Vol. 38, No. 2, 1971

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Indices for the Orlicz spaces

David W. Boyd

Vol. 38 (1971), No. 2, 315–323

The determination of the function spaces X which are intermediate in the weak sense between Lp and Lq has been shown, by the author, to depend on a pair of numbers (α,β) called the indices of the space. The indices depend on the function norm of X and on the properties of the underlying measure space: whether it has finite or infinite measure, is non-atomic or atomic. In this paper, formulas are given for the indices of an Orlicz space in case the measure space is non-atomic with finite or infinite measure, or else is purely atomic with atoms of equal measure. The indices for an Orlicz space over a non-atomic finite measure space turn out to be the reciprocals of the exponents of the space as introduced by Matuszewska and Orlicz, and generalized by Shimogaki. Some new results concerning submultiplicative functions are used in the proof of the main result.

Mathematical Subject Classification 2000
Primary: 46E30
Received: 17 June 1970
Published: 1 August 1971
David W. Boyd
Department of Mathematics
University of British Columbia
Vancouver BC V6T 1Z2