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].)

References

 [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
How to Cite This Entry:
Calderón couples. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calder%C3%B3n_couples&oldid=46188
This article was adapted from an original article by M. Cwikel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article