\backslash begin\{align\}$

0 &< \theta < 1, \\

1 &\le p, p_0, p_1, q, q_0, q_1 \le \infty, \\

s, &s_0, s_1 \in \mathbf{R}, \\

s_\theta &= (1 - \theta) s_0 + \theta s_1, \\[4pt]

\frac 1 {p_\theta} &= \frac{1 - \theta}{p_0} + \frac{\theta}{p_1}, \\[4pt]

\frac 1 {q_\theta} &= \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}.

\end{align}

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

:$\backslash begin\{align\}$

\left (H^{s_0}_{p_0}, H^{s_1}_{p_1} \right )_\theta &= H^{s_\theta}_{p_\theta}, && s_0 \ne s_1, 1 < p_0, p_1 < \infty. \\

\left (B^{s_0}_{p_0,q_0}, B^{s_1}_{p_1,q_1} \right)_\theta &= B^{s_\theta}_{p_\theta, q_\theta}, && s_0 \ne s_1.

\end{align}

Real interpolation between Sobolev spaces may give Besov spaces, except when {{math|s₀ {{=}} s₁}},

:$\backslash left\; (H^\{s\}\_\{p\_0\},\; H^\{s\}\_\{p\_1\}\; \backslash right)\_\{\backslash theta,\; p\_\backslash theta\}\; =\; H^\{s\}\_\{p\_\backslash theta\}.$

When {{math|s₀ ≠ s₁}} but {{math|p₀ {{=}} p₁}}, real interpolation between Sobolev spaces gives a Besov space:

:$\backslash left\; (H^\{s\_0\}\_p,\; H^\{s\_1\}\_p\; \backslash right)\_\{\backslash theta,\; q\}\; =\; B^\{s\_\backslash theta\}\_\{p,\; q\},\; \backslash qquad\; s\_0\; \backslash ne\; s\_1.$

Also,

:$\backslash begin\{align\}$

\left (B^{s_0}_{p,q_0}, B^{s_1}_{p,q_1} \right)_{\theta, q} &= B^{s_\theta}_{p,q}, && s_0 \ne s_1. \\

\left (B^s_{p,q_0}, B^s_{p, q_1} \right )_{\theta, q} &= B^{s}_{p, q_\theta}. \\

\left (B^{s_0}_{p_0,q_0}, B^{s_1}_{p_1,q_1} \right )_{\theta, q_\theta} &= B^{s_\theta}_{p_\theta, q_\theta}, && s_0 \ne s_1, p_\theta =q_\theta.

\end{align}

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

{{Reflist}}

• {{citation

| last = Calderón | first = Alberto P. | author-link = Alberto Calderón

| title = Intermediate spaces and interpolation, the complex method

| journal = Studia Math.

| volume = 24

| issue = 2 | year = 1964

| pages = 113–190

| doi = 10.4064/sm-24-2-113-190 | doi-access = free}}.

• {{citation

| last1 = Lions |first1 = Jacques-Louis.

| author-link1 = Jacques-Louis Lions

| last2 = Peetre |first2 = Jaak

| title = Sur une classe d'espaces d'interpolation

| language = French

| journal = Inst. Hautes Études Sci. Publ. Math.

| volume = 19

| year = 1964

| pages = 5–68

| doi=10.1007/bf02684796

|s2cid = 124471748

|url = http://www.numdam.org/item/PMIHES_1964__19__5_0/

}}.

• {{citation

| last1 = Bennett | first1 = Colin

| last2 = Sharpley |first2 = Robert

| title = Interpolation of operators

| series = Pure and Applied Mathematics

| volume = 129

| publisher = Academic Press, Inc., Boston, MA

| year = 1988

| pages = xiv+469

| isbn = 978-0-12-088730-9

}}.

• {{citation

| last1 = Bergh | first1 = Jöran

| last2 = Löfström | first2 = Jörgen

| title = Interpolation spaces. An introduction

| series = Grundlehren der Mathematischen Wissenschaften

| volume = 223

| publisher = Springer-Verlag

| location = Berlin-New York

| year = 1976

| pages = x+207

| isbn = 978-3-540-07875-3

}}.

• Leoni, Giovanni (2017). [http://bookstore.ams.org/gsm-181/ A First Course in Sobolev Spaces: Second Edition]. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. {{ISBN|978-1-4704-2921-8}}.
• {{citation

| last1 = Lindenstrauss

| first1 = Joram | author1-link = Joram Lindenstrauss

| last2 = Tzafriri | first2 = Lior

| title = Classical Banach spaces. II. Function spaces

| series = Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]

| volume = 97

| publisher = Springer-Verlag

| location = Berlin-New York

| year = 1979

| pages = x+243

| isbn = 978-3-540-08888-2

}}.

• {{citation|last=Tartar|first=Luc|title=An Introduction to Sobolev Spaces and Interpolation |publisher=Springer|year=2007| isbn=978-3-540-71482-8 }}.

{{Functional analysis}}

{{Topological vector spaces}}

Category:Banach spaces

Category:Fourier analysis

Category:Sobolev spaces