Strict inequalities for integrals of decreasingly rearranged functions

SynopsisIt is well known that iff,g,hare nonnegative functions andf*,g*,h* their symmetrically decreasing rearrangements, thenalso ifu* is a spherically decreasing rearrangement of a functionu,In this paper it is proved under suitable assumptions (including the assumption thathis already rearranged) that equality holds in (i) if and only iffandgare already rearranged, and, if 1 <p< ∞ equality holds in (ii) if and only ifuis already rearranged. We discuss (ii) both in ℝnand on the unit sphere.

A General Integral Inequality for the Derivative of an Equimeasurable Rearrangement

The theory of non-increasing (decreasing) equimeasurable rearrangements of functions was introduced by Hardy and Littlewood [6] in connection with their studies of fractional integrals and integral operators. Elementary properties of equimeasurable decreasing rearrangements are given in the monograph [7] of Hardy, Littlewood, and Polya on inequalities, while a more recent treatment is Okikiolu [9, § 5.4].

Variation Reducing Properties of Decreasing Rearrangements

One well-established characteristic of the operation of decreasing rearrangement is its variation reducing property. A systematic study of this property has been made in considerable detail by G.F.D. Duff in [5] and [6]. He proved some inequalities related to the operation of rearrangement in decreasing order showing that the total variation of a sequence or an absolutely continuous function is in general diminished by such rearrangement. He also showed that the Lp norm of the difference sequence (or the derivative function) is diminished by this rearrangement operation unless the given sequence (or absolutely continuous function) is already monotonie (or equal to a monotonie function almost everywhere).