View source for Calderón couples

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100102.png" /> be two Banach spaces (cf. [[Banach space|Banach space]]) embedded in a Hausdorff [[Topological vector space|topological vector space]]. Such a pair of spaces is termed a Banach couple or Banach pair. The theory of [[Interpolation of operators|interpolation of operators]] provides a variety of interpolation methods or interpolation functors for generating interpolation spaces with respect to any such couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100103.png" />, namely normed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100104.png" /> (cf. [[Normed space|Normed space]]) having the property that every [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100106.png" /> boundedly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100107.png" /> also maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100108.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c1100109.png" /> boundedly.

A fundamental problem in interpolation theory is the description of all interpolation spaces with respect to a given Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001010.png" />. In the 1960s, A.P. Calderón [[#References|[a4]]] and B.S. Mityagin [[#References|[a10]]] independently gave characterizations of all interpolation spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001011.png" /> with respect to the particular couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001012.png" />. Calderón showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001013.png" /> is an interpolation space if and only if it has the following monotonicity property: For every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001014.png" /> and every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001015.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001017.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001019.png" /> for some absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001020.png" />.

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001021.png" /> denotes the Peetre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001023.png" />-functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001024.png" /> with respect to the couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001025.png" />. In this particular case, where the couple is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001026.png" />, there is a concrete formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001027.png" /> (cf. [[Interpolation of operators|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001028.png" />.

The work of Calderón and Mityagin triggered a long series of papers by many mathematicians (many of these are listed in [[#References|[a2]]] and in [[#References|[a5]]]) in which it was shown that all the interpolation spaces of many other Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001029.png" /> can also be characterized via the Peetre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001030.png" />-functionals for those pairs, by a monotonicity condition exactly analogous to the one in Calderón's result above. The Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001031.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001033.png" />-monotone couples or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001035.png" />-pairs.)

It is also convenient to use the terminology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001037.png" />-space for any normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001038.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001039.png" /> as well as the above-mentioned monotonicity property with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001040.png" />-functional for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001041.png" />. By the important <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001043.png" />-divisibility theorem of Yu.A Brudnyi and N.Ya. Kruglyak [[#References|[a2]]], it follows that each such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001044.png" />-space necessarily coincides, to within equivalence of norms, with a space of the special form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001045.png" /> (as defined in [[Interpolation of operators|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001046.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001048.png" /> has to be relatively large.

Those Banach pairs which are known to be Calderón include pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001049.png" /> of weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001050.png" /> spaces for all choices of weight functions and for all exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001051.png" /> (the Sparr theorem, [[#References|[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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

In this last example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001053.png" /> can be taken to be an arbitrary Banach pair and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001055.png" /> can be arbitrary numbers. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001056.png" /> denotes the Lions–Peetre real-method interpolation space, consisting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001057.png" /> for which the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001058.png" /></td> </tr></table>

is finite.

By choosing particular pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001059.png" /> one obtains, as special cases of this last result, that various pairs of Besov spaces (cf. [[Imbedding theorems|Imbedding theorems]]) or Lorentz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001060.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001061.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001062.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001063.png" /> is a [[Sobolev space|Sobolev space]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001064.png" /> and also such simple pairs as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001066.png" />.

In [[#References|[a3]]], Brudnyi and A. Shteinberg consider whether pairs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001067.png" /> are Calderón, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001069.png" /> are interpolation functors (cf. [[Interpolation of operators|Interpolation of operators]]). Their results for the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001070.png" /> lead them to conjecture that the above-mentioned result about iterated pairs of the form (a1) cannot be extended, i.e., that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001071.png" /> is Calderón for every Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001072.png" /> if and only if both functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001073.png" /> are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001074.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001075.png" />-subcouples. One can ask whether this might in fact be true for all Calderón pairs.

