  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.
