Mathematical Research Letters

Volume 15 (2008)

Number 3

On multilinear spectral cluster estimates for manifolds with boundary

Pages: 419 – 426

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n3.a2

Authors

Matthew D. Blair (University of Rochester)

Hart F. Smith (University of Washington)

Christopher D. Sogge (Johns Hopkins University)

Abstract

We prove bilinear and trilinear estimates for the spectral cluster operator on two and three-dimensional compact manifolds with boundary. These are the natural analogs of earlier estimates for the boundaryless case of Burq, Gérard, and Tzvetkov~\cite{bgtbilin}, \cite{bgtmultilin}. Our theorem reduces to establishing inequalities over small cubes whose size depends on frequency. After rescaling, these inequalities follow from mixed $L^p$ norm estimates on squarefunctions associated to the wave equation.

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