Interpolation space

{{Short description|Vector space in mathematics}} 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 {{math|''L^p''}} 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,<ref>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}}.</ref> real interpolation,<ref>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}}.</ref> as well as other tools (see e.g. fractional derivative).

== The setting of interpolation == 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|(''X''₀, ''X''₁)}} of Banach spaces consists of two Banach spaces {{math|''X''₀}} and {{math|''X''₁}} that are continuously embedded in the same Hausdorff topological vector space {{mvar|Z}}.<ref>see {{harvtxt|Bennett|Sharpley|1988}}, pp.&nbsp;96&ndash;105.</ref> The embedding in a linear space {{mvar|Z}} allows to consider the two linear subspaces

:<math> X_0 \cap X_1</math>

and

:<math>X_0 + X_1 = \left \{ z \in Z : z = x_0 + x_1, \ x_0 \in X_0, \, x_1 \in X_1 \right \}.</math>

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of {{math|''X''₀}} and {{math|''X''₁}}. It depends in an essential way from the specific ''relative position'' that {{math|''X''₀}} and {{math|''X''₁}} occupy in a larger space {{mvar|Z}}.

One can define norms on {{math|''X''₀ ∩ ''X''₁}} and {{math|''X''₀ + ''X''₁}} by

:<math>\|x\|_{X_0 \cap X_1} := \max \left ( \left \|x \right \|_{X_0}, \left \|x \right \|_{X_1} \right ),</math> :<math>\|x\|_{X_0 + X_1} := \inf \left \{ \left \|x_0 \right \|_{X_0} + \left \|x_1 \right \|_{X_1} \ : \ x = x_0 + x_1, \; x_0 \in X_0, \; x_1 \in X_1 \right \}.</math>

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

:<math>X_0 \cap X_1 \subset X_0, \ X_1 \subset X_0 + X_1.</math>

Interpolation studies the family of spaces {{mvar|X}} that are '''intermediate spaces''' between {{math|''X''₀}} and {{math|''X''₁}} in the sense that

:<math>X_0 \cap X_1 \subset X \subset X_0 + X_1,</math>

where the two inclusions maps are continuous.

An example of this situation is the pair {{math|(''L''¹('''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|''L''¹('''R''')}} and {{math|''L''^∞('''R''')}}. More generally,

:<math>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 \infty,</math>

with continuous injections, so that, under the given condition, {{math|''L^p''('''R''')}} is intermediate between {{math|''L''^''p''₀('''R''')}} and {{math|''L''^''p''₁('''R''')}}.

:'''Definition.''' Given two compatible couples {{math|(''X''₀, ''X''₁)}} and {{math|(''Y''₀, ''Y''₁)}}, an '''interpolation pair''' is a couple {{math|(''X'', ''Y'')}} of Banach spaces with the two following properties: :*The space ''X'' is intermediate between {{math|''X''₀}} and {{math|''X''₁}}, and ''Y'' is intermediate between {{math|''Y''₀}} and {{math|''Y''₁}}. :*If {{math|''L''}} is any linear operator from {{math|''X''₀ + ''X''₁}} to {{math|''Y''₀ + ''Y''₁}}, which maps continuously {{math|''X''₀}} to {{math|''Y''₀}} and {{math|''X''₁}} to {{math|''Y''₁}}, 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 :<math>\|L\|_{X,Y} \leq C \|L\|_{X_0,Y_0}^{1-\theta} \; \|L\|_{X_1,Y_1}^{\theta}</math> 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}}'''.

== 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 <math>\mathcal{F}(X_0, X_1)</math> consists of all functions {{math|&thinsp;''f''&thinsp; : '''C''' → ''X''₀ + ''X''₁}}, 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|{&thinsp;''f''&thinsp;(''z'') : ''z'' ∈ ''S''} ⊂ ''X''₀ + ''X''₁}}, :{{math|{&thinsp;''f''&thinsp;(''it'') : ''t'' ∈ '''R'''} ⊂ ''X''₀}}, :{{math|{&thinsp;''f''&thinsp;(1 + ''it'') : ''t'' ∈ '''R'''} ⊂ ''X''₁}}.

