A Cauchy-Schwarz type inequality for bilinear integrals on positive measures. (English) Zbl 1066.26013

Summary: 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 \(C\geq0\), independent of \(\mu\) and \(\nu\), with \[ I(W;\mu,\nu)\leq C\sqrt{I(W;\mu,\mu)I(W;\nu,\nu)}\;. \] Our results apply to a much larger class of functions \(W\) than known before.


26D15 Inequalities for sums, series and integrals
43A35 Positive definite functions on groups, semigroups, etc.
35J20 Variational methods for second-order elliptic equations
60E15 Inequalities; stochastic orderings
42A82 Positive definite functions in one variable harmonic analysis
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