# 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 $[ 0, \infty )$.

How to Cite This Entry:
Boyd index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boyd_index&oldid=46142
This article was adapted from an original article by M. Braverman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article