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

Ackermann, Nils

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

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
DOI: https://doi.org/10.1090/S0002-9939-05-08082-2
Published electronically: April 15, 2005
MathSciNet review: 2146210
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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.

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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: https://doi.org/10.1090/S0002-9939-05-08082-2
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
Published electronically: 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.