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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 \[ I(W;\mu ,\nu ):=\int _{\mathbb {R}^n}\int _{\mathbb {R}^n}W(x-y) d\mu (x) d\nu (y)\;. \] We give conditions on $W$ such that there is a constant $C\ge 0$, independent of $\mu$ and $\nu$, with \[ I(W;\mu ,\nu )\le C\sqrt {I(W;\mu ,\mu )I(W;\nu ,\nu )}\;. \] Our results apply to a much larger class of functions $W$ 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

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.