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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.
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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" />.
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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).
+
Let  $  A _ {0} $
 +
and  $  A _ {1} $
 +
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 $  ( A _ {0} ,A _ {1} ) $,  
 +
namely normed spaces  $  A $(
 +
cf. [[Normed space|Normed space]]) having the property that every [[Linear operator|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.
  
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" />.
+
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 [[#References|[a4]]] and B.S. Mityagin [[#References|[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 $.
  
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.)
+
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|Interpolation of operators]] for further details).
  
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.
+
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 $.
  
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.
+
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  $  ( 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.)
  
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
+
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 [[#References|[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|Interpolation of operators]]). Thus, for Calderón pairs, all the interpolation spaces are of this relatively simple 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>
+
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.
  
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
+
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, [[#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/c11001058.png" /></td> </tr></table>
+
$$ \tag{a1 }
 +
( A _ {0} ,A _ {1} ) = \left ( ( B _ {0} ,B _ {1} ) _ {\theta _ {0}  ,q _ {0} }  ^ {K} , ( B _ {0} ,B _ {1} ) _ {\theta _ {1}  , q _ {1} }  ^ {K} \right ) .
 +
$$
 +
 
 +
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.
 
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.
+
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|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 <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 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|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 [[#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.
+
In [[#References|[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|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 [[#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.
+
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 $  X $
 +
for which $  ( X,L _  \infty  ) $
 +
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 $  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 <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.
+
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|Interpolation of operators]]) are not $  K $-
 +
spaces. But M. Mastyło and V.I. Ovchinnikov have found examples (see [[#References|[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 <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
+
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|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
  
<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>
+
$$
 +
K ( t,b;B _ {0} ,B _ {1} ) \leq  K ( t,a;A _ {0} ,A _ {1} )  \textrm{ for  all  }  t > 0
 +
$$
  
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" />.
+
implies that $  b \in B $
 +
with $  \| b \| _ {B} \leq  C \| a \| _ {A} $.
  
<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.
+
$  ( 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 [[#References|[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 [[#References|[a11]]] and [[#References|[a8]]] for some special cases. (Cf. also [[#References|[a1]]].)
 
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====
 
====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>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Arazy,  M. Cwikel,  "A new characterization of the interpolation spaces between $L^p$ and $L^q$"  ''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 $L^1$ and $L^\infty$ 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 $K$ and $J$ 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 $L^1 + L^\infty$ and $L^1 \cap L^\infty$"  ''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 $L_p$ spaces when they are not described by the $K$-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 $L^p$ spaces"  ''Studia Math.'' , '''62'''  (1978)  pp. 229–271</TD></TR></table>

Latest revision as of 09:05, 26 March 2023


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 $L^p$ and $L^q$" 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 $L^1$ and $L^\infty$ 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 $K$ and $J$ 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 $L^1 + L^\infty$ and $L^1 \cap L^\infty$" 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 $L_p$ spaces when they are not described by the $K$-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 $L^p$ 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=23204
This article was adapted from an original article by M. Cwikel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article