Boyd index
The lower and upper Boyd indices of a rearrangement-invariant function space on
or
are defined by the respective formulas [a1]
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and
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Here ,
, is the dilation operator, i.e.
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for a measurable function on
, while for an
on
,
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This operator is bounded in every rearrangement-invariant space and the expression
is its norm in
. The limits exist and
. Sometimes the indices are taken in the form
and
[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 and
, the conditions for a linear operator of a weak type to be bounded in
(cf. also Interpolation of operators).
A necessary and sufficient condition for some classical operators to be bounded in may be also obtained in terms of Boyd indices. For example, the Hardy–Littlewood operator
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is bounded in if and only if
[a3].
An important property of the class of rearrangement-invariant spaces with non-trivial Boyd indices was discovered in [a4]. Let be a rearrangement-invariant space on
and denote by
the space of all measurable functions on
such that
and
, where
is the decreasing rearrangement (cf. also Marcinkiewicz space) of
and
denotes the indicator of the set
. Put
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If the strong inequalities take place, then the spaces
and
are isomorphic. In other words,
admits a representation as a rearrangement-invariant space on
.
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 |
Boyd index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boyd_index&oldid=46142