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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 [[#References|[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|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} } } $[[#References|[a2]]].

There are many applications of Boyd indices. The first one was made by D.W. Boyd [[#References|[a1]]], who proved an interpolation theorem which gives, in terms of  $  \alpha _ {X} $
and  $  \beta _ {X} $, 
the conditions for a [[Linear operator|linear operator]] of a weak type to be bounded in  $  X $(
cf. also [[Interpolation of operators|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 $[[#References|[a3]]].

An important property of the class of rearrangement-invariant spaces with non-trivial Boyd indices was discovered in [[#References|[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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.W. Boyd,   "Indices of function spaces and their relationship to interpolation"  ''Canadian J. Math.'' , '''21'''  (1969)  pp. 1245–1254</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''II. Function spaces''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.G. Krein,   E.M. Semenov,   Yu.I. Petunin,   "Interpolation of linear operators" , ''Transl. Math. Monograph'' , '''54''' , Amer. Math. Soc.  (1982)  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.B. Johnson,   B. Maurey,   G. Schechtman,   L. Tzafriri,   "Symmetric structures in Banach spaces" , ''Memoirs'' , '''217''' , Amer. Math. Soc.  (1979)  pp. 1–298</TD></TR></table>