Namespaces
Variants
Actions

Difference between revisions of "Interpolation of operators"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
Obtaining from known properties of an operator in two or more spaces conclusions as to the properties of this operator in spaces that are in a certain sense intermediate. A Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519601.png" /> is a pair of Banach spaces (cf. [[Banach space|Banach space]]) that are algebraically and continuously imbedded in a separable [[Linear topological space|linear topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519602.png" />. One introduces the norm
+
<!--
 +
i0519601.png
 +
$#A+1 = 212 n = 0
 +
$#C+1 = 212 : ~/encyclopedia/old_files/data/I051/I.0501960 Interpolation of operators
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/i/i051/i051960/i0519603.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
on the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519604.png" />; on the arithmetical sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519605.png" /> the norm
+
Obtaining from known properties of an operator in two or more spaces conclusions as to the properties of this operator in spaces that are in a certain sense intermediate. A Banach pair  $  A , B $
 +
is a pair of Banach spaces (cf. [[Banach space|Banach space]]) that are algebraically and continuously imbedded in a separable [[Linear topological space|linear topological space]]  $  \mathfrak A $.  
 +
One introduces 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/i/i051/i051960/i0519606.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {A \cap B }  = \
 +
\max \{ \| x \| _ {A} , \| x \| _ {B} \}
 +
$$
  
is introduced. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519608.png" /> are Banach spaces. A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i0519609.png" /> is said to be intermediate for the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196011.png" />.
+
on the intersection  $  A \cap B $;
 +
on the arithmetical sum  $  A + B $
 +
the norm
  
A linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196012.png" />, acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196014.png" />, is called a bounded operator from the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196015.png" /> into the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196016.png" /> if its restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196017.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196018.png" />) is a bounded operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196019.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196020.png" /> (respectively, from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196021.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196022.png" />). A triple of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196023.png" /> is called an interpolation triple relative to the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196025.png" /> is intermediate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196026.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196027.png" /> is intermediate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196028.png" />), if every bounded operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196030.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196031.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196036.png" /> is called an interpolation space between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196038.png" />. For interpolation triples there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196039.png" /> such that
+
$$
 +
\| x \| _ {A+} B  = \
 +
\inf _ {x = u + v }
 +
\{ \| u \| _ {A} + \| v \| _ {B} \}
 +
$$
  
<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/i/i051/i051960/i05196040.png" /></td> </tr></table>
+
is introduced. The spaces  $  A \cap B $
 +
and  $  A + B $
 +
are Banach spaces. A Banach space  $  E $
 +
is said to be intermediate for the pair  $  A , B $
 +
if  $  A \cap B \subset  E \subset  A + B $.
  
The first interpolation theorem was obtained by M. Riesz (1926): The triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196041.png" /> is an interpolation triple for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196042.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196043.png" /> and if for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196044.png" />,
+
A linear mapping  $  T $,
 +
acting from  $  A + B $
 +
into  $  C + D $,
 +
is called a bounded operator from the pair  $  A , B $
 +
into the pair  $  C , D $
 +
if its restriction to  $  A $(
 +
respectively,  $  B $)
 +
is a bounded operator from  $  A $
 +
into  $  C $(
 +
respectively, from  $  B $
 +
into  $  D $).
 +
A triple of spaces  $  \{ A , B , E \} $
 +
is called an interpolation triple relative to the triple  $  \{ C , D , F \} $,
 +
where  $  E $
 +
is intermediate for  $  A , B $(
 +
respectively,  $  F $
 +
is intermediate for $  C , D $),
 +
if every bounded operator from  $  A , B $
 +
into  $  C , D $
 +
maps  $  E $
 +
into  $  F $.  
 +
If  $  A = C $,
 +
$  B = D $,
 +
$  E = F $,
 +
then  $  E $
 +
is called an interpolation space between  $  A $
 +
and  $  B $.  
 +
For interpolation triples there exists a constant  $  c $
 +
such that
  
<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/i/i051/i051960/i05196045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
\| T \| _ {E \rightarrow F }
 +
\leq  c \max
 +
\{ \| T \| _ {A \rightarrow C }  , \| T \| _ {B \rightarrow D }  \} .
 +
$$
  
The measures in the listed spaces may be different for each triple. Analogues of these theorems for other classes of families of spaces need not hold; e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196046.png" /> is not an interpolation space between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196048.png" />.
+
The first interpolation theorem was obtained by M. Riesz (1926): The triple $  \{ L _ {p _ {0}  } , L _ {p _ {1}  } , L _ {p _  \theta  } \} $
 +
is an interpolation triple for  $  \{ L _ {q _ {0}  } , L _ {q _ {1}  } , L _ {q _  \theta  } \} $
 +
if  $  1 \leq  p _ {0} , p _ {1} , q _ {0} , q _ {1} \leq  \infty $
 +
and if for a certain  $  \theta \in ( 0 , 1 ) $,
  
An interpolation functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196049.png" /> is a functor that assigns to each Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196050.png" /> an intermediate space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196051.png" />, where, moreover, for any two Banach pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196053.png" />, the triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196055.png" /> are interpolation for each other. There is a number of methods for constructing interpolation functors. Two of these gained the largest number of applications.
+
$$ \tag{1 }
  
==Peetre's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196056.png" />-method.==
+
\frac{1}{p} _  \theta  =
For a Banach pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196057.png" /> one constructs the functional
+
\frac{1 - \theta }{p _ {0} }
 +
+
  
