Boyd index

The lower and upper Boyd indices of a rearrangement-invariant function space $ X $ on $ [ 0, \infty ) $ or $ [ 0,1 ] $ are defined by the respective formulas [a1]

$$ \alpha _ {X} = {\lim\limits } _ {t \rightarrow 0 } { \frac{ { \mathop{\rm log} } \left \| {D _ {t} } \right \| _ {X} }{ { \mathop{\rm log} } t } } $$

and

$$ \beta _ {X} = {\lim\limits } _ {t \rightarrow \infty } { \frac{ { \mathop{\rm log} } \left \| {D _ {t} } \right \| _ {X} }{ { \mathop{\rm log} } t } } . $$

Here $ D _ {t} $, $ t > 0 $, is the dilation operator, i.e.

$$ D _ {t} f ( x ) = f \left ( { \frac{x}{t} } \right ) , $$

for a measurable function $ f $ on $ [ 0, \infty ) $, while for an $ f $ on $ [ 0,1 ] $,

$$ D _ {t} f ( x ) = \left \{ \begin{array}{l} {f ( { \frac{x}{t} } ) \ \textrm{ if } x \leq { \mathop{\rm min} } ( 1,t ) , } \\ {0 \ \textrm{ if } t < x \leq 1. } \end{array} \right . $$

This operator is bounded in every rearrangement-invariant space $ X $ and the expression $ \| {D _ {t} } \| _ {X} $ is its norm in $ X $. The limits exist and $ 0 \leq \alpha _ {X} \leq \beta _ {X} \leq 1 $. Sometimes the indices are taken in the form $ p _ {X} = {1 / {\beta _ {X} } } $ and $ q _ {X} = {1 / {\alpha _ {X} } } $[a2].

There are many applications of Boyd indices. The first one was made by D.W. Boyd [a1], who proved an interpolation theorem which gives, in terms of $ \alpha _ {X} $ and $ \beta _ {X} $, the conditions for a linear operator of a weak type to be bounded in $ X $( cf. also Interpolation of operators).

A necessary and sufficient condition for some classical operators to be bounded in $ X $ may be also obtained in terms of Boyd indices. For example, the Hardy–Littlewood operator

$$ Hf ( x ) = { \frac{1}{x} } \int\limits _ { 0 } ^ { x } {f ( t ) } {dt } $$

is bounded in $ X $ if and only if $ \beta _ {X} < 1 $[a3].

An important property of the class of rearrangement-invariant spaces with non-trivial Boyd indices was discovered in [a4]. Let $ X $ be a rearrangement-invariant space on $ [ 0,1 ] $ and denote by $ Y $ the space of all measurable functions on $ [ 0, \infty ) $ such that $ f ^ {*} \chi _ {[ 0,1 ] } \in X $ and $ f ^ {*} \chi _ {( 1, \infty ) } \in L _ {2} ( 1, \infty ) $, where $ f ^ {*} $ is the decreasing rearrangement (cf. also Marcinkiewicz space) of $ | f | $ and $ \chi _ {A} $ denotes the indicator of the set $ A $. Put

$$ \left \| f \right \| _ {Y} = $$

$$ = \max \left \{ \left \| {f ^ {*} \chi _ {[ 0,1 ] } } \right \| _ {X} , \left ( \sum _ {k = 0 } ^ \infty \left ( \int\limits _ { k } ^ { {k } + 1 } {f ^ {*} ( x ) } {dx } \right ) ^ {2} \right ) ^ {1/2 } \right \} . $$

If the strong inequalities $ 0 < \alpha _ {X} \leq \beta _ {X} < 1 $ take place, then the spaces $ X $ and $ Y $ are isomorphic. In other words, $ X $ admits a representation as a rearrangement-invariant space on $ [ 0, \infty ) $.

References

[a1]	D.W. Boyd, "Indices of function spaces and their relationship to interpolation" Canadian J. Math. , 21 (1969) pp. 1245–1254
[a2]	J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , II. Function spaces , Springer (1979)
[a3]	S.G. Krein, E.M. Semenov, Yu.I. Petunin, "Interpolation of linear operators" , Transl. Math. Monograph , 54 , Amer. Math. Soc. (1982) (In Russian)
[a4]	W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, "Symmetric structures in Banach spaces" , Memoirs , 217 , Amer. Math. Soc. (1979) pp. 1–298