Dynamical versions of Hardy’s uncertainty principle: A survey
HTML articles powered by AMS MathViewer
- by Aingeru Fernández-Bertolin and Eugenia Malinnikova HTML | PDF
- Bull. Amer. Math. Soc. 58 (2021), 357-375 Request permission
Abstract:
The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragmén–Lindelöf theorem. In this note we first describe the connection of the Hardy uncertainty to the Schrödinger equation, and give a new proof of Hardy’s result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been undeservedly forgotten. Then we survey the recent results on dynamical versions of Hardy’s theorem.References
- Shmuel Agmon, Unicité et convexité dans les problèmes différentiels, Séminaire de Mathématiques Supérieures, No. 13 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0252808
- Isaac Álvarez-Romero, Uncertainty principle for discrete Schrödinger evolution on graphs, Math. Scand. 123 (2018), no. 1, 51–71. MR 3843554, DOI 10.7146/math.scand.a-105369
- J. A. Barceló, L. Fanelli, S. Gutiérrez, A. Ruiz, and M. C. Vilela, Hardy uncertainty principle and unique continuation properties of covariant Schrödinger flows, J. Funct. Anal. 264 (2013), no. 10, 2386–2415. MR 3035060, DOI 10.1016/j.jfa.2013.02.017
- Arne Beurling, The collected works of Arne Beurling. Vol. 2, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989. Harmonic analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR 1057614
- Aline Bonami and Bruno Demange, A survey on uncertainty principles related to quadratic forms, Collect. Math. Vol. Extra (2006), 1–36. MR 2264204
- Aline Bonami, Bruno Demange, and Philippe Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoamericana 19 (2003), no. 1, 23–55. MR 1993414, DOI 10.4171/RMI/337
- Jean Bourgain and Carlos E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389–426. MR 2180453, DOI 10.1007/s00222-004-0435-7
- B. Cassano and L. Fanelli, Sharp Hardy uncertainty principle and Gaussian profiles of covariant Schrödinger evolutions, Trans. Amer. Math. Soc. 367 (2015), no. 3, 2213–2233. MR 3286512, DOI 10.1090/S0002-9947-2014-06383-6
- Sagun Chanillo, Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), no. 3, 325–331. MR 2352052, DOI 10.1007/s12044-007-0028-7
- M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, The Hardy uncertainty principle revisited, Indiana Univ. Math. J. 59 (2010), no. 6, 2007–2025. MR 2919746, DOI 10.1512/iumj.2010.59.4395
- Michael Cowling and John F. Price, Generalisations of Heisenberg’s inequality, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 443–449. MR 729369, DOI 10.1007/BFb0069174
- Maurice A. de Gosson, Two geometric interpretations of the multidimensional Hardy uncertainty principle, Appl. Comput. Harmon. Anal. 42 (2017), no. 1, 143–153. MR 3574565, DOI 10.1016/j.acha.2015.11.002
- B. Demange, Uncertainty principles and light cones, J. Fourier Anal. Appl. 21 (2015), no. 6, 1199–1250. MR 3421916, DOI 10.1007/s00041-015-9401-6
- Zhiwen Duan, Shuxia Han, and Peipei Sun, On unique continuation for Navier-Stokes equations, Abstr. Appl. Anal. , posted on (2015), Art. ID 597946, 16. MR 3335432, DOI 10.1155/2015/597946
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations 31 (2006), no. 10-12, 1811–1823. MR 2273975, DOI 10.1080/03605300500530446
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 883–907. MR 2443923, DOI 10.4171/JEMS/134
- Luis Escauriaza, Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 155 (2010), no. 1, 163–187. MR 2730375, DOI 10.1215/00127094-2010-053
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Uniqueness properties of solutions to Schrödinger equations, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 3, 415–442. MR 2917065, DOI 10.1090/S0273-0979-2011-01368-4
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys. 346 (2016), no. 2, 667–678. MR 3535897, DOI 10.1007/s00220-015-2500-z
- Aingeru Fernández-Bertolin, A discrete Hardy’s uncertainty principle and discrete evolutions, J. Anal. Math. 137 (2019), no. 2, 507–528. MR 3938012, DOI 10.1007/s11854-019-0002-1
- Aingeru Fernández-Bertolin, Convexity properties of discrete Schrödinger evolutions, J. Evol. Equ. 20 (2020), no. 1, 257–278. MR 4072656, DOI 10.1007/s00028-019-00524-6
- Aingeru Fernández-Bertolin and Philippe Jaming, Uniqueness for solutions of the Schrödinger equation on trees, Ann. Mat. Pura Appl. (4) 199 (2020), no. 2, 681–708. MR 4079656, DOI 10.1007/s10231-019-00896-z
- Aingeru Fernández Bertolin and Luis Vega, Uniqueness properties for discrete equations and Carleman estimates, J. Funct. Anal. 272 (2017), no. 11, 4853–4869. MR 3630642, DOI 10.1016/j.jfa.2017.03.006
- Aingeru Fernández-Bertolin and Jie Zhong, Hardy’s uncertainty principle and unique continuation property for stochastic heat equations, ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 9, 22. MR 4064470, DOI 10.1051/cocv/2019009
- A. Fernández-Bertolin, A. Grecu, and L. I. Ignat, Hardy uniqueness principle for the linear Schrödinger equation on quantum regular trees, arXiv:2005.06204 (2020).
