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
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
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].)
|[a1]||J. Arazy, M. Cwikel, "A new characterization of the interpolation spaces between and " Math. Scand. , 55 (1984) pp. 253–270|
|[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 and and the theorem of Marcinkiewicz" Studia Math. , 26 (1966) pp. 273–299|
|[a5]||M. Cwikel, P. Nilsson, "Interpolation of weighted Banach lattices" , Memoirs , Amer. Math. Soc. (to appear)|
|[a6]||M. Cwikel, J. Peetre, "Abstract and spaces" J. Math. Pures Appl. , 60 (1981) pp. 1–50|
|[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 and " J. Funct. Anal. , 107 (1992) pp. 343–351|
|[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 spaces when they are not described by the -method. Interpolation spaces and related topics" , Israel Math. Conf. Proc. Bar Ilan University , 5 , Amer. Math. Soc. (1992) pp. 187–206|
|[a12]||G. Sparr, "Interpolation of weighted spaces" Studia Math. , 62 (1978) pp. 229–271|
Calderón couples. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Calder%C3%B3n_couples&oldid=23204