Calderón couples

Let $ A _ {0} $ and $ A _ {1} $ 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 $ ( A _ {0} ,A _ {1} ) $, namely normed spaces $ A $( cf. Normed space) having the property that every linear operator $ T : {A _ {0} + A _ {1} } \rightarrow {A _ {0} + A _ {1} } $ such that $ T : {A _ {j} } \rightarrow {A _ {j} } $ boundedly for $ j = 0,1 $ also maps $ A $ to $ A $ boundedly.

A fundamental problem in interpolation theory is the description of all interpolation spaces with respect to a given Banach pair $ ( A _ {0} ,A _ {1} ) $. In the 1960s, A.P. Calderón [a4] and B.S. Mityagin [a10] independently gave characterizations of all interpolation spaces $ A $ with respect to the particular couple $ ( A _ {0} ,A _ {1} ) = ( L _ {1} , L _ \infty ) $. Calderón showed that $ A $ is an interpolation space if and only if it has the following monotonicity property: For every element $ a \in A $ and every element $ b \in A _ {0} + A _ {1} $, whenever $ K ( t,b ) \leq K ( t,a ) $ for all $ t > 0 $, it follows that $ b \in A $ and $ \| b \| _ {A} \leq C \| a \| _ {A} $ for some absolute constant $ C $.

Here, $ K ( t,f ) = K ( t,f;A _ {0} ,A _ {1} ) $ denotes the Peetre $ K $- functional of $ f $ with respect to the couple $ ( A _ {0} ,A _ {1} ) $. In this particular case, where the couple is $ ( L _ {1} ,L _ \infty ) $, there is a concrete formula for $ K ( t,x ) $( 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 $ A $.

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 $ ( A _ {0} ,A _ {1} ) $ can also be characterized via the Peetre $ K $- functionals for those pairs, by a monotonicity condition exactly analogous to the one in Calderón's result above. The Banach pairs $ ( A _ {0} ,A _ {1} ) $ 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, $ K $- monotone couples or $ {\mathcal C} $- pairs.)

It is also convenient to use the terminology $ K $- space for any normed space $ A $ satisfying $ A _ {0} \cap A _ {1} \subset A \subset A _ {0} + A _ {1} $ as well as the above-mentioned monotonicity property with respect to the $ K $- functional for $ ( A _ {0} ,A _ {1} ) $. By the important $ K $- divisibility theorem of Yu.A Brudnyi and N.Ya. Kruglyak [a2], it follows that each such $ K $- space necessarily coincides, to within equivalence of norms, with a space of the special form $ ( A _ {0} ,A _ {1} ) _ {G} ^ {K} $( 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 $ ( A _ {0} ,A _ {1} ) $ 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 $ A _ {0} $ and $ A _ {1} $ has to be relatively large.

Those Banach pairs which are known to be Calderón include pairs $ ( L _ {p _ {0} } ( w _ {0} ) ,L _ {p _ {1} } ( w _ {1} ) ) $ of weighted $ L _ {p} $ spaces for all choices of weight functions and for all exponents $ p _ {0} , p _ {1} \in [ 1, \infty ] $( 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

$$ \tag{a1 } ( A _ {0} ,A _ {1} ) = \left ( ( B _ {0} ,B _ {1} ) _ {\theta _ {0} ,q _ {0} } ^ {K} , ( B _ {0} ,B _ {1} ) _ {\theta _ {1} , q _ {1} } ^ {K} \right ) . $$

In this last example $ ( B _ {0} ,B _ {1} ) $ can be taken to be an arbitrary Banach pair and $ \theta _ {j} \in ( 0,1 ) $ and $ q _ {j} \in [ 1, \infty ] $ can be arbitrary numbers. Here, $ ( B _ {0} ,B _ {1} ) _ {\theta,q } ^ {K} $ denotes the Lions–Peetre real-method interpolation space, consisting of all elements $ b \in B _ {0} + B _ {1} $ for which the norm

$$ \left \| b \right \| = \left \{ \int\limits _ { 0 } ^ \infty {( t ^ {- \theta } K ( t,b;B _ {0} ,B _ {1} ) ) ^ {q} } { { \frac{dt }{t} } } \right \} ^ { {1 / q } } $$

is finite.

By choosing particular pairs $ ( B _ {0} ,B _ {1} ) $ one obtains, as special cases of this last result, that various pairs of Besov spaces (cf. Imbedding theorems) or Lorentz $ L _ {p,q } $ 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 $ ( L _ {p} ( \mathbf R ^ {n} ) , W _ {p} ^ {1} ( \mathbf R ^ {n} ) ) $ where $ p \neq 2 $( here $ W _ {p} ^ {1} $ is a Sobolev space) and $ ( C ( [ 0,1 ] ) , { \mathop{\rm Lip} } ( [ 0,1 ] ) ) $ and also such simple pairs as $ ( {\mathcal l} _ {1} \oplus {\mathcal l} _ {2} , {\mathcal l} _ \infty \oplus {\mathcal l} _ \infty ) $ and $ ( L _ {1} + L _ \infty ,L _ {1} \cap L _ \infty ) $.

