On the L^p norm of spectral clusters for compact manifolds with boundary

2006/05/26 by Hart F. Smith, Christopher D. Sogge · 1 citation
Mathematics · #math.AP #msc:35P99 #msc:35L20 #msc:42C99

paper · pdf

published as cta Math. 198 (2007), 107-153 · 39 pages, to appear in Acta Mathematica

arxiv created 2006/05/26 · arxiv updated 2013/01/29

Abstract

We use microlocal and paradifferential techniques to obtain L⁸ norm bounds for spectral clusters associated to elliptic second order operators on two-dimensional manifolds with boundary. The result leads to optimal L^q 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 L^q estimates in higher dimensions for a range of exponents q_n≤ q≤ ∞.

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