Semiclassical L^p Estimates of Quasimodes on Curved Hypersurfaces

Abstract

Let M be a compact manifold of dimension n, P = P(h) a semiclassical pseudodifferential operator on M, and u = u(h) an L^2 normalised family of functions such that Pu is O(h) in L^2(M) as h goes to 0. Let H be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L^p norms, p > 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch^d(n;k;p) where k=dimH (except for a logarithmic divergence in the case k = n-2; p = 2). When H is a hypersurface, i.e. k = n-1, we have d(n;n-1;2)=1/4, which is sharp when M is the round n-sphere and H is an equator. In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved with respect to the bicharacteristic flow of P. Under this assumption we improve the estimate from d=1/4 to 1/6, generalising work of Burq-Gerard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the Melrose-Taylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.