Home Strichartz inequality for orthonormal functions associated with special Hermite operator
Article
Licensed
Unlicensed Requires Authentication

Strichartz inequality for orthonormal functions associated with special Hermite operator

  • Sunit Ghosh , Shyam Swarup Mondal and Jitendriya Swain EMAIL logo
Published/Copyright: August 25, 2023

Abstract

In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions.


Communicated by Christopher D. Sogge


Acknowledgements

The first author thanks the Indian Institute of Technology Guwahati for the facilities provided during the period of this work. We thank the referee for the meticulous reading, thoughtful comments and efforts towards improving the manuscript.

References

[1] W. Beckner, Geometric inequalities in Fourier anaylsis, Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton 1991), Princeton Math. Ser. 42, Princeton University, Princeton (1995), 36–68. 10.1515/9781400852949.36Search in Google Scholar

[2] S. Beigi and M. M. Goodarzi, Operator-valued Schatten spaces and quantum entropies, preprint (2022), https://arxiv.org/abs/2207.06693. 10.1007/s11005-023-01712-9Search in Google Scholar

[3] F. A. Berezin, Convex functions of operators, Mat. Sb. (N. S.) 88(130) (1972), 268–276. Search in Google Scholar

[4] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. 10.1007/978-3-642-66451-9Search in Google Scholar

[5] N. Bez, Y. Hong, S. Lee, S. Nakamura and Y. Sawano, On the Strichartz estimates for orthonormal systems of initial data with regularity, Adv. Math. 354 (2019), Article ID 106736. 10.1016/j.aim.2019.106736Search in Google Scholar

[6] R. L. Frank, The Lieb–Thirring inequalities: Recent results and open problems, Nine mathematical Challenges—An Elucidation, Proc. Sympos. Pure Math. 104, American Mathematical Society, Providence (2021), 45–86. 10.1090/pspum/104/01877Search in Google Scholar

[7] 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

[8] 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

[9] H. Koch and D. Tataru, L p eigenfunction bounds for the Hermite operator, Duke Math. J. 128 (2005), no. 2, 369–392. 10.1215/S0012-7094-04-12825-8Search in Google Scholar

[10] E. H. Lieb, The classical limit of quantum spin systems, Comm. Math. Phys. 31 (1973), 327–340. 10.1007/BF01646493Search in Google Scholar

[11] E. H. Lieb, The stability of matter, Rev. Modern Phys. 48 (1976), no. 4, 553–569. 10.1103/RevModPhys.48.553Search in Google Scholar

[12] E. H. Lieb, An L p bound for the Riesz and Bessel potentials of orthonormal functions, J. Funct. Anal. 51 (1983), no. 2, 159–165. 10.1016/0022-1236(83)90023-XSearch in Google Scholar

[13] E. H. Lieb, The stability of matter: From atoms to stars, Bull. Amer. Math. Soc. (N. S.) 22 (1990), no. 1, 1–49. 10.1090/S0273-0979-1990-15831-8Search in Google Scholar

[14] E. H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University, Cambridge, 2010. 10.1017/CBO9780511819681Search in Google Scholar

[15] E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Princeton University, Princeton (1976), 269–303. 10.1515/9781400868940-014Search in Google Scholar

[16] E. H. Lieb and W. E. Thirring, Bound on kinetic energy of fermions which proves stability of matter, Phys. Pev. Lett. 35 (1983), 687–689. 10.1103/PhysRevLett.35.687Search in Google Scholar

[17] S. S. Mondal and J. Swain, Correction: Restriction theorem for the Fourier–Hermite transform and solution of the Hermite–Schrödinger equation, Adv. Oper. Theory 8 (2023), no. 3, Paper No. 47. 10.1007/s43036-023-00276-8Search in Google Scholar

[18] A. K. Nandakumaran and P. K. Ratnakumar, Schrödinger equation and the oscillatory semigroup for the Hermite operator, J. Funct. Anal. 224 (2005), no. 2, 371–385. 10.1016/j.jfa.2004.12.011Search in Google Scholar

[19] P. K. Ratnakumar, On Schrödinger propagator for the special Hermite operator, J. Fourier Anal. Appl. 14 (2008), no. 2, 286–300. 10.1007/s00041-008-9007-3Search in Google Scholar

[20] E. M. Stein, Oscillatory Integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis (Beijing 1984), Ann. of Math. Stud. 112, Princeton University, Princeton (1986), 307–355. 10.1515/9781400882090-007Search in Google Scholar

[21] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. 10.1215/S0012-7094-77-04430-1Search in Google Scholar

[22] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton University, Princeton, 1993. 10.1515/9780691213927Search in Google Scholar

[23] P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. 10.1090/S0002-9904-1975-13790-6Search in Google Scholar

[24] P. A. Tomas, Restriction theorems for the Fourier transform, Harmonic Analysis in Euclidean Spaces, Proc. Sympos. Pure Math. 35, American Mathematical Society, Providence (1979), 111–114. 10.1090/pspum/035.1/545245Search in Google Scholar

[25] L. Vega, Restriction theorems and the Schrödinger multiplier on the torus, Partial Differential Equations with Minimal Smoothness and Applications (Chicago 1990), IMA Vol. Math. Appl. 42, Springer, New York (1992), 199–211. 10.1007/978-1-4612-2898-1_18Search in Google Scholar

Received: 2023-04-04
Revised: 2023-05-05
Published Online: 2023-08-25
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0115/html
Scroll to top button