×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.