N.J. Kalton [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001076.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001077.png" /> is Calderón. Kalton's results, and also the following general negative result from [[#References|[a5]]], suggest that in some sense the Calderón property is very much linked to the spaces of the pair having some sort of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001078.png" /> structure or  "near-Lp"  structure. This result also shows that Sparr's theorem for weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001080.png" /> spaces cannot be sharpened: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001081.png" /> be a pair of saturated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001082.png" />-order continuous Banach lattices with the Fatou property on the non-atomic measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001083.png" />. Suppose that at least one of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001085.png" /> does not coincide, to within equivalence of norms, with a weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001086.png" /> space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001087.png" />. Then there exist weight functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001088.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001089.png" /> such that the weighted Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001090.png" /> is not Calderón.

In most known examples of Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001091.png" /> which are not Calderón, this happens because the complex interpolation spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001092.png" /> (see [[Interpolation of operators|Interpolation of operators]]) are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001093.png" />-spaces. But M. Mastyło and V.I. Ovchinnikov have found examples (see [[#References|[a9]]]) of non-Calderón couples for which all the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001094.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001095.png" />-spaces.

The notion of Calderón couples can also be considered in the wider context of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001096.png" /> mapping from the spaces of one Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001097.png" /> to a possibly different Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001098.png" />. In such a context one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c11001099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010100.png" /> are relative interpolation spaces if every linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010101.png" /> which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010102.png" /> boundedly into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010103.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010104.png" /> also maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010105.png" /> boundedly into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010106.png" />. (In the notation of [[Interpolation of operators|Interpolation of operators]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010107.png" /> is an interpolation triple relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010108.png" />.) One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010110.png" /> are relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010112.png" />-spaces if, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010114.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010115.png" />-functional inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010116.png" /></td> </tr></table>

implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010117.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010118.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010120.png" /> are said to be relative Calderón couples if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010122.png" /> are relative interpolation spaces if and only if they are relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010123.png" />-spaces. J. Peetre has shown (see [[#References|[a6]]]) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010124.png" /> is any pair of weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010125.png" /> spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010127.png" /> are relative Calderón couples for all Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010128.png" />. Dually, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010129.png" /> is an arbitrary pair of weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010130.png" /> spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010132.png" /> are relative Calderón couples for all Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010133.png" /> satisfying a mild  "closure"  condition. This latter result is another consequence of the Brudnyi–Kruglyak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010135.png" />-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 [[#References|[a11]]] and [[#References|[a8]]] for some special cases. (Cf. also [[#References|[a1]]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Arazy,   M. Cwikel,   "A new characterization of the interpolation spaces between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010136.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010137.png" />"  ''Math. Scand.'' , '''55'''  (1984)  pp. 253–270</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y.A. Brudnyi,   N.Ja. Krugljak,   "Real interpolation functors" , North-Holland  (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Brudnyi,   A. Shteinberg,   "Calderón couples of Lipschitz spaces"  ''J. Funct. Anal.'' , '''131'''  (1995)  pp. 459–498</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.P. Calderón,   "Spaces between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010139.png" /> and the theorem of Marcinkiewicz"  ''Studia Math.'' , '''26'''  (1966)  pp. 273–299</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Cwikel,   P. Nilsson,   "Interpolation of weighted Banach lattices" , ''Memoirs'' , Amer. Math. Soc.  (to appear)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Cwikel,   J. Peetre,   "Abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010141.png" /> spaces"  ''J. Math. Pures Appl.'' , '''60'''  (1981)  pp. 1–50</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N.J. Kalton,   "Calderón couples of re-arrangement invariant spaces"  ''Studia Math.'' , '''106'''  (1993)  pp. 233–277</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Maligranda,   V.I. Ovchinnikov,   "On interpolation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010143.png" />"  ''J. Funct. Anal.'' , '''107'''  (1992)  pp. 343–351</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V.I. Ovchinnikov,   "On the description of interpolation orbits in couples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010144.png" /> spaces when they are not described by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010145.png" />-method. Interpolation spaces and related topics" , ''Israel Math. Conf. Proc. Bar Ilan University'' , '''5''' , Amer. Math. Soc.  (1992)  pp. 187–206</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  G. Sparr,   "Interpolation of weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110010/c110010146.png" /> spaces"  ''Studia Math.'' , '''62'''  (1978)  pp. 229–271</TD></TR></table>