Interpolation of operators

Obtaining from known properties of an operator in two or more spaces conclusions as to the properties of this operator in spaces that are in a certain sense intermediate. A Banach pair is a pair of Banach spaces (cf. Banach space) that are algebraically and continuously imbedded in a separable linear topological space . One introduces the norm

on the intersection ; on the arithmetical sum the norm

is introduced. The spaces and are Banach spaces. A Banach space is said to be intermediate for the pair if .

A linear mapping , acting from into , is called a bounded operator from the pair into the pair if its restriction to (respectively, ) is a bounded operator from into (respectively, from into ). A triple of spaces is called an interpolation triple relative to the triple , where is intermediate for (respectively, is intermediate for ), if every bounded operator from into maps into . If , , , then is called an interpolation space between and . For interpolation triples there exists a constant such that

The first interpolation theorem was obtained by M. Riesz (1926): The triple is an interpolation triple for if and if for a certain ,

(1)

The measures in the listed spaces may be different for each triple. Analogues of these theorems for other classes of families of spaces need not hold; e.g., is not an interpolation space between and .

An interpolation functor is a functor that assigns to each Banach pair an intermediate space , where, moreover, for any two Banach pairs and , the triples and are interpolation for each other. There is a number of methods for constructing interpolation functors. Two of these gained the largest number of applications.

Peetre's -method.

For a Banach pair one constructs the functional

which is equivalent to the norm in for each . A Banach space of measurable functions on the semi-axis is called an ideal space if almost-everywhere on and imply and . One considers all elements from for which . They form the Banach space with the norm . The space is non-empty and is intermediate for if and only if the function belongs to . In this case is an interpolation functor. For some Banach pairs the function can be computed. This makes it possible to constructive effectively interpolation spaces. For :

where is a non-increasing right-continuous function on that is equi-measurable with the function . For :

where is the modulus of continuity (cf. Continuity, modulus of) of the function , and the sign denotes transition to the least convex majorant on . For (a Sobolev space),

where

One often takes the space with norm

as . The corresponding functor is denoted by . The Besov spaces

with play an important role in the theory of partial differential equations. A number of classical inequalities in analysis can be made more precise in terms of the Lorentz spaces

The complex method of Calderón–Lions.

Let be a Banach pair. Denote by the space of all functions defined in the strip of the complex plane, with values in , and having the following properties: 1) is continuous and bounded on in the norm of ; 2) is analytic inside in the norm of ; 3) is continuous and bounded in the norm of ; and 4) is continuous and bounded in the norm of . The space , , is defined as the set of all elements that can be represented as for . In it one introduces the norm

In this way the interpolation functor is defined. If , , then with . If and are two ideal spaces and if in at least one of them the norm is absolutely continuous, then consists of all functions for which for some , . If are two complex Hilbert spaces with , then is a family of spaces that have very important applications. It is called a Hilbert scale. If , , then (a Sobolev space of fractional index). For other methods of constructing interpolation functors, as well as on their relation to the theory of scales of Banach spaces, see [1], [3], [5], [8], [9].

In the theory of interpolation of operators, Marcinkiewicz' interpolation theorem on interpolation operators of weak type plays an important role. An operator from a Banach space into a space of measurable functions, e.g. on the semi-axis, is called an operator of weak type if . It is assumed here that and are non-decreasing functions (e.g. , ). Theorems of Marcinkiewicz type enable one to describe for operators of weak types and simultaneously (where is a Banach pair) the pairs of spaces for which . In many cases it is sufficient to check that the operator

(where is the Peetre functional for ) acts from into . If for all linear operators of weak types it has been shown that this functional acts from into , then this also holds for quasi-additive operators (i.e. with the property ) of weak types , . Many important operators in analysis (e.g. Hilbert's singular operator) are of weak types in natural spaces; hence the corresponding interpolation theorems have found numerous applications.

References

[1]	P. Butzer, H. Berens, "Semi-groups of operators and approximation" , Springer (1967)
[2]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[3]	S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[4]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[5]	E. Magenes, "Spazi di interpolazione ed equazioni a derivate parziali" , Conf. VII Congr. Union Mat. Italy, 1963 , Cremonese (1965) pp. 134–197
[6]	E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)
[7]	N.Ya. Vilenkin (ed.) et al. (ed.) , Functional analysis , Wolters-Noordhoff (1972) (Translated from Russian)
[8]	J. Bergh, B.I. Löfström, "Interpolation spaces" , Springer (1976)
[9]	H. Triebel, "Interpolation theory, function spaces, differential operators" , North-Holland (1978)

Comments

The theorem of M. Riesz mentioned in the main article is often called the M. Riesz convexity theorem. It has a somewhat more precise statement as follows (involving a bound on a certain norm for the bounded operator in question). Let be a linear operator mapping a linear space of complex-valued measurable functions on a measure space into measurable functions on another measure space . Assume contains all indicator functions of measurable sets and is such that whenever , then also all truncations (i.e. functions which coincide with in for certain and vanish elsewhere) belong to . The operator is said to be of type () if there is a constant such that

(a1)

The least for which (a1) holds is called the -norm of . The M. Riesz convexity theorem now states: If a linear operator is of types with -norms , , then is of type with -norm , provided and , satisfy (1). (The name "convexity theorem" derives from the fact that the -norm of , as a function of , is logarithmically convex.)

In the same setting, is called subadditive if

for almost-all and for . A subadditive operator is said to be of weak type () (where , ) if there is a constant such that

(a2)

for all . The least for which (a2) holds is called the weak ()-norm of . (Note that the left-hand side of (a2) is the so-called distribution function of .) For , (a2) must be replaced by .

A still further generalization is that of an operator of restricted weak type , cf. [6].

Singular integral operators (cf. Singular integral) often prove to be of some (weak) type (e.g. the Hilbert transform is of weak type ).

References

[a1]	C. Bennett, R.C. Sharpley, "Interpolation of operators" , Acad. Press (1988)