Wave packet methods and boundary value problems
speaker: Hart Smith (University of Washington Seattle)abstract: Our lectures concerned the Lp norm of eigenfunctions for metrics of limited differentiability. The results also apply to functions with spectrum contained in a unit range of frequencies. The question of interest is to bound the Lp norm in terms of a power of the frequency, given that the L2 norm is bounded by 1. Sogge established the sharp exponent for compact Riemannian manifolds with smooth metrics, and our first result discussed was that the same exponent is valid for metrics which are only twice continuously differentiable. We then discussed the case of metrics in Holder classes between Lipschitz and $C2$, giving both examples showing that the exponents can be strictly larger, as well as establishing the best possible exponent for a range of $p$ near 2.
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