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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[1] Nils Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423 – 443. · Zbl 1059.35037
[2] B. Buffoni, L. Jeanjean, and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), no. 1, 179 – 186. · Zbl 0789.35052
[3] Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. · Zbl 0152.20602
[4] L. Mattner, Strict definiteness of integrals via complete monotonicity of derivatives, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3321 – 3342. · Zbl 0901.26009
[5] Zoltán Sasvári, Positive definite and definitizable functions, Mathematical Topics, vol. 2, Akademie Verlag, Berlin, 1994. · Zbl 0815.43003
[6] James Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math. 6 (1976), no. 3, 409 – 434. · Zbl 0337.42017
[7] Chuanming Zong, Strange phenomena in convex and discrete geometry, Universitext, Springer-Verlag, New York, 1996. · Zbl 0865.52001
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