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It is notable that when interpolating with the real method between ''A''<sub>0</sub> = {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>0</sub>,''q''<sub> 0</sub></sub>}} and ''A''<sub>1</sub> = {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>1</sub>,''q''<sub> 1</sub></sub>}}, only the values of θ<sub>0</sub> and θ<sub>1</sub> matter. Also, ''A''<sub>0</sub> and ''A''<sub>1</sub> can be complex interpolation spaces between ''X''<sub>0</sub> and ''X''<sub>1</sub>, with parameters θ<sub>0</sub> and θ<sub>1</sub> respectively.
== Duality ==
Let {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} be a compatible couple, and assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. In this case, the restriction map from the (continuous) [[Dual space#Continuous dual space|dual]] {{nowrap|''X'' ′<sub>''j''</sub>}} of ''X''<sub>''j''</sub>, {{nowrap| ''j'' {{=}} 0, 1}}, to the dual of {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is one-to-one. It follows that the pair of duals {{nowrap|(''X'' ′<sub>0</sub>, ''X'' ′<sub>1</sub>)}} is a compatible couple continuously embedded in the dual {{nowrap|(''X''<sub>0</sub> ∩ ''X''<sub>1</sub>) ′}}.
For the complex interpolation method, the following duality result holds:
'''Theorem.'''<ref name="Cald">see 12.1 and 12.2, p. 121 in {{harvtxt|Calderón|1964}}.</ref>
Let {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} be a compatible couple of complex Banach spaces, and assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. If ''X''<sub>0</sub> and ''X''<sub>1</sub> are [[Reflexive space|reflexive]], then the dual of the complex interpolation space is obtained by interpolating the duals,
:<math> ( (X_0, X_1)_\theta )' = (X'_0, X'_1)_\theta, \quad 0 < \theta < 1.</math>
In general, the dual of the space {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub>}} is equal<ref name="Cald"></ref>
to {{nowrap|(''X'' ′<sub>0</sub>, ''X'' ′<sub>1</sub>)<sup>θ</sup>}}, 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 for reflexive spaces.
For the real interpolation method, the duality holds provided that the parameter ''q'' is finite:
'''Theorem.'''<ref>see Théorème 3.1, p. 23 in {{harvtxt|Lions|Peetre|1964}}.</ref>
Let {{nowrap| 0 < θ < 1}}, {{nowrap|1 ≤ ''q'' < ∞}} and {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} a compatible couple of real Banach spaces. Assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. Then
:<math> ( (X_0, X_1)_{\theta, q} )' = (X'_0, X'_1)_{\theta, q'}, \ \ \text{where} \ \ 1/q' = 1 - 1/q.</math>
== See also ==
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Revision as of 11:59, 28 May 2013
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 Lp and also on a certain space Lq, then it is also continuous on the space Lr, for any intermediate r between p and q. In other words, Lr is a space which is intermediate, or between Lp and Lq.
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,[1] real interpolation,[2] as well as other tools (see e.g. fractional derivative).
The setting of interpolation
Suppose that a Banach space X is given as linear subspace of a Hausdorff topological vector space Z. The space X is said to be continuously embedded in Z when the inclusion map from X into Z is continuous. Assume that two Banach spaces X0 and X1 are given, and that they are both continuously embedded in a Hausdorff topological vector space Z. This is called a compatible couple of Banach spaces.[3] One can define norms on X0 ∩ X1 and X0 + X1 by
Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:
The space Z plays no further role, it was merely a tool that allows to make sense of X0 + X1. However the interpolation depends in an essential way from the specific manner in which some elements in X0 and in X1 have been identified as being the same elements in X0 ∩ X1. The interest now is in the family of spaces X that are intermediate spaces between X0 and X1 in the following sense:
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 space Z of measurable functions on the line, equipped with the topology of convergence in measure. In this situation, the spaces Lp(R), for 1 ≤ p ≤ ∞ are intermediate between L1(R) and L∞(R). More generally, when 1 ≤ p0 ≤ p ≤ p1 ≤ ∞,
with continous injections.
- Definition. Given two compatible couples (X0, X1) and (Y0, Y1), an interpolation pair is a couple (X, Y) of Banach spaces with the two following properties:
- The space X is intermediate between X0 and X1, and Y is intermediate between Y0 and Y1.
- If L is a linear operator from X0 + X1 into Y0 + Y1, which is continuous from X0 to Y0 and from X1 to Y1, then it is also continuous from X to Y.
