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

Ackermann, Nils

doi:10.1090/S0002-9939-05-08082-2

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

Author: Nils Ackermann
Journal: Proc. Amer. Math. Soc. 133 (2005), 2647-2656
MSC (2000): Primary 26D15; Secondary 43A35, 35J20, 60E15
Posted: April 15, 2005
MathSciNet review: 2146210
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Abstract | References | Similar Articles | Additional Information

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

$\begin{displaymath}I(W;\mu,\nu):=\int_{\mathbb{R} ^n}\int_{\mathbb{R} ^n}W(x-y)\,d\mu(x)\,d\nu(y)\;. \end{displaymath}$

We give conditions on

such that there is a constant $C\ge0$ , independent of $\mu$ and $\nu$ , with

$\begin{displaymath}I(W;\mu,\nu)\le C\sqrt{I(W;\mu,\mu)I(W;\nu,\nu)}\;. \end{displaymath}$

Our results apply to a much larger class of functions

than known before.

1. Nils Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423–443. MR 2088936 (2005i:35075), http://dx.doi.org/10.1007/s00209-004-0663-y
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. MR 1145940 (93k:35086), http://dx.doi.org/10.1090/S0002-9939-1993-1145940-X
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. MR 0226496 (37 #2085)
4. L. Mattner, Strict definiteness of integrals via complete monotonicity of derivatives, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3321–3342. MR 1422615 (97m:26026), http://dx.doi.org/10.1090/S0002-9947-97-01966-1
5. Zoltán Sasvári, Positive definite and definitizable functions, Mathematical Topics, vol. 2, Akademie Verlag, Berlin, 1994. MR 1270018 (95c:43005)
6. James Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math. 6 (1976), no. 3, 409–434. MR 0430674 (55 #3679)
7. Chuanming Zong, Strange phenomena in convex and discrete geometry, Universitext, Springer-Verlag, New York, 1996. MR 1416567 (97m:52001)

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Additional Information

Nils Ackermann
Affiliation: Justus-Liebig-Universität, Mathematisches Institut, Arndtstr. 2, D-35392 Giessen, Germany
Email: nils.ackermann@math.uni-giessen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08082-2
PII: S 0002-9939(05)08082-2
Keywords: Integral inequalities, positive definite functions, Cauchy-Schwarz inequality
Received by editor(s): June 18, 2003
Received by editor(s) in revised form: April 21, 2004
Posted: April 15, 2005
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.