On multilinear spectral cluster estimates for manifolds with boundary

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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Published 1 January 2008