<math>\mathcal{F}(X_0, X_1)</math> is a Banach space under the norm

:<math>\|f\|_{\mathcal{F}(X_0, X_1)} = \max \left\{ \sup_{t \in \mathbf{R}} \|f(it)\|_{X_0}, \; \sup_{t \in \mathbf{R}}\|f(1 + it)\|_{X_1} \right\}.</math>

'''Definition.'''<ref>see p.&nbsp;88 in {{harvtxt|Bergh|Löfström|1976}}.</ref> For {{math|0 < ''θ'' < 1}}, the '''complex interpolation space''' {{math|(''X''₀, ''X''₁)_''θ''}} is the linear subspace of {{math|''X''₀ + ''X''₁}} consisting of all values ''f''(''θ'') when ''f'' varies in the preceding space of functions,

:<math>(X_0, X_1)_\theta = \left \{ x \in X_0 + X_1 : x = f(\theta), \; f \in \mathcal{F}(X_0, X_1) \right \}.</math>

The norm on the complex interpolation space {{math|(''X''₀, ''X''₁)_''θ''}} is defined by

:<math>\ \|x\|_\theta = \inf \left \{ \|f\|_{\mathcal{F}(X_0, X_1)} \ :\ f(\theta) = x, \; f \in \mathcal{F}(X_0, X_1) \right \}.</math>

Equipped with this norm, the complex interpolation space {{math|(''X''₀, ''X''₁)_''θ''}} is a Banach space.

:'''Theorem.'''<ref>see Theorem 4.1.2, p.&nbsp;88 in {{harvtxt|Bergh|Löfström|1976}}.</ref> Given two compatible couples of Banach spaces {{math|(''X''₀, ''X''₁)}} and {{math|(''Y''₀, ''Y''₁)}}, the pair {{math|((''X''₀, ''X''₁)_''θ'', (''Y''₀, ''Y''₁)_''θ'')}} is an exact interpolation pair of exponent {{mvar|θ}}, i.e., if {{math|''T'' : ''X''₀ + ''X''₁ → ''Y''₀ + ''Y''₁}}, is a linear operator bounded from {{math|''X_j''}} to {{math|''Y_j'', ''j'' {{=}} 0, 1}}, then {{mvar|T}} is bounded from {{math|(''X''₀, ''X''₁)_''θ''}} to {{math|(''Y''₀, ''Y''₁)_''θ''}} and <math display="block"> \|T\|_\theta \le \|T\|_0^{1 - \theta} \|T\|_1^\theta. </math>

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

:<math>\left( L^{p_0}(R, \Sigma, \mu), L^{p_1}(R, \Sigma, \mu) \right)_\theta = L^p(R, \Sigma, \mu), \qquad \frac{1}{p} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1},</math>

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 {{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.

=== K-method === The K-method of real interpolation<ref>see pp.&nbsp;293–302 in {{harvtxt|Bennett|Sharpley|1988}}.</ref> can be used for Banach spaces over the field {{math|'''R'''}} of real numbers.

'''Definition.''' Let {{math|(''X''₀, ''X''₁)}} be a compatible couple of Banach spaces. For {{math|''t'' > 0}} and every {{math|''x'' ∈ ''X''₀ + ''X''₁}}, let

:<math>K(x, t; X_0, X_1) = \inf \left \{ \left \|x_0 \right \|_{X_0} + t \left \|x_1 \right \|_{X_1} \ :\ x = x_0 + x_1, \; x_0 \in X_0, \, x_1 \in X_1 \right \}.</math>

Changing the order of the two spaces results in:<ref>see Proposition&nbsp;1.2, p.&nbsp;294 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>

:<math>K(x, t; X_0, X_1) = t K \left (x, t^{-1}; X_1, X_0 \right).</math>

Let

:<math>\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}</math>

The K-method of real interpolation consists in taking {{math|''K''_''θ'',''q''(''X''₀, ''X''₁) }} to be the linear subspace of {{math|''X''₀ + ''X''₁}} consisting of all {{mvar|x}} such that {{math|{{!!}}''x''{{!!}}_{''θ'',''q'';''K''} < ∞}}.