<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/i/i051/i051960/i05196058.png" /></td> </tr></table>
+
\frac \theta {p _ {1} }
 +
,\ \
  
which is equivalent to the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196059.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196060.png" />. A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196061.png" /> of measurable functions on the semi-axis is called an ideal space if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196062.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196064.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196066.png" />. One considers all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196067.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196068.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196069.png" />. They form the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196070.png" /> with the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196071.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196072.png" /> is non-empty and is intermediate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196073.png" /> if and only if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196074.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196075.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196076.png" /> is an interpolation functor. For some Banach pairs the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196077.png" /> can be computed. This makes it possible to constructive effectively interpolation spaces. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196078.png" />:
+
\frac{1}{q} _  \theta  = \
  
<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/i/i051/i051960/i05196079.png" /></td> </tr></table>
+
\frac{1 - \theta }{q _ {0} }
 +
+
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196080.png" /> is a non-increasing right-continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196081.png" /> that is equi-measurable with the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196082.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196083.png" />:
+
\frac \theta {q _ {1} }
 +
.
 +
$$
  
<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/i/i051/i051960/i05196084.png" /></td> </tr></table>
+
The measures in the listed spaces may be different for each triple. Analogues of these theorems for other classes of families of spaces need not hold; e.g.,  $  C  ^ {1} ( 0 , 1 ) $
 +
is not an interpolation space between  $  C ( 0 , 1 ) $
 +
and  $  C  ^ {2} ( 0 , 1 ) $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196085.png" /> is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196086.png" />, and the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196087.png" /> denotes transition to the least convex majorant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196088.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196089.png" /> (a [[Sobolev space|Sobolev space]]),
+
An interpolation functor  $  F $
 +
is a functor that assigns to each Banach pair  $  A , B $
 +
an intermediate space  $  F ( A , B ) $,
 +
where, moreover, for any two Banach pairs  $  A , B $
 +
and  $  C , D $,  
 +
the triples  $  \{ A , B , F ( A , B ) \} $
 +
and $  \{ C , D , F ( C , D ) \} $
 +
are interpolation for each other. There is a number of methods for constructing interpolation functors. Two of these gained the largest number of applications.
  
<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/i/i051/i051960/i05196090.png" /></td> </tr></table>
+
==Peetre's  $  K $-method.==
 +
For a Banach pair  $  A , B $
 +
one constructs the functional
 +
 
 +
$$
 +
K ( t , x )  = \
 +
\inf _ {x = u + v }
 +
\{ \| u \| _ {A} + t \| v \| _ {B} \} ,
 +
$$
 +
 
 +
which is equivalent to the norm in  $  A + B $
 +
for each  $  t $.
 +
A Banach space  $  G $
 +
of measurable functions on the semi-axis is called an ideal space if  $  | f( t) | \leq  | g ( t) | $
 +
almost-everywhere on  $  ( 0 , \infty ) $
 +
and  $  g \in G $
 +
imply  $  f \in G $
 +
and  $  \| f \| _ {G} \leq  \| g \| _ {G} $.
 +
One considers all elements  $  x $
 +
from  $  A + B $
 +
for which  $  K ( t , x ) \in G $.
 +
They form the Banach space  $  ( A , B ) _ {G}  ^ {K} $
 +
with the norm  $  \| x \| _ {( A , B ) _ {G}  ^ {K} } = \| K ( t , x ) \| _ {G} $.
 +
The space  $  ( A, B ) _ {G}  ^ {K} $
 +
is non-empty and is intermediate for  $  A , B $
 +
if and only if the function  $  \min \{ t , 1 \} $
 +
belongs to  $  G $.
 +
In this case  $  F ( A , B ) = ( A , B ) _ {G}  ^ {K} $
 +
is an interpolation functor. For some Banach pairs the function  $  K ( t , x ) $
 +
can be computed. This makes it possible to constructive effectively interpolation spaces. For  $  L _ {1} , L _  \infty  $:
 +
 
 +
$$
 +
K ( t , x )  = \
 +
\int\limits _ { 0 } ^ { 1 }  x  ^ {*} ( \tau )  d \tau ,
 +
$$
 +
 
 +
where  $  x  ^ {*} ( t) $
 +
is a non-increasing right-continuous function on  $  ( 0, \infty ) $
 +
that is equi-measurable with the function  $  x $.  
 +
For  $  C , C  ^ {1} $:
 +
 
 +
$$
 +
K ( t , x )  = 
 +
\frac{1}{2}
 +
\widehat \omega  ( 2 t , x ) ,
 +
$$
 +
 
 +
where  $  \omega ( t , x ) $
 +
is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of the function  $  x $,
 +
and the sign  $  \widehat{ {}}  $
 +
denotes transition to the least convex majorant on  $  ( 0 , \infty ) $.  
 +
For  $  L _ {p} ( \mathbf R  ^ {n} ) , W _ {p}  ^ {l} ( \mathbf R  ^ {n} ) $(
 +
a [[Sobolev space|Sobolev space]]),
 +
 
 +
$$
 +
K ( t , x )  = \
 +
\left \{
 +
\begin{array}{ll}
 +
\omega _ {l,p} ( t  ^ {1/p} , x ) + t  \| x \| _ {L _ {p}  } ,  & t < 1 ,  \\
 +
\| x \| _ {L _ {p}  } ,  &t \geq  1 ,  \\
 +
\end{array}
 +
 
