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Let and
be two Banach spaces (cf. Banach space) embedded in a Hausdorff topological vector space. Such a pair of spaces is termed a Banach couple or Banach pair. The theory of interpolation of operators provides a variety of interpolation methods or interpolation functors for generating interpolation spaces with respect to any such couple
, namely normed spaces
(cf. Normed space) having the property that every linear operator
such that
boundedly for
also maps
to
boundedly.
A fundamental problem in interpolation theory is the description of all interpolation spaces with respect to a given Banach pair . In the 1960s, A.P. Calderón [a4] and B.S. Mityagin [a10] independently gave characterizations of all interpolation spaces
with respect to the particular couple
. Calderón showed that
is an interpolation space if and only if it has the following monotonicity property: For every element
and every element
, whenever
for all
, it follows that
and
for some absolute constant
.
Here, denotes the Peetre
-functional of
with respect to the couple
. In this particular case, where the couple is
, there is a concrete formula for
(cf. Interpolation of operators for further details).
Mityagin's result, though of course ultimately equivalent to Calderón's, is formulated differently, in terms of the effect of measure-preserving transformations and multiplication by unimodular functions on elements of .
The work of Calderón and Mityagin triggered a long series of papers by many mathematicians (many of these are listed in [a2] and in [a5]) in which it was shown that all the interpolation spaces of many other Banach pairs can also be characterized via the Peetre
-functionals for those pairs, by a monotonicity condition exactly analogous to the one in Calderón's result above. The Banach pairs
for which such a characterization holds are often referred to as Calderón couples or Calderón pairs. (They are also sometimes referred to using other terminology, such as Calderón–Mityagin couples,
-monotone couples or
-pairs.)
It is also convenient to use the terminology -space for any normed space
satisfying
as well as the above-mentioned monotonicity property with respect to the
-functional for
. By the important
-divisibility theorem of Yu.A Brudnyi and N.Ya. Kruglyak [a2], it follows that each such
-space necessarily coincides, to within equivalence of norms, with a space of the special form
(as defined in Interpolation of operators). Thus, for Calderón pairs, all the interpolation spaces are of this relatively simple form.
So, one can remark that, roughly speaking, for a Banach pair to be Calderón, the class of its interpolation spaces has to be relatively small, and correspondingly, the family of linear operators which are bounded on both
and
has to be relatively large.
Those Banach pairs which are known to be Calderón include pairs of weighted
spaces for all choices of weight functions and for all exponents
(the Sparr theorem, [a12]). Other examples include all Banach pairs of Hilbert spaces, various pairs of Hardy spaces, or of Lorentz or Marcinkiewicz spaces and all "iterated" pairs of the form
![]() | (a1) |
In this last example can be taken to be an arbitrary Banach pair and
and
can be arbitrary numbers. Here,
denotes the Lions–Peetre real-method interpolation space, consisting of all elements
for which the norm
![]() |
is finite.
By choosing particular pairs one obtains, as special cases of this last result, that various pairs of Besov spaces (cf. Imbedding theorems) or Lorentz
spaces or Schatten operator ideals are all Calderón.
In parallel with all these positive results it has also been shown that many Banach pairs fail to be Calderón. These include where
(here
is a Sobolev space) and
and also such simple pairs as
and
.
In [a3], Brudnyi and A. Shteinberg consider whether pairs of the form are Calderón, where
and
are interpolation functors (cf. Interpolation of operators). Their results for the pair
lead them to conjecture that the above-mentioned result about iterated pairs of the form (a1) cannot be extended, i.e., that
is Calderón for every Banach pair
if and only if both functors
are of the form
. As they also remark, all Calderón pairs which have so far (1996) been identified are either couples of Banach lattices of measurable functions on a given measure space, or are obtained from such lattice couples as partial retracts or
-subcouples. One can ask whether this might in fact be true for all Calderón pairs.
