Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L2 normalized family of functions such that P(h)u(h) is O(h) in L2(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the Lp norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch−δ(n,k,p) where k=dim H (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 δ(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 (in the sense of Definition 2.6 below) with respect to the bicharacteristic flow of P. Under this assumption we improve the estimate from δ=1/4 to 1/6, generalizing work of Burq–Gérard–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.
Anantharaman, N., Nonnenmacher, S.: Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier (Grenoble) 57(7), 2465–2523 (2007). Festival Yves Colin de Verdière
Bourgain, J., Rudnick, Z.: Restriction of toral eigenfunctions to hypersurfaces. C. R. Math. 347(21–22), 1249–1253 (2009)
Burq, N., Gérard, P., Tzvetkov, N.: The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds, pp. 21–52. Pubbl. Cent. Ric. Mat. Ennio Giorgi. Scuola Norm. Sup., Pisa (2004)