In [a3], Brudnyi and A. Shteinberg consider whether pairs of the form $ ( F _ {0} ( B _ {0} ,B _ {1} ) ,F _ {1} ( B _ {0} ,B _ {1} ) ) $ are Calderón, where $ F _ {0} $ and $ F _ {1} $ are interpolation functors (cf. Interpolation of operators). Their results for the pair $ ( B _ {0} ,B _ {1} ) = ( C ( [ 0,1 ] ) , { \mathop{\rm Lip} } ( [ 0,1 ] ) ) $ lead them to conjecture that the above-mentioned result about iterated pairs of the form (a1) cannot be extended, i.e., that $ ( F _ {0} ( B _ {0} ,B _ {1} ) ,F _ {1} ( B _ {0} ,B _ {1} ) ) $ is Calderón for every Banach pair $ ( B _ {0} ,B _ {1} ) $ if and only if both functors $ F _ {j} $ are of the form $ F _ {j} ( B _ {0} ,B _ {1} ) = ( B _ {0} ,B _ {1} ) _ {\theta _ {j} ,q _ {j} } ^ {K} $. 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 $ K $- 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 $ X $ for which $ ( X,L _ \infty ) $ 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 $ L _ {p} $ structure or "near-Lp" structure. This result also shows that Sparr's theorem for weighted $ L _ {p} $ spaces cannot be sharpened: Let $ {( X _ {0} ,X _ {1} ) } $ be a pair of saturated $ \sigma $- order continuous Banach lattices with the Fatou property on the non-atomic measure space $ ( \Omega, \Sigma, \mu ) $. Suppose that at least one of the spaces $ X _ {0} $ and $ X _ {1} $ does not coincide, to within equivalence of norms, with a weighted $ L ^ {p} $ space on $ \Omega $. Then there exist weight functions $ {w _ {j} } : \Omega \rightarrow {( 0, \infty ) } $ for $ j = 0,1 $ such that the weighted Banach pair $ ( X _ {0} ( w _ {0} ) ,X _ {1} ( w _ {1} ) ) $ is not Calderón.

In most known examples of Banach pairs $ ( A _ {0} ,A _ {1} ) $ which are not Calderón, this happens because the complex interpolation spaces $ [ A _ {0} ,A _ {1} ] _ \alpha $( see Interpolation of operators) are not $ K $- spaces. But M. Mastyło and V.I. Ovchinnikov have found examples (see [a9]) of non-Calderón couples for which all the spaces $ [ A _ {0} ,A _ {1} ] _ \alpha $ are $ K $- spaces.

The notion of Calderón couples can also be considered in the wider context of operators $ T $ mapping from the spaces of one Banach pair $ ( A _ {0} ,A _ {1} ) $ to a possibly different Banach pair $ ( B _ {0} ,B _ {1} ) $. In such a context one says that $ A $ and $ B $ are relative interpolation spaces if every linear mapping $ T : {A _ {0} + A _ {1} } \rightarrow {B _ {0} + B _ {1} } $ which maps $ A _ {j} $ boundedly into $ B _ {j} $ for $ j = 0,1 $ also maps $ A $ boundedly into $ B $. (In the notation of Interpolation of operators, $ \{ A _ {0} ,A _ {1} ,A \} $ is an interpolation triple relative to $ \{ B _ {0} ,B _ {1} ,B \} $.) One says that $ A $ and $ B $ are relative $ K $- spaces if, for all $ a \in A $ and $ b \in B _ {0} + B _ {1} $, the $ K $- functional inequality

$$ K ( t,b;B _ {0} ,B _ {1} ) \leq K ( t,a;A _ {0} ,A _ {1} ) \textrm{ for all } t > 0 $$

implies that $ b \in B $ with $ \| b \| _ {B} \leq C \| a \| _ {A} $.

$ ( A _ {0} ,A _ {1} ) $ and $ ( B _ {0} ,B _ {1} ) $ are said to be relative Calderón couples if $ A $ and $ B $ are relative interpolation spaces if and only if they are relative $ K $- spaces. J. Peetre has shown (see [a6]) that if $ ( B _ {0} ,B _ {1} ) $ is any pair of weighted $ L _ \infty $ spaces, then $ ( A _ {0} ,A _ {1} ) $ and $ ( B _ {0} ,B _ {1} ) $ are relative Calderón couples for all Banach pairs $ ( A _ {0} ,A _ {1} ) $. Dually, if $ ( A _ {0} ,A _ {1} ) $ is an arbitrary pair of weighted $ L _ {1} $ spaces, then $ ( A _ {0} ,A _ {1} ) $ and $ ( B _ {0} ,B _ {1} ) $ are relative Calderón couples for all Banach pairs $ ( B _ {0} ,B _ {1} ) $ satisfying a mild "closure" condition. This latter result is another consequence of the Brudnyi–Kruglyak $ K $- 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].)

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