N.J. Kalton [a7] has given very extensive results about pairs of rearrangement-invariant spaces which are or are not Calderón, including a characterization of all rearrangement-invariant spaces for which
is Calderón. Kalton's results, and also the following general negative result from [a5], suggest that in some sense the Calderón property is very much linked to the spaces of the pair having some sort of
structure or "near-Lp" structure. This result also shows that Sparr's theorem for weighted
spaces cannot be sharpened: Let
be a pair of saturated
-order continuous Banach lattices with the Fatou property on the non-atomic measure space
. Suppose that at least one of the spaces
and
does not coincide, to within equivalence of norms, with a weighted
space on
. Then there exist weight functions
for
such that the weighted Banach pair
is not Calderón.
In most known examples of Banach pairs which are not Calderón, this happens because the complex interpolation spaces
(see Interpolation of operators) are not
-spaces. But M. Mastyło and V.I. Ovchinnikov have found examples (see [a9]) of non-Calderón couples for which all the spaces
are
-spaces.
The notion of Calderón couples can also be considered in the wider context of operators mapping from the spaces of one Banach pair
to a possibly different Banach pair
. In such a context one says that
and
are relative interpolation spaces if every linear mapping
which maps
boundedly into
for
also maps
boundedly into
. (In the notation of Interpolation of operators,
is an interpolation triple relative to
.) One says that
and
are relative
-spaces if, for all
and
, the
-functional inequality
![]() |
implies that with
.
and
are said to be relative Calderón couples if
and
are relative interpolation spaces if and only if they are relative
-spaces. J. Peetre has shown (see [a6]) that if
is any pair of weighted
spaces, then
and
are relative Calderón couples for all Banach pairs
. Dually, if
is an arbitrary pair of weighted
spaces, then
and
are relative Calderón couples for all Banach pairs
satisfying a mild "closure" condition. This latter result is another consequence of the Brudnyi–Kruglyak
-divisibility theorem.
Finally, given that there are so many cases of couples whose interpolation spaces cannot be all characterized by a Calderón-style condition, one must also seek alternative ways to characterize interpolation spaces. See [a11] and [a8] for some special cases. (Cf. also [a1].)
References
[a1] | J. Arazy, M. Cwikel, "A new characterization of the interpolation spaces between ![]() ![]() |
[a2] | Y.A. Brudnyi, N.Ja. Krugljak, "Real interpolation functors" , North-Holland (1991) |
[a3] | Y. Brudnyi, A. Shteinberg, "Calderón couples of Lipschitz spaces" J. Funct. Anal. , 131 (1995) pp. 459–498 |
[a4] | A.P. Calderón, "Spaces between ![]() ![]() |
[a5] | M. Cwikel, P. Nilsson, "Interpolation of weighted Banach lattices" , Memoirs , Amer. Math. Soc. (to appear) |
[a6] | M. Cwikel, J. Peetre, "Abstract ![]() ![]() |
[a7] | N.J. Kalton, "Calderón couples of re-arrangement invariant spaces" Studia Math. , 106 (1993) pp. 233–277 |
[a8] | L. Maligranda, V.I. Ovchinnikov, "On interpolation between ![]() ![]() |
[a9] | M. Mastyło, V.I. Ovchinnikov, "On the relation between complex and real methods of interpolation" Studia Math. (to appear) (Preprint Report 056/1996, Dept. Math. Comput. Sci. Adam Mickiewicz Univ., Poznan, 1996) |
[a10] | B.S. Mityagin, "An interpolation theorem for modular spaces" , Proc. Conf. Interpolation Spaces and Allied Topics in Analysis, Lund, 1983 , Lecture Notes in Mathematics , 1070 , Springer (1984) pp. 10–23 (In Russian) Mat. Sbornik , 66 (1965) pp. 472–482 |
[a11] | V.I. Ovchinnikov, "On the description of interpolation orbits in couples of ![]() ![]() |
[a12] | G. Sparr, "Interpolation of weighted ![]() |
Calderón couples. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calder%C3%B3n_couples&oldid=22225