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

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

2006-05-26

arxiv.org/abs/math/0605682v1

Mathematics

Analysis of PDEs

39 pages, to appear in Acta Mathematica

Scientific paper

We use microlocal and paradifferential techniques to obtain $L^8$ 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 $2\le q\le\infty$, for $L^2$-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 $\bar{q}_n\le q\le \infty$.

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