Boyd index

The lower and upper Boyd indices of a rearrangement-invariant function space on or are defined by the respective formulas [a1]

and

Here , , is the dilation operator, i.e.

for a measurable function on , while for an on ,

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

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

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