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

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- 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.## References

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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