Cases of Equality in the Riesz Rearrangement Inequality 1 Statement of the Results Minkowski Sum of a and B. It Is Deened By

We determine the cases of equality in the Riesz rearrangement inequality ZZ f (y)g(x ? y)h(x) dydx ZZ f (y)g (x ? y)h (x) dydx where f , g , and h are the spherically decreasing rearrangements of the functions f , g, and h on R n. We apply our results to the weak Young inequality. The Riesz rearrangement inequality states that the functional I(f; g; h) := Z f gh dx = ZZ f(y)g(x?y)h(x) dydx (1:1) never decreases under spherical rearrangement, that is, 1 for any triple (f; g; h) of nonnegative measurable functions on R n for which the right hand side is deened. The spherically decreasing rearrangement, f , of a nonnegative measurable function f is the spherically decreasing function equimeasurable to f. We will deene it by f (x) = sup n s > 0 j (N s (f)) ! n jxj n o ; where N s (f) := n x 2 R n j f(x) > s o is the level set of f at height s, and ! n denotes the measure of the unit ball in R n. That is, the level sets of f are the centered balls of equal measure as the corresponding level sets of f. This deenition makes sense if all level sets corresponding to positive values of f have nite measure, for example, if f is in L p for some p < 1. In this paper, we determine the cases of equality in (1.2). A triple of functions that satisses (1.2) with equality will be called an optimizing triple, or optimizer, of the inequality. There are many optimizers of (1.2). One reason is that I is invariant under a large group of aane transformations: For any linear map, L, of determinant 1, and vectors a, b, and c = a + b in R n , we have where g ? denotes the function deened by g ? (x) := g(?x). Clearly, any triple of functions that is equivalent to a triple of spherically decreasing functions under these symmetries is an optimizer. There is a second reason to expect many optimizers. Consider the case when f and g have compact support. Then also the convolution f g has compact support. If h is the characteristic function of a set that contains the support of f g, then f; g; h produce equality in (1.2) regardless …