The interpolation pair (X, Y) is said to be of exponent θ (with 0 < θ < 1) if there exists a constant C such that
for all operators L as above. The notation ||L||A, B is for the norm of the operator L as a map from A to B. If C = 1 (which is the smallest possible), one says that (X, Y) is an exact interpolation pair of exponent θ.
There are many ways of obtaining interpolation spaces (and the Riesz-Thorin theorem is an example of this for Lp spaces). A method for arbitrary complex Banach spaces is the complex interpolation method, while the real interpolation method applies also to real Banach spaces.
Complex interpolation
If the field of scalars is the field of complex numbers, properties of complex analytic functions may be used to define an interpolation space. For a compatible couple (X0, X1) of Banach spaces, the complex interpolation method consists in looking at the space of analytic functions f with values in X0+X1, defined on the open strip 0 < Re z < 1 in the complex plane, continuous on the closed strip 0 ≤ Re z ≤ 1, such that f(z) is bounded in X0 + X1 and that
- y ∈ R → f (iy) is bounded in X0 and y ∈ R → f (1+ iy) is bounded in X1.
The following norm is defined on this space of functions:
Definition. For 0 < θ < 1, the complex interpolation space (X0, X1)θ is the linear subspace of X0 + X1 consisting of all values f(θ) when f varies in the preceding space of functions,
The norm on the complex interpolation space (X0, X1)θ is defined by
Equipped with this norm, the complex interpolation space (X0, X1)θ is a Banach space.
Theorem. Given two compatible couples (X0, X1) and (Y0, Y1), the pair ((X0, X1)θ, (Y0, Y1)θ) is an exact interpolation pair of exponent θ, i.e., if T is a linear operator from X0 + X1 to Y0 + Y1, bounded from Xj to Yj, j = 0, 1, then T is bounded from (X0, X1)θ to (Y0, Y1)θ and
The family of Lp spaces (consisting of complex valued functions) behaves well under complex interpolation. If (X, Σ, μ) is an arbitrary measure space, if 1 ≤ p0, p1 ≤ ∞ and 0 < θ < 1, then
with equality of norms.
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 θ 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 form the dual couple by the J-method.
The K-method
The K-method of real interpolation[4] can be used for Banach spaces over the field R of real numbers.
Definition. Let (X0, X1) be a compatible couple of Banach spaces. For t > 0 and every u ∈ X0 + X1, let
Changing the order of the two spaces results[5] in
Let
and
The K-method of real interpolation consists in taking Kθ, q (X0, X1) to be the linear subspace of X0 + X1 consisting of all u such that ‖u ‖θ,q ; K < ∞.
Example
An important example is that of the couple (L1(X, Σ, μ), L∞(X, Σ, μ)), where the functional K(t, f ; L1, L∞) can be computed explicitely. The measure μ is supposed σ-finite. In this context, the best way of cutting the function f ∈ L1 + L∞ as sum of two functions f0 in L1 and f1 in L∞ is, for some s > 0 to be chosen as function of t, to let f1(x) be given for all x ∈ X by
The optimal choice of s leads to the formula[6]
where f * is the decreasing rearrangement of f.
The J-method
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. For t > 0 and for every vector u ∈ X0 ∩ X1, let
A vector u in X0 + X1 belongs to the interpolation space Jθ, q (X0, X1) if and only if it can be written as
where v(t) is measurable with values in X0 ∩ X1 and such that
The norm of u in Jθ, q (X0, X1) is given by the formula
Relations between the interpolation methods
The two real interpolation methods are equivalent when 0 < θ < 1.[7]
- Theorem. Let (X0, X1) be a compatible couple of Banach spaces. If 0 < θ < 1 and 1 ≤ q ≤ ∞, then
- with equivalence of norms.
The theorem covers degenerate cases that have not been excluded: if for example X0 and 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 0 < θ < 1, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters θ and q. The notation for this real interpolation space is (X0, X1)θ, q. One has that
For a given value of θ, the real interpolation spaces increase with q:[8] if 0 < θ < 1 and 1 ≤ q ≤ r ≤ ∞, the following continuous inclusion holds true:
Theorem. Given 0 < 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 exponent θ.[9]
Real interpolation between Lp spaces gives[10]
the family of Lorentz spaces. Assuming 0 < θ < 1 and 1 ≤ q ≤ ∞, one has that
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 q = p, the Lorentz space Lp,p is equal to Lp, up to renorming. When q = ∞, the Lorentz space Lp,∞ is equal to weak-Lp.