 +
\right .$$
  
 
where
 
where
  
<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/i/i051/i051960/i05196091.png" /></td> </tr></table>
+
$$
 +
\omega _ {l,p} ( t , x )  = \
 +
\sup  \left \{ {
 +
\| \Delta _ {h}  ^ {l} x ( s) \| _ {L _ {p}  } } : {
 +
| h | \leq  t } \right \}
 +
.
 +
$$
  
 
One often takes the space with norm
 
One often takes the space with 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/i/i051/i051960/i05196092.png" /></td> </tr></table>
+
$$
 +
\| f \| _ {G}  = \
 +
\left \{
 +
\int\limits _ { 0 } ^  \infty 
 +
t ^ {- \theta } | f ( t) |  ^ {q}
 +
\frac{dt}{t}
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196093.png" />. The corresponding functor is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196094.png" />. The Besov spaces
+
\right \}  ^ {1/q} ,\ \
 +
0 < \theta < 1 ,\ \
 +
1 \leq  q \leq  \infty ,
 +
$$
  
<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/i/i051/i051960/i05196095.png" /></td> </tr></table>
+
as  $  G $.  
 +
The corresponding functor is denoted by  $  ( A , B ) _ {\theta , p }  ^ {K} $.  
 +
The Besov spaces
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196096.png" /> play an important role in the theory of partial differential equations. A number of classical inequalities in analysis can be made more precise in terms of the Lorentz spaces
+
$$
 +
B _ {p,q}  ^ {m}  = ( L _ {p} , W _ {p}  ^ {l} ) _ {\theta , q }  ^ {K}
 +
$$
  
<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/i/i051/i051960/i05196097.png" /></td> </tr></table>
+
with  $  m = \theta l $
 +
play an important role in the theory of partial differential equations. A number of classical inequalities in analysis can be made more precise in terms of the Lorentz spaces
 +
 
 +
$$
 +
L _ {r,q}  = ( L _ {1} , L _  \infty  ) _ {\theta , q }  ^ {K } ,\ \
 +
r =  
 +
\frac{1}{1 - \theta }
 +
.
 +
$$
  
 
==The complex method of Calderón–Lions.==
 
==The complex method of Calderón–Lions.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196098.png" /> be a Banach pair. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i05196099.png" /> the space of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960100.png" /> defined in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960101.png" /> of the complex plane, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960102.png" />, and having the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960103.png" /> is continuous and bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960104.png" /> in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960105.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960106.png" /> is analytic inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960107.png" /> in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960108.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960109.png" /> is continuous and bounded in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960110.png" />; and 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960111.png" /> is continuous and bounded in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960112.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960114.png" />, is defined as the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960115.png" /> that can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960116.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960117.png" />. In it one introduces the norm
+
Let $  A , B $
 +
be a Banach pair. Denote by $  \Phi ( A , B ) $
 +
the space of all functions $  \phi ( z) $
 +
defined in the strip $  \Pi = \{ {z } : {0 \leq  \mathop{\rm Re}  z \leq  1 } \} $
 +
of the complex plane, with values in $  A + B $,  
 +
and having the following properties: 1) $  \phi ( z) $
 +
is continuous and bounded on $  \Pi $
 +
in the norm of $  A + B $;  
 +
2) $  \phi ( z) $
 +
is analytic inside $  \Pi $
 +
in the norm of $  A + B $;  
 +
3) $  \phi ( i \tau ) $
 +
is continuous and bounded in the norm of $  A $;  
 +
and 4) $  \phi ( 1 + i \tau ) $
 +
is continuous and bounded in the norm of $  B $.  
 +
The space $  [ A , B ] _  \alpha  $,  
 +
0 \leq  \alpha \leq  1 $,  
 +
is defined as the set of all elements $  x \in A + B $
 +
that can be represented as $  x = \phi ( \alpha ) $
 +
for  $  \phi \in \Phi ( A , B ) $.
 +
In it one introduces the norm
 +
 
 +
$$
 +
\| x \| _ {[ A , B ] _  \alpha  }
 +
= \inf _ {\phi ( \alpha ) = x } \
 +
\| \phi \| _ {\Phi ( A , B ) }  .
 +
$$
 +
 
 +
In this way the interpolation functor  $  [ A , B ] _  \alpha  $
 +
is defined. If  $  A = L _ {p _ {0}  } , B = L _ {p _ {1}  } $,
 +
$  p _ {0} , p _ {1} \leq  \infty $,
 +
then  $  [ L _ {p _ {0}  } , L _ {p _ {1}  } ] _  \alpha  = L _ {p} $
 +
with  $  1/p = ( 1 - \alpha ) / p _ {0} + \alpha / p _ {1} $.  
 +
If  $  G _ {0} $
 +
and  $  G _ {1} $
 +
are two ideal spaces and if in at least one of them the norm is absolutely continuous, then  $  [ G _ {0} , G _ {1} ] _  \alpha  $
 +
consists of all functions  $  x ( t) $
 +
for which  $  | x ( t) | = | x _ {0} ( t) | ^ {1 - \alpha } | x _ {1} ( t) |  ^  \alpha  $
 +
for some  $  x _ {0} \in G _ {0} $,
 +
$  x _ {1} \in G _ {1} $.  
 +
If  $  H _ {0} , H _ {1} $
 +
are two complex Hilbert spaces with  $  H _ {1} \subset  H _ {0} $,
 +
then  $  [ H _ {0} , H _ {1} ] _  \epsilon  $
 +
is a family of spaces that have very important applications. It is called a Hilbert scale. If  $  H _ {0} = L _ {2} $,
 +
$  H _ {2} = W _ {2}  ^ {l} $,
 +
then  $  [ H _ {0} , H _ {1} ] _  \alpha  = W _ {2} ^ {\alpha l } $(
 +
a Sobolev space of fractional index). For other methods of constructing interpolation functors, as well as on their relation to the theory of scales of Banach spaces, see [[#References|[1]]], [[#References|[3]]], [[#References|[5]]], [[#References|[8]]], [[#References|[9]]].
  
