  The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain
integrals involving products of two or three functions increase under symmetric
decreasing rearrangement. It is known that these inequalities extend to
integrands of the form F(u_1,..., u_m) where F is supermodular; in particular,
they hold when F has nonnegative mixed second derivatives. This paper concerns
the regularity assumptions on F and the equality cases. It is shown here that
extended Hardy-Littlewood and Riesz inequalities are valid for supermodular
integrands that are just Borel measurable. Under some nondegeneracy conditions,
all equality cases are equivalent to radially decreasing functions under
transformations that leave the functionals invariant (i.e., measure-preserving
maps for the Hardy-Littlewood inequality, translations for the Riesz
inequality). The proofs rely on monotone changes of variables in the spirit of
Sklar's theorem.
