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.
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,
[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
[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
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Open orbits and primitive zero ideals for solvable Lie algebras
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
- Perturbation of domain for the linear parabolic equation
- K-theory of flag Bott manifolds
- Some results on Seshadri constants of vector bundles
- Strichartz inequality for orthonormal functions associated with special Hermite operator
- The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds
- On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
- Boundedness of commutators of rough Hardy operators on grand variable Herz spaces
- Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets
- An alternative proof of Tataru’s dispersive estimates
- The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions
- Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction
- Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case
- A class of quaternionic Fourier orthonormal bases
- Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields
- Normalized solutions for scalar field equation involving multiple critical nonlinearities
Articles in the same Issue
- Frontmatter
- Open orbits and primitive zero ideals for solvable Lie algebras
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
- Perturbation of domain for the linear parabolic equation
- K-theory of flag Bott manifolds
- Some results on Seshadri constants of vector bundles
- Strichartz inequality for orthonormal functions associated with special Hermite operator
- The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds
- On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
- Boundedness of commutators of rough Hardy operators on grand variable Herz spaces
- Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets
- An alternative proof of Tataru’s dispersive estimates
- The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions
- Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction
- Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case
- A class of quaternionic Fourier orthonormal bases
- Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields
- Normalized solutions for scalar field equation involving multiple critical nonlinearities