<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/i/i051/i051960/i051960118.png" /></td> </tr></table>
+
In the theory of interpolation of operators, Marcinkiewicz' interpolation theorem on interpolation operators of weak type plays an important role. An operator  $  T $
 +
from a Banach space  $  A $
 +
into a space of measurable functions, e.g. on the semi-axis, is called an operator of weak type  $  ( A , \psi ) $
 +
if  $  ( T x )  ^ {*} ( t) \leq  ( c / \psi ( t) )  \| x \| _ {A} $.
 +
It is assumed here that  $  \psi ( t) $
 +
and  $  t / \psi ( t) $
 +
are non-decreasing functions (e.g. $  \psi ( t) = t  ^  \alpha  $,
 +
$  0 \leq  \alpha \leq  1 $).  
 +
Theorems of Marcinkiewicz type enable one to describe for operators  $  T $
 +
of weak types  $  ( A _ {0} , \psi _ {0} ) $
 +
and  $  ( A _ {1} , \psi _ {1} ) $
 +
simultaneously (where  $  A _ {0} , A _ {1} $
 +
is a Banach pair) the pairs of spaces  $  A , E $
 +
for which  $  T A \subset  E $.  
 +
In many cases it is sufficient to check that the operator
  
In this way the interpolation functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960119.png" /> is defined. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960121.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960123.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960125.png" /> are two ideal spaces and if in at least one of them the norm is absolutely continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960126.png" /> consists of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960127.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960128.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960130.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960131.png" /> are two complex Hilbert spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960132.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960133.png" /> is a family of spaces that have very important applications. It is called a Hilbert scale. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960136.png" /> (a Sobolev space of fractional index). For other methods of constructing interpolation functors, as well as on their relation to the theory of scales of Banach spaces, see [[#References|[1]]], [[#References|[3]]], [[#References|[5]]], [[#References|[8]]], [[#References|[9]]].
+
$$
  
In the theory of interpolation of operators, Marcinkiewicz' interpolation theorem on interpolation operators of weak type plays an important role. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960137.png" /> from a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960138.png" /> into a space of measurable functions, e.g. on the semi-axis, is called an operator of weak type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960140.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960141.png" />. It is assumed here that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960143.png" /> are non-decreasing functions (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960145.png" />). Theorems of Marcinkiewicz type enable one to describe for operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960146.png" /> of weak types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960148.png" /> simultaneously (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960149.png" /> is a Banach pair) the pairs of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960150.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960151.png" />. In many cases it is sufficient to check that the operator
+
\frac{1}{\psi _ {0} ( t) }
 +
K
 +
\left (
  
<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/i/i051/i051960/i051960152.png" /></td> </tr></table>
+
\frac{\psi _ {0} ( t) }{\psi _ {1} ( t) }
 +
, x \right )
 +
$$
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960153.png" /> is the Peetre functional for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960154.png" />) acts from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960155.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960156.png" />. If for all linear operators of weak types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960157.png" /> it has been shown that this functional acts from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960158.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960159.png" />, then this also holds for quasi-additive operators (i.e. with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960160.png" />) of weak types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960162.png" />. Many important operators in analysis (e.g. Hilbert's singular operator) are of weak types in natural spaces; hence the corresponding interpolation theorems have found numerous applications.
+
(where $  K ( t , x ) $
 +
is the Peetre functional for $  A _ {0} , A _ {1} $)  
 +
acts from $  A $
 +
into $  E $.  
 +
If for all linear operators of weak types $  ( A _ {i} , \psi _ {i} ) $
 +
it has been shown that this functional acts from $  A $
 +
into $  E $,  
 +
then this also holds for quasi-additive operators (i.e. with the property $  | T ( x + y ) ( t) | \leq  b ( | T x ( t) | + | T y ( t) | ) $)  
 +
of weak types $  ( A _ {i} , \psi _ {i} ) $,  
 +
$  i = 0 , 1 $.  
 +
Many important operators in analysis (e.g. Hilbert's singular operator) are of weak types in natural spaces; hence the corresponding interpolation theorems have found numerous applications.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semi-groups of operators and approximation" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Magenes,  "Spazi di interpolazione ed equazioni a derivate parziali" , ''Conf. VII Congr. Union Mat. Italy, 1963'' , Cremonese  (1965)  pp. 134–197</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.Ya. Vilenkin (ed.)  et al. (ed.) , ''Functional analysis'' , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Bergh,  B.I. Löfström,  "Interpolation spaces" , Springer  (1976)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Triebel,  "Interpolation theory, function spaces, differential operators" , North-Holland  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semi-groups of operators and approximation" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Magenes,  "Spazi di interpolazione ed equazioni a derivate parziali" , ''Conf. VII Congr. Union Mat. Italy, 1963'' , Cremonese  (1965)  pp. 134–197</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.Ya. Vilenkin (ed.)  et al. (ed.) , ''Functional analysis'' , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Bergh,  B.I. Löfström,  "Interpolation spaces" , Springer  (1976)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Triebel,  "Interpolation theory, function spaces, differential operators" , North-Holland  (1978)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The theorem of M. Riesz mentioned in the main article is often called the M. Riesz convexity theorem. It has a somewhat more precise statement as follows (involving a bound on a certain norm for the bounded operator in question). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960163.png" /> be a linear operator mapping a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960164.png" /> of complex-valued measurable functions on a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960165.png" /> into measurable functions on another measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960166.png" />. Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960167.png" /> contains all indicator functions of measurable sets and is such that whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960168.png" />, then also all truncations (i.e. functions which coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960169.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960170.png" /> for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960171.png" /> and vanish elsewhere) belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960172.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960173.png" /> is said to be of type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960175.png" />) if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960176.png" /> such that
+
The theorem of M. Riesz mentioned in the main article is often called the M. Riesz convexity theorem. It has a somewhat more precise statement as follows (involving a bound on a certain norm for the bounded operator in question). Let $  T $
 +
be a linear operator mapping a linear space $  D $
 +
of complex-valued measurable functions on a [[Measure space|measure space]] $  ( M , {\mathcal M} , \mu ) $
 +
into measurable functions on another measure space $  ( N , {\mathcal N} , \nu ) $.  
 +
Assume $  D $
 +
contains all indicator functions of measurable sets and is such that whenever $  f \in D $,  
 +
then also all truncations (i.e. functions which coincide with $  f $
 +
in $  c _ {1} < | f ( x) | \leq  c _ {2} $
 +
for certain $  c _ {1} , c _ {2} > 0 $
 +
and vanish elsewhere) belong to $  D $.  
 +
The operator $  T $
 +
is said to be of type ( $  p , q $)  
 +
if there is a constant $  C $
 +
such that
  
