# 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.

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)

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=47394
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article