==== Example ==== An important example is that of the couple {{math|(''L''¹('''R''', Σ, ''μ''), ''L''^∞('''R''', Σ, ''μ''))}}, where the functional {{math|''K''(''t'', ''f''&thinsp;; ''L''¹, ''L''^∞)}} can be computed explicitly. The measure {{mvar|μ}} is supposed {{mvar|σ}}-finite. In this context, the best way of cutting the function {{math|&thinsp;''f''&thinsp; ∈ ''L''¹ + ''L''^∞}} as sum of two functions {{math|&thinsp;''f''₀ ∈ ''L''¹&thinsp;}} and {{math|&thinsp;''f''₁ ∈ ''L''^∞&thinsp;}} is, for some {{math|''s'' > 0}} to be chosen as function of {{mvar|t}}, to let {{math|&thinsp;''f''₁(''x'')}} be given for all {{math|''x'' ∈ '''R'''}} by

:<math>f_1(x) = \begin{cases} f(x) & |f(x)| < s, \\ \frac{s f(x)}{|f(x)|} & \text{otherwise} \end{cases}</math>

The optimal choice of {{mvar|s}} leads to the formula<ref>see p.&nbsp;298 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>

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

where {{math|&thinsp;''f''^&nbsp;∗}} is the decreasing rearrangement of {{math|&thinsp;''f''&thinsp;}}.

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

'''Definition.''' Let {{math|(''X''₀, ''X''₁)}} be a compatible couple of Banach spaces. For {{math|''t'' > 0}} and for every vector {{math|''x'' ∈ ''X''₀ ∩ ''X''₁}}, let <math display="block">J(x, t; X_0, X_1) = \max \left ( \|x\|_{X_0}, t \|x\|_{X_1} \right ).</math>

A vector {{mvar|x}} in {{math|''X''₀ + ''X''₁}} belongs to the interpolation space {{math|''J''_''θ'',''q''(''X''₀, ''X''₁)}} if and only if it can be written as

:<math>x = \int_0^\infty v(t) \, \frac{dt}{t},</math>

where {{math|''v''(''t'')}} is measurable with values in {{math|''X''₀ ∩ ''X''₁}} and such that

:<math>\Phi(v) = \left( \int_0^\infty \left( t^{-\theta} J(v(t), t; X_0, X_1) \right)^q \, \tfrac{dt}{t} \right)^{\frac{1}{q}} < \infty.</math>

The norm of {{mvar|x}} in {{math|''J''_''θ'',''q''(''X''₀, ''X''₁)}} is given by the formula

:<math>\|x\|_{\theta,q;J} := \inf_v \left\{ \Phi(v) \ :\ x = \int_0^\infty v(t) \, \tfrac{dt}{t} \right\}.</math>

=== Relations between the interpolation methods === The two real interpolation methods are equivalent when {{math|0 < ''θ'' < 1}}.<ref>see Theorem&nbsp;2.8, p.&nbsp;314 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>

:'''Theorem.''' Let {{math|(''X''₀, ''X''₁)}} be a compatible couple of Banach spaces. If {{math|0 < ''θ'' < 1}} and {{math|1 ≤ ''q'' ≤ ∞}}, then <math display="block">J_{\theta,q}(X_0, X_1) = K_{\theta,q}(X_0, X_1),</math> with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if {{math|''X''₀}} and {{math|''X''₁}} 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|(''X''₀, ''X''₁)_''θ'',''q''}}. One has that

:<math>(X_0, X_1)_{\theta, q} = (X_1, X_0)_{1 - \theta, q}, \qquad 0 < \theta < 1, 1 \le q \le \infty.</math>