<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/i/i051/i051960/i051960177.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\| T f \| _ {L _ {q}  ( N) }  \leq  \
 +
C  \| f \| _ {L _ {p}  ( M) } \ \
 +
\textrm{ for  all  }  f \in D \cap L _ {p} ( M) .
 +
$$
  
The least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960178.png" /> for which (a1) holds is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960180.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960181.png" />. The M. Riesz convexity theorem now states: If a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960182.png" /> is of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960183.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960184.png" />-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960186.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960187.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960188.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960189.png" />-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960190.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960191.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960193.png" /> satisfy (1). (The name  "convexity theorem"  derives from the fact that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960194.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960195.png" />, as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960196.png" />, is logarithmically convex.)
+
The least $  C $
 +
for which (a1) holds is called the $  ( p , q ) $-
 +
norm of $  T $.  
 +
The M. Riesz convexity theorem now states: If a linear operator $  T $
 +
is of types $  ( p _ {i} , q _ {i} ) $
 +
with $  ( p _ {i} , q _ {i} ) $-
 +
norms $  k _ {i} $,  
 +
$  i = 0 , 1 $,
 +
then $  T $
 +
is of type $  ( p _  \theta  , q _  \theta  ) $
 +
with $  ( p _  \theta  , q _  \theta  ) $-
 +
norm $  k _  \theta  \leq  k _ {0} ^ {1 - \theta } k _ {1}  ^  \theta  $,  
 +
provided 0 \leq  \theta \leq  1 $
 +
and $  p _  \theta  $,  
 +
$  q _  \theta  $
 +
satisfy (1). (The name  "convexity theorem"  derives from the fact that the $  ( p _  \theta  , q _  \theta  ) $-
 +
norm of $  T $,  
 +
as a function of $  \theta $,  
 +
is logarithmically convex.)
  
In the same setting, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960197.png" /> is called subadditive if
+
In the same setting, $  T $
 +
is called subadditive if
  
<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/i/i051/i051960/i051960198.png" /></td> </tr></table>
+
$$
 +
| ( T ( f _ {1} + f _ {2} ) ) ( x) |  \leq  \
 +
| ( T f _ {1} ) ( x) | +
 +
| ( T f _ {2} ) ( x) |
 +
$$
  
for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960199.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960200.png" />. A subadditive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960201.png" /> is said to be of weak type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960203.png" />) (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960205.png" />) if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960206.png" /> such that
+
for almost-all $  x \in N $
 +
and for $  f _ {1} , f _ {2} \in D $.  
 +
A subadditive operator $  T $
 +
is said to be of weak type ( $  p , q $)  
 +
(where $  1 \leq  p \leq  \infty $,  
 +
$  1\leq  q < \infty $)  
 +
if there is a constant $  k $
 +
such that
  
<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/i/i051/i051960/i051960207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\nu ( \{ {x \in N } : {| ( T f  ) ( x) | > s } \}
 +
)  \leq  \
 +
\left (
 +
\frac{k  \| f \| _ {L _ {p}  } }{s}
 +
\right ) ^ {q}
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960208.png" />. The least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960209.png" /> for which (a2) holds is called the weak (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960211.png" />)-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960212.png" />. (Note that the left-hand side of (a2) is the so-called distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960213.png" />.) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960214.png" />, (a2) must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960215.png" />.
+
for all $  f \in L _ {p} ( M) \cap D $.  
 +
The least $  k $
 +
for which (a2) holds is called the weak ( $  p , q $)-
 +
norm of $  T $.  
 +
(Note that the left-hand side of (a2) is the so-called distribution function of $  T f $.)  
 +
For $  q = \infty $,  
 +
(a2) must be replaced by $  \| T f \| _ {L _ {q}  } \leq  k  \| f \| _ {L _ {p}  } $.
  
A still further generalization is that of an operator of restricted weak type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960217.png" />, cf. [[#References|[6]]].
+
A still further generalization is that of an operator of restricted weak type $  ( p , q ) $,
 +
cf. [[#References|[6]]].
  
