A note on constructing families of sharp examples for 𝐿^{𝑝} growth of eigenfunctions and quasimodes

Tacy, Melissa

doi:10.1090/proc/14028

A note on constructing families of sharp examples for $L^{p}$ growth of eigenfunctions and quasimodes
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Proc. Amer. Math. Soc. 146 (2018), 2909-2924 Request permission

Abstract:

In this note we analyse $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular, we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition, we produce a family of flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check $L^{p}$ growth estimates against the simple flat model examples. Finally, we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples.

References

N. Burq, P. Gérard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445–486 (English, with English and French summaries). MR 2322684, DOI 10.1215/S0012-7094-07-13834-1
N. Burq and C. Zuily, Concentration of Laplace eigenfunctions and stabilization of weakly damped wave equation, Comm. Math. Phys. 345 (2016), no. 3, 1055–1076. MR 3519589, DOI 10.1007/s00220-016-2591-1
Xuehua Chen and Christopher D. Sogge, A few endpoint geodesic restriction estimates for eigenfunctions, Comm. Math. Phys. 329 (2014), no. 2, 435–459. MR 3210140, DOI 10.1007/s00220-014-1959-3
Z. Guo, X. Han, and M. Tacy, $L^p$ bilinear quasimode estimates. arXiv:1503.00413, 2015.
Andrew Hassell and Melissa Tacy, Semiclassical $L^p$ estimates of quasimodes on curved hypersurfaces, J. Geom. Anal. 22 (2012), no. 1, 74–89. MR 2868958, DOI 10.1007/s12220-010-9191-7
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
Herbert Koch, Daniel Tataru, and Maciej Zworski, Semiclassical $L^p$ estimates, Ann. Henri Poincaré 8 (2007), no. 5, 885–916. MR 2342881, DOI 10.1007/s00023-006-0324-2
C. Sogge, Problems related to the concentration of eigenfunctions, arxiv.org/abs/1510.07723, 2015.
Christopher D. Sogge, Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–138. MR 930395, DOI 10.1016/0022-1236(88)90081-X
Melissa Tacy, Semiclassical $L^p$ estimates of quasimodes on submanifolds, Comm. Partial Differential Equations 35 (2010), no. 8, 1538–1562. MR 2754054, DOI 10.1080/03605301003611006