×

Bounds for spectral projectors on tori. (English) Zbl 1502.42008

The authors provide estimates of the norm from \(L^2\) to \(L^p\), \(p\in[2,\infty]\), of spectral projections on thin spherical shells of the Laplace-Beltrami operator \(\Delta\) on the \(d\)-dimensional torus \(\mathbb R^d/(\mathbb Ze_1+\dots+\mathbb Ze_d)\), where \(e_1,\dots,e_d\) is a basis of \(\mathbb R^d\). This amounts to considering the norm of operators \(P^\chi_{\lambda,\delta}=\chi((\sqrt{-Q(\nabla)}-\lambda)/\delta)\) on the standard torus \(\mathbb T^d=\mathbb R^d/\mathbb Z^d\), where \(Q\) is a positive definite quadratic form on \(\mathbb R^d\), \(\chi\) is a non-negative cutoff function with support in \([-1,1]\) and equal to 1 on \([-1/2,1/2]\), \(\lambda\geq1\), and \(0<\delta<1\). It is in fact “essentially independent” of the cutoff function and “essentially equivalent”, “up to possibly logarithmic factors”, of estimates for the resolvents of \(\Delta\) and the \(L^p\) norm of its eigenfunctions. An appendix provides for ease of reference the estimate in the case of \(\mathbb R^d\) with the Euclidean metric: let \(\sigma(p)=d-1-2d/p\); let \(p_{ST}=2(d+1)/(d-1)\) be the Stein-Thomas exponent; note that \((d-1)(1/2-1/p_{ST})=\sigma(p_{ST})\) and \((d+1)(1/2-1/p_{ST})=1\); then the norm of \(P^\chi_{\lambda,\delta}\) is bounded up to a constant by \(\lambda^{\sigma(p)/2}\delta^{1/2}\) if \(p\geq p_{ST}\), and by \(\lambda^{(d-1)(1/2-1/p)/2}\delta^{(d+1)(1/2-1/p)/2}\) if \(2\leq p\leq p_{ST}\); two examples show that each bound is optimal. These examples may be adapted to conjecture that the sought-after norm in the case of the torus is bounded up to a constant by \((\lambda\delta)^{(d-1)(1/2-1/p)/2}+\lambda^{\sigma(p)/2}\delta^{1/2}\) when \(\delta\geq\lambda^{-1}\). The authors provide a theorem, Theorem 6.1, that states their improvements on the range of validity of this conjecture; they show that it entails the conjecture “up to subpolynomial losses” in three cases: (1) \(2\leq p\leq p_{ST}\); (2) \(p\geq p_{ST}\) and \(\delta\) is “large”, i.e.\(\delta\lambda>\lambda^{2(p-2)/(dp-dp_{ST}+p-2)}\); (3) \(d=3\). Their method of proof combines a refinement of E. Landau [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1915, 209–243 (1915; JFM 45.0312.02)], to count the caps of the spherical shell that contain many lattice points using results provided by J. W. S. Cassels [An introduction to the geometry of numbers. Springer, Cham (1959; Zbl 0086.26203)], with the \(\ell^2\) decoupling theorem of J. Bourgain and C. Demeter [Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)].

MSC:

