Mathematical Research Letters

Volume 15, Issue 3, May 2008 pp. 419-426.

On multilinear spectral cluster estimates for manifolds with boundary

Authors: Matthew D. Blair (1), Hart F. Smith (2), and Christopher D. Sogge (3)
Author institution: University of Rochester (1), University of Washington (2), and Johns Hopkins University (3)

Summary: 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, G\'erard, 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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