Singular integral operators (cf. [[Singular integral|Singular integral]]) often prove to be of some (weak) type (e.g. the [[Hilbert transform|Hilbert transform]] is of weak type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051960/i051960218.png" />).
+
Singular integral operators (cf. [[Singular integral|Singular integral]]) often prove to be of some (weak) type (e.g. the [[Hilbert transform|Hilbert transform]] is of weak type $  ( 1 , 1 ) $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bennett,  R.C. Sharpley,  "Interpolation of operators" , Acad. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bennett,  R.C. Sharpley,  "Interpolation of operators" , Acad. Press  (1988)</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


Obtaining from known properties of an operator in two or more spaces conclusions as to the properties of this operator in spaces that are in a certain sense intermediate. A Banach pair $ A , B $ is a pair of Banach spaces (cf. Banach space) that are algebraically and continuously imbedded in a separable linear topological space $ \mathfrak A $. One introduces the norm

$$ \| x \| _ {A \cap B } = \ \max \{ \| x \| _ {A} , \| x \| _ {B} \} $$

on the intersection $ A \cap B $; on the arithmetical sum $ A + B $ the norm

$$ \| x \| _ {A+} B = \ \inf _ {x = u + v } \{ \| u \| _ {A} + \| v \| _ {B} \} $$

is introduced. The spaces $ A \cap B $ and $ A + B $ are Banach spaces. A Banach space $ E $ is said to be intermediate for the pair $ A , B $ if $ A \cap B \subset E \subset A + B $.

A linear mapping $ T $, acting from $ A + B $ into $ C + D $, is called a bounded operator from the pair $ A , B $ into the pair $ C , D $ if its restriction to $ A $( respectively, $ B $) is a bounded operator from $ A $ into $ C $( respectively, from $ B $ into $ D $). A triple of spaces $ \{ A , B , E \} $ is called an interpolation triple relative to the triple $ \{ C , D , F \} $, where $ E $ is intermediate for $ A , B $( respectively, $ F $ is intermediate for $ C , D $), if every bounded operator from $ A , B $ into $ C , D $ maps $ E $ into $ F $. If $ A = C $, $ B = D $, $ E = F $, then $ E $ is called an interpolation space between $ A $ and $ B $. For interpolation triples there exists a constant $ c $ such that

$$ \| T \| _ {E \rightarrow F } \leq c \max \{ \| T \| _ {A \rightarrow C } , \| T \| _ {B \rightarrow D } \} . $$

The first interpolation theorem was obtained by M. Riesz (1926): The triple $ \{ L _ {p _ {0} } , L _ {p _ {1} } , L _ {p _ \theta } \} $ is an interpolation triple for $ \{ L _ {q _ {0} } , L _ {q _ {1} } , L _ {q _ \theta } \} $ if $ 1 \leq p _ {0} , p _ {1} , q _ {0} , q _ {1} \leq \infty $ and if for a certain $ \theta \in ( 0 , 1 ) $,

$$ \tag{1 } \frac{1}{p} _ \theta = \frac{1 - \theta }{p _ {0} } + \frac \theta {p _ {1} } ,\ \ \frac{1}{q} _ \theta = \ \frac{1 - \theta }{q _ {0} } + \frac \theta {q _ {1} } . $$

The measures in the listed spaces may be different for each triple. Analogues of these theorems for other classes of families of spaces need not hold; e.g., $ C ^ {1} ( 0 , 1 ) $ is not an interpolation space between $ C ( 0 , 1 ) $ and $ C ^ {2} ( 0 , 1 ) $.

An interpolation functor $ F $ is a functor that assigns to each Banach pair $ A , B $ an intermediate space $ F ( A , B ) $, where, moreover, for any two Banach pairs $ A , B $ and $ C , D $, the triples $ \{ A , B , F ( A , B ) \} $ and $ \{ C , D , F ( C , D ) \} $ are interpolation for each other. There is a number of methods for constructing interpolation functors. Two of these gained the largest number of applications.

Peetre's $ K $-method.

For a Banach pair $ A , B $ one constructs the functional

$$ K ( t , x ) = \ \inf _ {x = u + v } \{ \| u \| _ {A} + t \| v \| _ {B} \} , $$

which is equivalent to the norm in $ A + B $ for each $ t $. A Banach space $ G $ of measurable functions on the semi-axis is called an ideal space if $ | f( t) | \leq | g ( t) | $ almost-everywhere on $ ( 0 , \infty ) $ and $ g \in G $ imply $ f \in G $ and $ \| f \| _ {G} \leq \| g \| _ {G} $. One considers all elements $ x $ from $ A + B $ for which $ K ( t , x ) \in G $. They form the Banach space $ ( A , B ) _ {G} ^ {K} $ with the norm $ \| x \| _ {( A , B ) _ {G} ^ {K} } = \| K ( t , x ) \| _ {G} $. The space $ ( A, B ) _ {G} ^ {K} $ is non-empty and is intermediate for $ A , B $ if and only if the function $ \min \{ t , 1 \} $ belongs to $ G $. In this case $ F ( A , B ) = ( A , B ) _ {G} ^ {K} $ is an interpolation functor. For some Banach pairs the function $ K ( t , x ) $ can be computed. This makes it possible to constructive effectively interpolation spaces. For $ L _ {1} , L _ \infty $:

