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 normalized family of functions such that P(h)u(h) is O(h) in L 2(M) as h↓0. Let HM 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 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.

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Correspondence to Melissa Tacy.

Additional information

This research was supported in part by Australian Research Council Discovery Grant DP0771826, and an Australian Postgraduate Award.

Communicated by Michael Taylor.

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Hassell, A., Tacy, M. Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces. J Geom Anal 22, 74–89 (2012). https://doi.org/10.1007/s12220-010-9191-7

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Keywords

  • Eigenfunction estimates
  • L p estimates
  • Semiclassical analysis
  • Pseudodifferential operators
  • Restriction to hypersurfaces

Mathematics Subject Classification (2000)

  • 35Pxx
  • 58J40