- Xin Gao, On Beurling’s uncertainty principle, Bull. Lond. Math. Soc. 48 (2016), no. 2, 341–348. MR 3483071, DOI 10.1112/blms/bdw006
- Philippe Jaming, Yurii Lyubarskii, Eugenia Malinnikova, and Karl-Mikael Perfekt, Uniqueness for discrete Schrödinger evolutions, Rev. Mat. Iberoam. 34 (2018), no. 3, 949–966. MR 3850274, DOI 10.4171/RMI/1011
- G. H. Hardy, A Theorem Concerning Fourier Transforms, J. London Math. Soc. 8 (1933), no. 3, 227–231. MR 1574130, DOI 10.1112/jlms/s1-8.3.227
- Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780, DOI 10.1007/978-3-642-78377-7
- Haakan Hedenmalm, Heisenberg’s uncertainty principle in the sense of Beurling, J. Anal. Math. 118 (2012), no. 2, 691–702. MR 3000695, DOI 10.1007/s11854-012-0048-9
- Lars Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat. 29 (1991), no. 2, 237–240. MR 1150375, DOI 10.1007/BF02384339
- V. A. Kondrat′ev and E. M. Landis, Qualitative theory of second-order linear partial differential equations, Partial differential equations, 3 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 99–215, 220 (Russian). MR 1133457, DOI 10.1134/S0081543814050101
- A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov, The Landis conjecture on exponential decay, arXiv:2007.07034 (2020).
- Yurii Lyubarskii and Eugenia Malinnikova, Sharp uniqueness results for discrete evolutions, Non-linear partial differential equations, mathematical physics, and stochastic analysis, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018, pp. 423–436. MR 3823854
- V. Z. Meshkov, On the possible rate of decay at infinity of solutions of second order partial differential equations, Math. USSR Sbornik 72 (1992), 343–360.
- G. W. Morgan, A Note on Fourier Transforms, J. London Math. Soc. 9 (1934), no. 3, 187–192. MR 1574180, DOI 10.1112/jlms/s1-9.3.187
- F. L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra i Analiz 5 (1993), no. 4, 3–66 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 4, 663–717. MR 1246419
- Edward Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12 (1961), 995. MR 259149, DOI 10.1090/S0002-9939-1961-0259149-4
- Elmar Pauwels and Maurice de Gosson, On the prolate spheroidal wave functions and Hardy’s uncertainty principle, J. Fourier Anal. Appl. 20 (2014), no. 3, 566–576. MR 3217488, DOI 10.1007/s00041-014-9319-4
- Barry Simon, Harmonic analysis, A Comprehensive Course in Analysis, Part 3, American Mathematical Society, Providence, RI, 2015. MR 3410783, DOI 10.1090/simon/003
- A. Sitaram, M. Sundari, and S. Thangavelu, Uncertainty principles on certain Lie groups, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), no. 2, 135–151. MR 1350473, DOI 10.1007/BF02880360
- Sundaram Thangavelu, An introduction to the uncertainty principle, Progress in Mathematics, vol. 217, Birkhäuser Boston, Inc., Boston, MA, 2004. Hardy’s theorem on Lie groups; With a foreword by Gerald B. Folland. MR 2008480, DOI 10.1007/978-0-8176-8164-7
- Terence Tao, An epsilon of room, I: real analysis, Graduate Studies in Mathematics, vol. 117, American Mathematical Society, Providence, RI, 2010. Pages from year three of a mathematical blog. MR 2760403, DOI 10.1090/gsm/117
Additional Information
- Aingeru Fernández-Bertolin
- Affiliation: Universidad del País Vasco /Euskal Herriko Unibertsitatea (UPV/EHU), Dpto. de Matemáticas, Apartado 644, 48080 Bilbao, Spain
- ORCID: 0000-0002-5772-7249
- Email: aingeru.fernandez@ehu.eus
- Eugenia Malinnikova
- Affiliation: Department of Mathematics, Stanford University, Stanford, California; and Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway
- MR Author ID: 630914
- ORCID: 0000-0002-6126-1592
- Email: eugeniam@stanford.edu
- Received by editor(s): August 13, 2020
- Published electronically: June 3, 2021
- Additional Notes: The first author was partially supported by ERCEA Advanced Grant 2014 669689 - HADE, by the project PGC2018-094528-B-I00 (AEI/FEDER, UE) and acronym “IHAIP”, and by the Basque Government through the project IT1247-19.
The second author was partially supported by NSF grant DMS-1956294 and by the Research Council of Norway, project 275113. - © Copyright 2021 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 58 (2021), 357-375
- MSC (2020): Primary 42A38, 35B05
- DOI: https://doi.org/10.1090/bull/1729
- MathSciNet review: 4273105