$$ K ( t , x ) = \ \int\limits _ { 0 } ^ { 1 } x ^ {*} ( \tau ) d \tau , $$

where $ x ^ {*} ( t) $ is a non-increasing right-continuous function on $ ( 0, \infty ) $ that is equi-measurable with the function $ x $. For $ C , C ^ {1} $:

$$ K ( t , x ) = \frac{1}{2} \widehat \omega ( 2 t , x ) , $$

where $ \omega ( t , x ) $ is the modulus of continuity (cf. Continuity, modulus of) of the function $ x $, and the sign $ \widehat{ {}} $ denotes transition to the least convex majorant on $ ( 0 , \infty ) $. For $ L _ {p} ( \mathbf R ^ {n} ) , W _ {p} ^ {l} ( \mathbf R ^ {n} ) $( a Sobolev space),

$$ K ( t , x ) = \ \left \{ \begin{array}{ll} \omega _ {l,p} ( t ^ {1/p} , x ) + t \| x \| _ {L _ {p} } , & t < 1 , \\ \| x \| _ {L _ {p} } , &t \geq 1 , \\ \end{array} \right .$$

where

$$ \omega _ {l,p} ( t , x ) = \ \sup \left \{ { \| \Delta _ {h} ^ {l} x ( s) \| _ {L _ {p} } } : { | h | \leq t } \right \} . $$

One often takes the space with norm

$$ \| f \| _ {G} = \ \left \{ \int\limits _ { 0 } ^ \infty t ^ {- \theta } | f ( t) | ^ {q} \frac{dt}{t} \right \} ^ {1/q} ,\ \ 0 < \theta < 1 ,\ \ 1 \leq q \leq \infty , $$

as $ G $. The corresponding functor is denoted by $ ( A , B ) _ {\theta , p } ^ {K} $. The Besov spaces

$$ B _ {p,q} ^ {m} = ( L _ {p} , W _ {p} ^ {l} ) _ {\theta , q } ^ {K} $$

with $ m = \theta l $ play an important role in the theory of partial differential equations. A number of classical inequalities in analysis can be made more precise in terms of the Lorentz spaces

$$ L _ {r,q} = ( L _ {1} , L _ \infty ) _ {\theta , q } ^ {K } ,\ \ r = \frac{1}{1 - \theta } . $$

The complex method of Calderón–Lions.

Let $ A , B $ be a Banach pair. Denote by $ \Phi ( A , B ) $ the space of all functions $ \phi ( z) $ defined in the strip $ \Pi = \{ {z } : {0 \leq \mathop{\rm Re} z \leq 1 } \} $ of the complex plane, with values in $ A + B $, and having the following properties: 1) $ \phi ( z) $ is continuous and bounded on $ \Pi $ in the norm of $ A + B $; 2) $ \phi ( z) $ is analytic inside $ \Pi $ in the norm of $ A + B $; 3) $ \phi ( i \tau ) $ is continuous and bounded in the norm of $ A $; and 4) $ \phi ( 1 + i \tau ) $ is continuous and bounded in the norm of $ B $. The space $ [ A , B ] _ \alpha $, $ 0 \leq \alpha \leq 1 $, is defined as the set of all elements $ x \in A + B $ that can be represented as $ x = \phi ( \alpha ) $ for $ \phi \in \Phi ( A , B ) $. In it one introduces the norm

$$ \| x \| _ {[ A , B ] _ \alpha } = \inf _ {\phi ( \alpha ) = x } \ \| \phi \| _ {\Phi ( A , B ) } . $$

In this way the interpolation functor $ [ A , B ] _ \alpha $ is defined. If $ A = L _ {p _ {0} } , B = L _ {p _ {1} } $, $ p _ {0} , p _ {1} \leq \infty $, then $ [ L _ {p _ {0} } , L _ {p _ {1} } ] _ \alpha = L _ {p} $ with $ 1/p = ( 1 - \alpha ) / p _ {0} + \alpha / p _ {1} $. If $ G _ {0} $ and $ G _ {1} $ are two ideal spaces and if in at least one of them the norm is absolutely continuous, then $ [ G _ {0} , G _ {1} ] _ \alpha $ consists of all functions $ x ( t) $ for which $ | x ( t) | = | x _ {0} ( t) | ^ {1 - \alpha } | x _ {1} ( t) | ^ \alpha $ for some $ x _ {0} \in G _ {0} $, $ x _ {1} \in G _ {1} $. If $ H _ {0} , H _ {1} $ are two complex Hilbert spaces with $ H _ {1} \subset H _ {0} $, then $ [ H _ {0} , H _ {1} ] _ \epsilon $ is a family of spaces that have very important applications. It is called a Hilbert scale. If $ H _ {0} = L _ {2} $, $ H _ {2} = W _ {2} ^ {l} $, then $ [ H _ {0} , H _ {1} ] _ \alpha = W _ {2} ^ {\alpha l } $( a Sobolev space of fractional index). For other methods of constructing interpolation functors, as well as on their relation to the theory of scales of Banach spaces, see [1], [3], [5], [8], [9].

