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.
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