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. If 0 < θ < 1, then
The reiteration theorem
An intermediate space X of the compatible couple (X0, X1) is said to be of class θ[11] if
with continuous injections. Beside all real interpolation spaces (X0, X1)θ, q with parameter θ and 1 ≤ q ≤ ∞, the complex interpolation space (X0, X1)θ is an intermediate space of class θ of the compatible couple (X0, X1).
The reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a convex combination a = (1 - θ) x0 + θ x1 : taking a further convex combination of two convex combinations gives another convex combination.
Theorem.[12] Let A0, A1 be intermediate spaces of the compatible couple (X0, X1), of class θ0 and θ1 respectively, with 0 < θ0, θ1 < 1 and θ0 ≠ θ1. When 0 < θ < 1 and 1 ≤ q ≤ ∞, one has
It is notable that when interpolating with the real method between A0 = (X0, X1)θ0,q 0 and A1 = (X0, X1)θ1,q 1, only the values of θ0 and θ1 matter. Also, A0 and A1 can be complex interpolation spaces between X0 and X1, with parameters θ0 and θ1 respectively.
Duality
Let (X0, X1) be a compatible couple, and assume that X0 ∩ X1 is dense in X0 and in X1. In this case, the restriction map from the (continuous) dual X ′j of Xj, j = 0, 1, to the dual of X0 ∩ X1 is one-to-one. It follows that the pair of duals (X ′0, X ′1) is a compatible couple continuously embedded in the dual (X0 ∩ X1) ′.
For the complex interpolation method, the following duality result holds:
Theorem.[13] Let (X0, X1) be a compatible couple of complex Banach spaces, and assume that X0 ∩ X1 is dense in X0 and in X1. If X0 and X1 are reflexive, 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[13]
to (X ′0, X ′1)θ, 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 for reflexive spaces.
For the real interpolation method, the duality holds provided that the parameter q is finite:
Theorem.[14] Let 0 < θ < 1, 1 ≤ q < ∞ and (X0, X1) a compatible couple of real Banach spaces. Assume that X0 ∩ X1 is dense in X0 and in X1. Then
See also
Notes
- ^ The seminal papers in this direction are Lions, Jacques-Louis (1960), "Une construction d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 251: 1853–1855 and Calderón (1964).
- ^ first defined in Lions, Jacques-Louis; Peetre, Jaak (1961), "Propriétés d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 253: 1747–1749, developed in 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 Peetre, Jaak (1963), "Nouvelles propriétés d'espaces d'interpolation", C. R. Acad. Sci. Paris (in French), 256: 1424–1426, and
Peetre, Jaak (1968), A theory of interpolation of normed spaces, Notas de Matemática, vol. 39, Rio de Janeiro: Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, pp. iii+86
{{citation}}
: CS1 maint: extra punctuation (link). - ^ see Bennett & Sharpley (1988), pp. 96–105.
- ^ see pp. 293–302 in Bennett & Sharpley (1988).
- ^ see Proposition 1.2, p. 294 in Bennett & Sharpley (1988).
- ^ see p. 298 in Bennett & Sharpley (1988).
- ^ see Theorem 2.8, p. 314 in Bennett & Sharpley (1988).
- ^ see Proposition 1.10, p. 301 in Bennett & Sharpley (1988)
- ^ see Theorem 1.12, pp. 301–302 in Bennett & Sharpley (1988).
- ^ see Theorem 1.9, p. 300 in Bennett & Sharpley (1988).
- ^ see Definition 2.2, pp. 309–310 in Bennett & Sharpley (1988)
- ^ see Theorem 2.4, p. 311 in Bennett & Sharpley (1988)
- ^ a b see 12.1 and 12.2, p. 121 in Calderón (1964).
- ^ see Théorème 3.1, p. 23 in Lions & Peetre (1964).
References
- Calderón, Alberto P. (1964), "Intermediate spaces and interpolation, the complex method", Studia Math., 24: 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.
- Bergh, Jöran; Löfström, Jörgen (1976), Interpolation Spaces: An Introduction, Springer-Verlag, ISBN 3-540-07875-4
- Tartar, Luc (2007), An Introduction to Sobolev Spaces and Interpolation, Springer, ISBN 978-3-540-71482-8
- Bennett, Colin; Sharpley, Robert (1988), Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, pp. xiv+469, ISBN 0-12-088730-4
{{citation}}
: CS1 maint: extra punctuation (link)