In the theory of interpolation of operators, Marcinkiewicz' interpolation theorem on interpolation operators of weak type plays an important role. An operator $ T $ from a Banach space $ A $ into a space of measurable functions, e.g. on the semi-axis, is called an operator of weak type $ ( A , \psi ) $ if $ ( T x ) ^ {*} ( t) \leq ( c / \psi ( t) ) \| x \| _ {A} $. It is assumed here that $ \psi ( t) $ and $ t / \psi ( t) $ are non-decreasing functions (e.g. $ \psi ( t) = t ^ \alpha $, $ 0 \leq \alpha \leq 1 $). Theorems of Marcinkiewicz type enable one to describe for operators $ T $ of weak types $ ( A _ {0} , \psi _ {0} ) $ and $ ( A _ {1} , \psi _ {1} ) $ simultaneously (where $ A _ {0} , A _ {1} $ is a Banach pair) the pairs of spaces $ A , E $ for which $ T A \subset E $. In many cases it is sufficient to check that the operator

$$ \frac{1}{\psi _ {0} ( t) } K \left ( \frac{\psi _ {0} ( t) }{\psi _ {1} ( t) } , x \right ) $$

(where $ K ( t , x ) $ is the Peetre functional for $ A _ {0} , A _ {1} $) acts from $ A $ into $ E $. If for all linear operators of weak types $ ( A _ {i} , \psi _ {i} ) $ it has been shown that this functional acts from $ A $ into $ E $, then this also holds for quasi-additive operators (i.e. with the property $ | T ( x + y ) ( t) | \leq b ( | T x ( t) | + | T y ( t) | ) $) of weak types $ ( A _ {i} , \psi _ {i} ) $, $ i = 0 , 1 $. Many important operators in analysis (e.g. Hilbert's singular operator) are of weak types in natural spaces; hence the corresponding interpolation theorems have found numerous applications.

References

[1] P. Butzer, H. Berens, "Semi-groups of operators and approximation" , Springer (1967)
[2] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[3] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[4] J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[5] E. Magenes, "Spazi di interpolazione ed equazioni a derivate parziali" , Conf. VII Congr. Union Mat. Italy, 1963 , Cremonese (1965) pp. 134–197
[6] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)
[7] N.Ya. Vilenkin (ed.) et al. (ed.) , Functional analysis , Wolters-Noordhoff (1972) (Translated from Russian)
[8] J. Bergh, B.I. Löfström, "Interpolation spaces" , Springer (1976)
[9] H. Triebel, "Interpolation theory, function spaces, differential operators" , North-Holland (1978)

Comments

The theorem of M. Riesz mentioned in the main article is often called the M. Riesz convexity theorem. It has a somewhat more precise statement as follows (involving a bound on a certain norm for the bounded operator in question). Let $ T $ be a linear operator mapping a linear space $ D $ of complex-valued measurable functions on a measure space $ ( M , {\mathcal M} , \mu ) $ into measurable functions on another measure space $ ( N , {\mathcal N} , \nu ) $. Assume $ D $ contains all indicator functions of measurable sets and is such that whenever $ f \in D $, then also all truncations (i.e. functions which coincide with $ f $ in $ c _ {1} < | f ( x) | \leq c _ {2} $ for certain $ c _ {1} , c _ {2} > 0 $ and vanish elsewhere) belong to $ D $. The operator $ T $ is said to be of type ( $ p , q $) if there is a constant $ C $ such that

$$ \tag{a1 } \| T f \| _ {L _ {q} ( N) } \leq \ C \| f \| _ {L _ {p} ( M) } \ \ \textrm{ for all } f \in D \cap L _ {p} ( M) . $$

The least $ C $ for which (a1) holds is called the $ ( p , q ) $- norm of $ T $. The M. Riesz convexity theorem now states: If a linear operator $ T $ is of types $ ( p _ {i} , q _ {i} ) $ with $ ( p _ {i} , q _ {i} ) $- norms $ k _ {i} $, $ i = 0 , 1 $, then $ T $ is of type $ ( p _ \theta , q _ \theta ) $ with $ ( p _ \theta , q _ \theta ) $- norm $ k _ \theta \leq k _ {0} ^ {1 - \theta } k _ {1} ^ \theta $, provided $ 0 \leq \theta \leq 1 $ and $ p _ \theta $, $ q _ \theta $ satisfy (1). (The name "convexity theorem" derives from the fact that the $ ( p _ \theta , q _ \theta ) $- norm of $ T $, as a function of $ \theta $, is logarithmically convex.)

In the same setting, $ T $ is called subadditive if

$$ | ( T ( f _ {1} + f _ {2} ) ) ( x) | \leq \ | ( T f _ {1} ) ( x) | + | ( T f _ {2} ) ( x) | $$

for almost-all $ x \in N $ and for $ f _ {1} , f _ {2} \in D $. A subadditive operator $ T $ is said to be of weak type ( $ p , q $) (where $ 1 \leq p \leq \infty $, $ 1\leq q < \infty $) if there is a constant $ k $ such that

$$ \tag{a2 } \nu ( \{ {x \in N } : {| ( T f ) ( x) | > s } \} ) \leq \ \left ( \frac{k \| f \| _ {L _ {p} } }{s} \right ) ^ {q} $$

for all $ f \in L _ {p} ( M) \cap D $. The least $ k $ for which (a2) holds is called the weak ( $ p , q $)- norm of $ T $. (Note that the left-hand side of (a2) is the so-called distribution function of $ T f $.) For $ q = \infty $, (a2) must be replaced by $ \| T f \| _ {L _ {q} } \leq k \| f \| _ {L _ {p} } $.

A still further generalization is that of an operator of restricted weak type $ ( p , q ) $, cf. [6].

Singular integral operators (cf. Singular integral) often prove to be of some (weak) type (e.g. the Hilbert transform is of weak type $ ( 1 , 1 ) $).

References

[a1] C. Bennett, R.C. Sharpley, "Interpolation of operators" , Acad. Press (1988)
How to Cite This Entry:
Interpolation of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_of_operators&oldid=17969
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article