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

Abstract

If W: R n → [0, oo] is Borel measurable, define for σ-finite positive Borel measures μ, ν on R n the bilinear integral expression I(W; μ, ν):= ∫ R n∫ R W(x - y) dμ(x) dv(y) We give conditions on W such that there is a constant C > 0, independent of μ and ν, with I(W; μ, v) ≤ CI(W;μ, μ)I(W; v, ν). Our results apply to a much larger class of functions W than known before.