A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If . : R-n -> [0.infinity] is Borel measurable, define for .-finite positive Borel measures ... on R-n the bilinear integral expression .(. ; ...) := integral(R)(n) integral(R)(n) .(. - .) .. (.) .. (.). We give conditions on (.) such that there is a constant (.) >= 0, independent of (.) and ., with .(. ; ...) <= root.(.(. ; ...) . (. ; ...). Our results apply to a much larger class of functions (.) than known before.