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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Cauchy-Schwarz type inequality for bilinear integrals on positive measures
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by Nils Ackermann PDF
Proc. Amer. Math. Soc. 133 (2005), 2647-2656 Request permission

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
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2146210