Acta Mathematica

On the Lp norm of spectral clusters for compact manifolds with boundary

Hart F. Smith and Christopher D. Sogge

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We use microlocal and paradifferential techniques to obtain L8 norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal Lq bounds, in the range 2⩽q⩽∞, for L2 - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp Lq estimates in higher dimensions for a range of exponents q̅nq⩽∞.


The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.

Article information

Acta Math., Volume 198, Number 1 (2007), 107-153.

Received: 15 November 2005
First available in Project Euclid: 31 January 2017

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2007 © Institut Mittag-Leffler


Smith, Hart F.; Sogge, Christopher D. On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math. 198 (2007), no. 1, 107--153. doi:10.1007/s11511-007-0014-z.

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