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

:<math>(X_0, X_1)_{\theta, q} \subset (X_0, X_1)_{\theta, r}.</math>

:'''Theorem.''' Given {{math|0 < ''θ'' < 1}}, {{math|1 ≤ ''q'' ≤ ∞}} and two compatible couples {{math|(''X''₀, ''X''₁)}} and {{math|(''Y''₀, ''Y''₁)}}, the pair {{math|((''X''₀, ''X''₁)_''θ'',''q'', (''Y''₀, ''Y''₁)_''θ'',''q'')}} is an exact interpolation pair of exponent {{mvar|θ}}.<ref>see Theorem&nbsp;1.12, pp.&nbsp;301–302 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>

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|(''X''₀, ''X''₁)}} be a compatible couple of Banach spaces. If {{math|0 < ''θ'' < 1}}, then <math display="block">(X_0, X_1)_{\theta, 1} \subset (X_0, X_1)_\theta \subset (X_0, X_1)_{\theta, \infty}.</math>

==== Examples ==== When {{math|''X''₀ {{=}} ''C''([0, 1])}} and {{math|''X''₁ {{=}} ''C''¹([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''; ''X''₀, ''X''₁)}} of this couple is equivalent to

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

Only values {{math|0 < ''t'' < 1}} are interesting here.

Real interpolation between {{math|''L^p''}} spaces gives<ref>see Theorem&nbsp;1.9, p.&nbsp;300 in {{harvtxt|Bennett|Sharpley|1988}}.</ref> the family of Lorentz spaces. Assuming {{math|0 < ''θ'' < 1}} and {{math|1 ≤ ''q'' ≤ ∞}}, one has:

:<math> \left ( L^1(\mathbf{R}, \Sigma, \mu), L^\infty(\mathbf{R}, \Sigma, \mu) \right)_{\theta, q} = L^{p, q}(\mathbf{R}, \Sigma, \mu), \qquad \text{where } \tfrac{1}{p} = 1 - \theta,</math>

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 weak-{{math|''L^p''}}.

== The reiteration theorem == An intermediate space {{mvar|X}} of the compatible couple {{math|(''X''₀, ''X''₁)}} is said to be of '''class ''θ''''' if <ref>see Definition 2.2, pp.&nbsp;309&ndash;310 in {{harvtxt|Bennett|Sharpley|1988}}</ref>

:<math>(X_0, X_1)_{\theta,1} \subset X \subset (X_0, X_1)_{\theta,\infty},</math>

with continuous injections. Beside all real interpolation spaces {{math|(''X''₀, ''X''₁)_''θ'',''q''}} with parameter {{mvar|θ}} and {{math|1 ≤ ''q'' ≤ ∞}}, the complex interpolation space {{math|(''X''₀, ''X''₁)_''θ''}} is an intermediate space of class {{mvar|θ}} of the compatible couple {{math|(''X''₀, ''X''₁)}}.

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'')''x''₀ + ''θx''₁}}: taking a further convex combination of two convex combinations gives another convex combination.

:'''Theorem.'''<ref>see Theorem 2.4, p.&nbsp;311 in {{harvtxt|Bennett|Sharpley|1988}}</ref> Let {{math|''A''₀, ''A''₁}} be intermediate spaces of the compatible couple {{math|(''X''₀, ''X''₁)}}, of class {{math|''θ''₀}} and {{math|''θ''₁}} respectively, with {{math|0 < ''θ''₀ ≠ ''θ''₁ < 1}}. When {{math|0 < ''θ'' < 1}} and {{math|1 ≤ ''q'' ≤ ∞}}, one has <math display="block">(A_0, A_1)_{\theta, q} = (X_0, X_1)_{\eta, q}, \qquad \eta = (1 - \theta) \theta_0 + \theta \theta_1.</math>

It is notable that when interpolating with the real method between {{math|''A''₀ {{=}} (''X''₀, ''X''₁)_{''θ''₀,''q''₀}}} and {{math|''A''₁ {{=}} (''X''₀, ''X''₁)_{''θ''₁,''q''₁}}}, only the values of {{math|''θ''₀}} and {{math|''θ''₁}} matter. Also, {{math|''A''₀}} and {{math|''A''₁}} can be complex interpolation spaces between {{math|''X''₀}} and {{math|''X''₁}}, with parameters {{math|''θ''₀}} and {{math|''θ''₁}} respectively.

