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
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on the intersection ; on the arithmetical sum
the norm
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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
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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
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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
:
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where is a non-increasing right-continuous function on
that is equi-measurable with the function
. For
:
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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
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One often takes the space with norm
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as . The corresponding functor is denoted by
. The Besov spaces
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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
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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
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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
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(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) |
Interpolation of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_of_operators&oldid=17969