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

Abstract:

If $W\colon \mathbb {R}^n \to [0,\infty ]$ is Borel measurable, define for $\sigma$-finite positive Borel measures $\mu ,\nu$ on $\mathbb {R}^n$ the bilinear integral expression \[ I(W;\mu ,\nu ):=\int _{\mathbb {R}^n}\int _{\mathbb {R}^n}W(x-y) d\mu (x) d\nu (y)\;. \] We give conditions on $W$ such that there is a constant $C\ge 0$, independent of $\mu$ and $\nu$, with \[ I(W;\mu ,\nu )\le C\sqrt {I(W;\mu ,\mu )I(W;\nu ,\nu )}\;. \] Our results apply to a much larger class of functions $W$ than known before.