Mathematical Research Letters

Volume 14 (2007)

Number 1

Sharp $L^q$ bounds on spectral clusters for Holder metrics

Pages: 77 – 85



Herbert Koch (Universität Bonn)

Hart F. Smith (University of Washington)

Daniel Tataru (University of California at Berkeley)


We establish $L^q$ bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity $C^s$, where $0\le s\le 1$. We also produce examples which show that these bounds are best possible for the case $q=\infty$, and for $2\le q\le q_n$.

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