Journal of Geometric Analysis

, Volume 22, Issue 1, pp 74–89 | Cite as

Semiclassical Lp Estimates of Quasimodes on Curved Hypersurfaces

Article

Abstract

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 HM 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 Chδ(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.

Keywords

Eigenfunction estimates Lp estimates Semiclassical analysis Pseudodifferential operators Restriction to hypersurfaces 

Mathematics Subject Classification (2000)

35Pxx 58J40 

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Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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