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Convolution estimates for singular measures and some global nonlinear Brascamp—Lieb inequalities

Part of: Manifolds Functions of several variables

Published online by Cambridge University Press:  08 October 2015

Herbert Koch  and
Stefan Steinerberger
Herbert Koch
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany (stefan.steinerberger@yale.edu)
Stefan Steinerberger*
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany (stefan.steinerberger@yale.edu)
*
*Present address: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA
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Abstract

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We give an L2 x L2 → L2 convolution estimate for singular measures supported on transversal hypersurfaces in ℝn, which improves earlier results of Bejenaru et al. as well as Bejenaru and Herr. The quantities arising are relevant to the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp–Lieb inequalities with explicit constants in the same spirit.

Keywords

convolutionBrascamp–LiebLoomis–Whitneybilinear estimate

MSC classification

Secondary: 26B15: Integration 49Q15: Geometric measure and integration theory, integral and normal currents
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 6 , December 2015 , pp. 1223 - 1237
Copyright
Copyright © Royal Society of Edinburgh 2015