Home Improved spectral cluster bounds for orthonormal systems
Article
Licensed
Unlicensed Requires Authentication

Improved spectral cluster bounds for orthonormal systems

  • Tianyi Ren and An Zhang EMAIL logo
Published/Copyright: December 15, 2023

Abstract

We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) to [ λ 2 , ( λ + ϵ ( λ ) ) 2 ) , where ϵ ( λ ) is a function of λ that goes to 0 as λ goes to . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].

MSC 2020: 58J50; 35P15

Communicated by Christopher D. Sogge


Funding statement: Tianyi Ren is supported in part by the Fundamental Research Funds for the Central Universities Grant No. KG16-2508-01. An Zhang is supported in part by NSFC grants No. 11801536, and the Fundamental Research Funds for the Central Universities Grant No. YWF-22-T-204 and No. KG16248701..

Acknowledgements

The authors would like to thank the anonymous referee for the typos they pointed out and the suggestions they provided that help to make this article clearer.

References

[1] V. G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956), 327–344. 10.1007/BF01473886Search in Google Scholar

[2] N. Bez, S. Lee and S. Nakamura, Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates, Forum Math. Sigma 9 (2021), Paper No. e1. 10.1017/fms.2020.64Search in Google Scholar

[3] M. D. Blair, X. Huang, Y. Sire and C. D. Sogge, Uniform Sobolev estimates on compact manifolds involving singular potentials, Rev. Mat. Iberoam. 38 (2022), no. 4, 1239–1286. 10.4171/rmi/1300Search in Google Scholar

[4] M. D. Blair, X. Huang and C. D. Sogge, Improved spectral projection estimates, preprint (2022), https://arxiv.org/abs/2211.17266. Search in Google Scholar

[5] M. D. Blair and C. D. Sogge, Logarithmic improvements in L p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature, Invent. Math. 217 (2019), no. 2, 703–748. 10.1007/s00222-019-00873-6Search in Google Scholar

[6] J. Bourgain and C. Demeter, The proof of the l 2 decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. 10.4007/annals.2015.182.1.9Search in Google Scholar

[7] J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 (2015), no. 3, 1483–1527. 10.1007/s00220-014-2077-ySearch in Google Scholar

[8] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992. 10.1007/978-1-4757-2201-7Search in Google Scholar

[9] R. L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials. III, Trans. Amer. Math. Soc. 370 (2018), no. 1, 219–240. 10.1090/tran/6936Search in Google Scholar

[10] R. L. Frank, M. Lewin, E. H. Lieb and R. Seiringer, Strichartz inequality for orthonormal functions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1507–1526. 10.4171/jems/467Search in Google Scholar

[11] R. L. Frank and J. Sabin, Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates, Amer. J. Math. 139 (2017), no. 6, 1649–1691. 10.1353/ajm.2017.0041Search in Google Scholar

[12] R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 (2017), 157–192. 10.1016/j.aim.2017.06.023Search in Google Scholar

[13] P. Germain and S. L. Rydin Myerson, Bounds for spectral projectors on tori, Forum Math. Sigma 10 (2022), Paper No. e24. 10.1017/fms.2022.18Search in Google Scholar

[14] J. Hickman, Uniform L p resolvent estimates on the torus, Math. Res. Rep. 1 (2020), 31–45. 10.5802/mrr.1Search in Google Scholar

[15] E. Hlawka, Über Integrale auf konvexen Körpern. I, Monatsh. Math. 54 (1950), 1–36. 10.1007/BF01304101Search in Google Scholar

[16] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. 10.1007/BF02391913Search in Google Scholar

[17] X. Huang, Y. Sire and C. Zhang, Spectral cluster estimates for Schrödinger operators of relativistic type, J. Math. Pures Appl. (9) 155 (2021), 32–61. 10.1016/j.matpur.2021.08.004Search in Google Scholar

[18] B. M. Levitan, On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325–352. Search in Google Scholar

[19] E. H. Lieb and W. E. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35 (1975), no. 11, 687–689. 10.1103/PhysRevLett.35.687Search in Google Scholar

[20] S. S. Mondal and M. Song, Orthonormal strichartz inequalities for the ( k , a ) -generalized laguerre operator and Dunkl operator, preprint (2022), https://arxiv.org/abs/2208.12015. Search in Google Scholar

[21] W. Müller, Lattice points in large convex bodies, Monatsh. Math. 128 (1999), no. 4, 315–330. 10.1007/s006050050066Search in Google Scholar

[22] S. Nakamura, The orthonormal Strichartz inequality on torus, Trans. Amer. Math. Soc. 373 (2020), no. 2, 1455–1476. 10.1090/tran/7982Search in Google Scholar

[23] T. Ren, Y. Xi and C. Zhang, An endpoint version of uniform Sobolev inequalities, Forum Math. 30 (2018), no. 5, 1279–1289. 10.1515/forum-2018-0042Search in Google Scholar

[24] B. Simon, Trace Ideals and Their Applications, Math. Surveys Monogr. 120, American Mathematical Society, Providence, 2005. Search in Google Scholar

[25] C. 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. 10.1016/0022-1236(88)90081-XSearch in Google Scholar

[26] C. D. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, Ann. of Math. Stud. 188, Princeton University, Princeton, 2014. 10.1515/9781400850549Search in Google Scholar

[27] C. D. Sogge, Fourier Integrals in Classical Analysis, 2nd ed., Cambridge Tracts in Math. 210, Cambridge University, Cambridge, 2017. 10.1017/9781316341186Search in Google Scholar

Received: 2023-07-17
Revised: 2023-11-01
Published Online: 2023-12-15
Published in Print: 2024-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0254/html?lang=en
Scroll to top button