Interpolation of operators

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.[edit]

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.[edit]

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.

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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 ) $).

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