There is also a reiteration theorem for the complex method.

:'''Theorem.'''<ref>see 12.3, p.&nbsp;121 in {{harvtxt|Calderón|1964}}.</ref> Let {{math|(''X''₀, ''X''₁)}} be a compatible couple of complex Banach spaces, and assume that {{math|''X''₀ ∩ ''X''₁}} is dense in {{math|''X''₀}} and in {{math|''X''₁}}. Let {{math|''A''₀ {{=}} (''X''₀, ''X''₁)_''θ''₀}} and {{math|''A''₁ {{=}} (''X''₀, ''X''₁)_''θ''₁}}, where {{math|0 ≤ ''θ''₀ ≤ ''θ''₁ ≤ 1}}. Assume further that {{math|''X''₀ ∩ ''X''₁}} is dense in {{math|''A''₀ ∩ ''A''₁}}. Then, for every {{math|0 ≤ ''θ'' ≤ 1}}, <math display="block"> \left( \left (X_0, X_1 \right )_{\theta_0}, \left (X_0, X_1 \right )_{\theta_1} \right)_\theta = (X_0, X_1)_\eta, \qquad \eta = (1 - \theta) \theta_0 + \theta \theta_1.</math>

The density condition is always satisfied when {{math|''X''₀ ⊂ ''X''₁}} or {{math|''X''₁ ⊂ ''X''₀}}.

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

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

:'''Theorem.'''<ref name="Cald">see 12.1 and 12.2, p.&nbsp;121 in {{harvtxt|Calderón|1964}}.</ref> Let {{math|(''X''₀, ''X''₁)}} be a compatible couple of complex Banach spaces, and assume that {{math|''X''₀ ∩ ''X''₁}} is dense in {{math|''X''₀}} and in {{math|''X''₁}}. If {{math|''X''₀}} and {{math|''X''₁}} are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals, <math display="block"> ( (X_0, X_1)_\theta )' = \left(X'_0, X'_1 \right )_\theta, \qquad 0 < \theta < 1.</math>

In general, the dual of the space {{math|(''X''₀, ''X''₁)_''θ''}} is equal<ref name="Cald" /> to <math> \left (X'_0, X'_1 \right )^{\theta},</math> a space defined by a variant of the complex method.<ref>Theorem 4.1.4, p.&nbsp;89 in {{harvtxt|Bergh|Löfström|1976}}.</ref> The upper-&theta; and lower-&theta; methods do not coincide in general, but they do if at least one of ''X''₀, ''X''₁ is a reflexive space.<ref>Theorem 4.3.1, p.&nbsp;93 in {{harvtxt|Bergh|Löfström|1976}}.</ref>

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

:'''Theorem.'''<ref>see Théorème&nbsp;3.1, p.&nbsp;23 in {{harvtxt|Lions|Peetre|1964}}, or Theorem&nbsp;3.7.1, p.&nbsp;54 in {{harvtxt|Bergh|Löfström|1976}}.</ref> Let {{math|0 < ''θ'' < 1, 1 ≤ ''q'' < ∞}} and {{math|(''X''₀, ''X''₁)}} a compatible couple of real Banach spaces. Assume that {{math|''X''₀ ∩ ''X''₁}} is dense in {{math|''X''₀}} and in {{math|''X''₁}}. Then <math display="block"> \left ( \left (X_0, X_1 \right )_{\theta, q} \right )' = \left (X'_0, X'_1 \right )_{\theta, q'},</math> where <math>\tfrac{1}{q'} = 1 - \tfrac{1}{q}.</math>

== Discrete definitions == 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.<ref>see chap.&nbsp;II in {{harvtxt|Lions|Peetre|1964}}.</ref> This series is obtained by breaking {{math|(0, ∞)}} into pieces {{math|(2^''n'', 2^''n''+1)}} of equal mass for the measure {{math|{{sfrac|d''t''|''t''}}}},

:<math> \|x\|_{\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}}.</math>

In the special case where {{math|''X''₀}} is continuously embedded in {{math|''X''₁}}, one can omit the part of the series with negative indices {{mvar|n}}. In this case, each of the functions {{math|''x'' → ''K''(''x'', 2^''n''; ''X''₀, ''X''₁)}} defines an equivalent norm on {{math|''X''₁}}.

