## 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.

### MSC:

 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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### References:

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