42B15 Multipliers for harmonic analysis in several variables
11H06 Lattices and convex bodies (number-theoretic aspects)
11L07 Estimates on exponential sums
11P21 Lattice points in specified regions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Blair, M. D. and Sogge, C. D.. Logarithmic improvements in \({L}^p\) bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature. Invent. Math., 217(2):703-748, 2019. · Zbl 1428.35248
[2] Bourgain, J.. Eigenfunction bounds for the Laplacian on the \(n\)-torus. Internat. Math. Res. Notices, (3):61-66, 1993. · Zbl 0779.58039
[3] Bourgain, J.. Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math., 193(1):441-458, 2013. · Zbl 1271.42039
[4] Bourgain, J., Burq, N., and Zworski, M.. Control for Schrödinger operators on 2-tori: rough potentials. J. Eur. Math. Soc. (JEMS), 15(5):1597-1628, 2013. · Zbl 1279.35016
[5] Bourgain, J. and Demeter, C.. Improved estimates for the discrete Fourier restriction to the higher dimensional sphere. Illinois J. Math., 57(1):213-227, 2013. · Zbl 1319.42006
[6] Bourgain, J. and Demeter, C.. New bounds for the discrete Fourier restriction to the sphere in 4D and 5D. Int. Math. Res. Not. IMRN, (11):3150-3184, 2015. · Zbl 1322.42013
[7] Bourgain, J. and Demeter, C.. The proof of the \({l}^2\) decoupling conjecture. Ann. of Math. (2), 182(1):351-389, 2015. · Zbl 1322.42014
[8] Bourgain, J., Shao, P., Sogge, C. D., and Yao, X.. On \({L}^p\)-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds. Comm. Math. Phys., 333(3):1483-1527, 2015. · Zbl 1396.58025
[9] Cassels, J. W. S.. An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 99 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959. · Zbl 0086.26203
[10] Chamizo, F., Cristóbal, E., and Ubis, A.. Lattice points in rational ellipsoids. J. Math. Anal. Appl., 350(1):283-289, 2009. · Zbl 1254.11092
[11] Cuenin, J.-C.. From spectral cluster to uniform resolvent estimates on compact manifolds. arXiv preprint (2011.07254).
[12] Ferreira, D. Dos Santos, Kenig, C. E., and Salo, M.. On \({L}^p\) resolvent estimates for Laplace-Beltrami operators on compact manifolds. Forum Math., 26(3):815-849, 2014. · Zbl 1302.35382
[13] Germain, P. and Leger, T.. Spectral projectors, resolvent, and fourier restriction on the hyperbolic space. arXiv preprint (2104.04126).
[14] Götze, F.. Lattice point problems and values of quadratic forms. Invent. Math., 157(1):195-226, 2004. · Zbl 1090.11063
[15] Guo, J.. On lattice points in large convex bodies. Acta Arith., 151(1):83-108, 2012. · Zbl 1314.11061
[16] Hickman, J.. Uniform \({l}^p\) resolvent estimates on the torus. Math. Res. Rep., 1: 31-45, 2020. · Zbl 1496.35173
[17] Huxley, M. N.. Exponential sums and lattice points. III. Proc. London Math. Soc. (3), 87(3):591-609, 2003. · Zbl 1065.11079
[18] Kenig, C. E., Ruiz, A., and Sogge, C. D.. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J., 55(2):329-347, 1987. · Zbl 0644.35012
[19] Krätzel, E.. Analytische Funktionen in der Zahlentheorie, volume 139 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teubner, Stuttgart, 2000. · Zbl 0962.11001
[20] Landau, E.. Ueber die Anzahl der Gitterpunkte in Gewissen Bereichen. (Zweite Abhandlung). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1915(1): 209-243, 1915. · JFM 45.0312.02
[21] Nowak, W. G.. Integer Points in Large Bodies, pages 583-599. Springer International Publishing, Cham, 2014. · Zbl 1360.11091
[22] Shen, Z.. On absolute continuity of the periodic Schrödinger operators. Internat. Math. Res. Notices, (1):1-31, 2001. · Zbl 0998.35007
[23] Sogge, C. D.. Fourier integrals in classical analysis, volume 105 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993. · Zbl 0783.35001
[24] Sogge, C. D., Toth, J. A., and Zelditch, S.. About the blowup of quasimodes on Riemannian manifolds. J. Geom. Anal., 21(1):150-173, 2011. · Zbl 1214.58012
[25] Stein, E. M.. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series.Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[26] Tao, T.. 247b, notes 2: decoupling theory. https://terrytao.wordpress.com/2020/04/13/247b-notes-2-decoupling-theory/.
[27] Tomas, P. A.. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 81:477-478, 1975. · Zbl 0298.42011
[28] Walfisz, A.. Über Gitterpunkte in vierdimensionalen Ellipsoiden. Math. Z., 72:259-278, 1959/60. · JFM 50.0116.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.