The interpolation space {{math|(''X''₀, ''X''₁)_''θ'',''q''}} is a "diagonal subspace" of an {{math|''ℓ^&thinsp;q''}}-sum of a sequence of Banach spaces (each one being isomorphic to {{math|''X''₀ + ''X''₁}}). Therefore, when {{mvar|q}} is finite, the dual of {{math|(''X''₀, ''X''₁)_''θ'',''q''}} is a quotient of the {{math|''ℓ^&thinsp;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|(''X''₀, ''X''₁)_''θ'',''q''}}:

:<math> \|x'\|_{\theta, p; J} \simeq \inf \left\{ \left( \sum_{n \in \mathbf{Z}} \left( 2^{\theta n} \max \left (\left \|x'_n \right \|_{X'_0}, 2^{-n} \left \|x'_n \right\|_{X'_1} \right ) \right)^p \right)^{\frac{1}{p}} \ : \ x' = \sum_{n \in \mathbf{Z}} x'_n \right\}.</math>

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.'''<ref>see chap.&nbsp;5, Théorème&nbsp;2.2, p.&nbsp;37 in {{harvtxt|Lions|Peetre|1964}}.</ref> If the linear operator {{mvar|T}} is compact from {{math|''X''₀}} to a Banach space {{mvar|Y}} and bounded from {{math|''X''₁}} to {{mvar|Y}}, then {{mvar|T}} is compact from {{math|(''X''₀, ''X''₁)_''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.'''<ref>{{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&ndash;327 | doi = 10.1016/0022-1236(74)90044-5| doi-access = free }}, see also Theorem 2.g.11, p.&nbsp;224 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> 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 {{math|''ℓ^&thinsp;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

:<math>a_n, b_n > 0, \ \ \sum_{n=1}^\infty \min(a_n, b_n) < \infty.</math>

The interpolation space {{math|''K''(''X''₀, ''X''₁, ''Y'', {''a_n''}, {''b_n''})}} consists of the vectors {{mvar|x}} in {{math|''X''₀ + ''X''₁}} such that<ref>{{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&ndash;84 | publisher = North-Holland | location = Amsterdam | year = 2001}}, and section&nbsp;2.g in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>

:<math>\|x\|_{K(X_0, X_1)} = \sup_{m \ge 1} \left \| \sum_{n=1}^m a_n K \left (x, \tfrac{b_n}{a_n}; X_0, X_1 \right) \, y_n \right\|_Y < \infty,</math>

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.'''<ref>see Theorem&nbsp;3.b.1, p.&nbsp;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}}.</ref> 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'',<ref>Theorem 6.4.5, p.&nbsp;152 in {{harvtxt|Bergh|Löfström|1976}}.</ref>

:<math>\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}</math>

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

:<math>\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}</math>

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

:<math>\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}</math>

Real interpolation between Sobolev spaces may give Besov spaces, except when {{math|''s''₀ {{=}} ''s''₁}},

:<math>\left (H^{s}_{p_0}, H^{s}_{p_1} \right)_{\theta, p_\theta} = H^{s}_{p_\theta}.</math>

When {{math|''s''₀ ≠ ''s''₁}} but {{math|''p''₀ {{=}} ''p''₁}}, real interpolation between Sobolev spaces gives a Besov space:

:<math>\left (H^{s_0}_p, H^{s_1}_p \right)_{\theta, q} = B^{s_\theta}_{p, q}, \qquad s_0 \ne s_1.</math>

Also,

:<math>\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}</math>

== See also == * Fundamental lemma of interpolation theory * Riesz–Thorin theorem * Marcinkiewicz interpolation theorem

== Notes == {{Reflist}}

== References == *{{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.&nbsp;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