Semiclassical \(L^p\) estimates of quasimodes on curved hypersurfaces. (English) Zbl 1256.35218
Summary: 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 \(h\downarrow 0\). Let \(H\subset M\) be a compact submanifold of \(M\). In [Commun. Partial Differ. Equations 35, No. 8, 1538–1562 (2010; Zbl 1205.35352)], the second-named author proved estimates on the \(L^p\) norms, \(p\geq 2\), 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 \(Ch^{-\delta (n,k,p)}\) 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 \(\delta (n,n-1,2)=1/4\), 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 \(\delta =1/4\) to \(1/6\), generalizing work of N. Burq, P. Gérard and N. Tzvetkov [in: Phase space analysis of partial differential equations. Vol. I. Proceedings of the research trimester, Centro di Ricerca Matematica “Ennio de Giorgi”, Pisa, Italy, 2004. Pisa: Scuola Normale Superiore. Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi, 21–52 (2004; Zbl 1084.35086)] and R. Hu [Forum Math. 21, No. 6, 1021–1052 (2009; Zbl 1187.35147)] for Laplace eigenfunctions. To do this we apply the R. B. Melrose and M. E. Taylor theorem [Adv. Math. 55, 242–315 (1985; Zbl 0591.58034)], as adapted by Y. Pan and C. D. Sogge [Colloq. Math. 60/61, No. 2, 413–419 (1990; Zbl 0746.58076)], for Fourier integral operators with folding canonical relations.
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 \(\delta =1/4\) to \(1/6\), generalizing work of N. Burq, P. Gérard and N. Tzvetkov [in: Phase space analysis of partial differential equations. Vol. I. Proceedings of the research trimester, Centro di Ricerca Matematica “Ennio de Giorgi”, Pisa, Italy, 2004. Pisa: Scuola Normale Superiore. Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi, 21–52 (2004; Zbl 1084.35086)] and R. Hu [Forum Math. 21, No. 6, 1021–1052 (2009; Zbl 1187.35147)] for Laplace eigenfunctions. To do this we apply the R. B. Melrose and M. E. Taylor theorem [Adv. Math. 55, 242–315 (1985; Zbl 0591.58034)], as adapted by Y. Pan and C. D. Sogge [Colloq. Math. 60/61, No. 2, 413–419 (1990; Zbl 0746.58076)], for Fourier integral operators with folding canonical relations.
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
Keywords:
eigenfunction estimates; \(L^p\) estimates; semiclassical analysis; pseudodifferential operators; restriction to hypersurfacesReferences:
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