a1 Northwestern University, Evanston, Illinois, U.S.A.
a2 Oxford University, Mathematical Institute, Oxford, England
It is well known that if f, g, h are nonnegative functions and f*, g*, h* their symmetrically decreasing rearrangements, then
also if u* is a spherically decreasing rearrangement of a function u,
In this paper it is proved under suitable assumptions (including the assumption that h is already rearranged) that equality holds in (i) if and only if f and g are already rearranged, and, if 1 < p < ∞ equality holds in (ii) if and only if u is already rearranged. We discuss (ii) both in ℝn and on the unit sphere.
(Received January 15 1985)
(Revised July 18 1985)
* This work is partially supported by National Science Foundation Grant MCS 791 5171, U.S. Army Grant DAJA 37-81-C-0220, and a grant from the U.K. Science and Engineering Research Council.