Improved Critical Eigenfunction Restriction Estimates on Riemannian Surfaces with Nonpositive Curvature
Tóm tắt
We show that one can obtain improved L
4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge’s strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L
2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved
$${L^\infty}$$
estimate of Bérard to obtain an improved weak-type L
4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767–781, 2011). This estimate improves the L
4 restriction estimate of Burq et al. (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009) by a power of
$${(\log\log\lambda)^{-1}}$$
. Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of
$${(\log\lambda)^{-1}}$$
by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover
$${\mathbb{H}^2}$$
.
Tài liệu tham khảo
Bak J.G., Seeger A.: Extensions of the Stein-Tomas theorem. Math. Res. Lett. 18(4), 767–781 (2011)
Bérard P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)
Blair, M.D., Sogge, C.D.: Concerning Toponogov’s Theorem and logarithmic improvement of estimates of eigenfunctions. arXiv:1510.07726 Preprint
Bourgain J.: Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 2, 145–187 (1991)
Burq N., Gérard P., Tzvetkov N.: Restriction of the Laplace-Beltrami eigenfunctions to submanifolds. Duke Math. J. 138, 445–486 (2007)
Chen X.: An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature. Trans. Am. Math. Soc. 367, 4019–4039 (2015)
Chen X., Sogge C.D.: A few endpoint geodesic restriction estimate for eigenfunctions. Commun. Math. Phys. 329(3), 435–459 (2014)
Hassell A., Tacy M.: Improvement of eigenfunction estimates on manifolds of non-positive curvature. Forum Math. 27, 1435–1451 (2015)
Hu R.: L p norm estimates of eigenfunctions restricted to submanifolds. Forum Math. 6, 1021–1052 (2009)
Sogge C.D.: Concerning the L p norm of spectral cluster of second-order elliptic operators on compact manifolds. J. Funct. Anal. 77, 123–138 (1988)
Sogge, C.D.: Fourier integrals in classical analysis. Cambridge tracts in mathematics, vol. 105. Cambridge University Press, Cambridge (1993)
Sogge, C.D.: Hangzhou lectures on eigenfunctions of the Laplacian. Annals of mathematics studies, vol. 188. Princeton University Press, Princeton (2014)
Sogge, C.D.: Improved critical eigenfunction estimates on manifolds of nonpositive curvature. arXiv:1512.03725 (Preprint)
Sogge, C.D.: Problems related to the concentration of eigenfunctions. arXiv:1510.07723 (Preprint)
Sogge, C.D., Zelditch, S.: On eigenfunction restriction estimates and L 4-bounds for compact surfaces with nonpositive curvature. In: Advances in anlysis: the legacy of Elias M. Stein, Priceton Mathematical Series, pp. 447–461. Princeton University Press, Princeton (2014)