Sharp uncertainty principles on general Finsler manifolds

Huang, Libing; Kristály, Alexandru; Zhao, Wei

doi:10.1090/tran/8178

Sharp uncertainty principles on general Finsler manifolds
HTML articles powered by AMS MathViewer

by Libing Huang, Alexandru Kristály and Wei Zhao PDF

Trans. Amer. Math. Soc. 373 (2020), 8127-8161 Request permission

Abstract:

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg, and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non- Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

References

Adimurthi, Nirmalendu Chaudhuri, and Mythily Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2002), no. 2, 489–505. MR 1862130, DOI 10.1090/S0002-9939-01-06132-9
J. C. Álvarez Paiva and G. Berck, What is wrong with the Hausdorff measure in Finsler spaces, Adv. Math. 204 (2006), no. 2, 647–663. MR 2249627, DOI 10.1016/j.aim.2005.06.007
J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1–48. MR 2132656, DOI 10.4171/prims/123
D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675, DOI 10.1007/978-1-4612-1268-3
David Bao, Colleen Robles, and Zhongmin Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004), no. 3, 377–435. MR 2106471
G. Barbatis, S. Filippas, and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169–2196. MR 2048514, DOI 10.1090/S0002-9947-03-03389-0
Elvise Berchio, Debdip Ganguly, and Gabriele Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, J. Funct. Anal. 272 (2017), no. 4, 1661–1703. MR 3590248, DOI 10.1016/j.jfa.2016.11.018
L. Berwald, Parallelübertragung in allgemeinen Räumen, Atti Congr. Intern. Mat. Bologna 4(1928), 262-270.
Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), no. 10, 883–891 (French, with English and French summaries). MR 1489943, DOI 10.1016/S0021-7824(97)89976-X
L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
Xinyue Cheng and Zhongmin Shen, A class of Finsler metrics with isotropic $S$-curvature, Israel J. Math. 169 (2009), 317–340. MR 2460908, DOI 10.1007/s11856-009-0013-1
Shiing-Shen Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc. 43 (1996), no. 9, 959–963. MR 1400859
Manfredo Perdigão do Carmo and Changyu Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818–826. MR 2041783, DOI 10.1112/S0010437X03000745
Lorenzo D’Ambrosio and Serena Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 3, 449–475. MR 3208450, DOI 10.1016/j.anihpc.2013.04.004
Wolfgang Erb, Uncertainty principles on compact Riemannian manifolds, Appl. Comput. Harmon. Anal. 29 (2010), no. 2, 182–197. MR 2652457, DOI 10.1016/j.acha.2009.08.012
Csaba Farkas, Alexandru Kristály, and Csaba Varga, Singular Poisson equations on Finsler-Hadamard manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1219–1241. MR 3396410, DOI 10.1007/s00526-015-0823-4
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
Stathis Filippas and Achilles Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002), no. 1, 186–233. MR 1918494, DOI 10.1006/jfan.2001.3900
Patrick Foulon and Vladimir S. Matveev, Zermelo deformation of Finsler metrics by Killing vector fields, Electron. Res. Announc. Math. Sci. 25 (2018), 1–7. MR 3808237, DOI 10.3934/era.2018.25.001
R. L. Frank, Sobolev inequalities and uncertainty principles in mathematical physics, Part 1, Link: http://www.math.caltech.edu/~rlfrank/sobweb1.pdf. 1-35.
Nassif Ghoussoub and Amir Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA 105 (2008), no. 37, 13746–13751. MR 2443723, DOI 10.1073/pnas.0803703105
Nassif Ghoussoub and Amir Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann. 349 (2011), no. 1, 1–57. MR 2753796, DOI 10.1007/s00208-010-0510-x
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
Libing Huang and Xiaohuan Mo, On geodesics of Finsler metrics via navigation problem, Proc. Amer. Math. Soc. 139 (2011), no. 8, 3015–3024. MR 2801641, DOI 10.1090/S0002-9939-2011-10726-3
Libing Huang and Qiong Xue, Affine vector fields on Finsler manifolds, Manuscripta Math. 161 (2020), no. 3-4, 293–302. MR 4060482, DOI 10.1007/s00229-018-1093-1
Alexandru Kristály, Sharp uncertainty principles on Riemannian manifolds: the influence of curvature, J. Math. Pures Appl. (9) 119 (2018), 326–346 (English, with English and French summaries). MR 3862150, DOI 10.1016/j.matpur.2017.09.002
Alexandru Kristály, A sharp Sobolev interpolation inequality on Finsler manifolds, J. Geom. Anal. 25 (2015), no. 4, 2226–2240. MR 3427122, DOI 10.1007/s12220-014-9510-5
A. Kristály, Sharp uncertainty principles on Finsler manifolds: the effect of curvature. arXiv:1311.6418v2.
Alexandru Kristály and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, Math. Ann. 357 (2013), no. 2, 711–726. MR 3096522, DOI 10.1007/s00208-013-0918-1
Alexandru Kristály and Dušan Repovš, Quantitative Rellich inequalities on Finsler-Hadamard manifolds, Commun. Contemp. Math. 18 (2016), no. 6, 1650020, 17. MR 3547105, DOI 10.1142/S0219199716500206
Alexandru Kristály and Imre J. Rudas, Elliptic problems on the ball endowed with Funk-type metrics, Nonlinear Anal. 119 (2015), 199–208. MR 3334184, DOI 10.1016/j.na.2014.09.015
Ismail Kombe and Murad Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203. MR 2538592, DOI 10.1090/S0002-9947-09-04642-X
Ismail Kombe and Murad Özaydin, Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035–5050. MR 3074365, DOI 10.1090/S0002-9947-2013-05763-7
Jean Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, NUMDAM, [place of publication not identified], 1933 (French). MR 3533002
Elliott H. Lieb, The stability of matter: from atoms to stars, 4th ed., Springer, Berlin, 2005. Selecta of Elliott H. Lieb; Edited by W. Thirring, and with a preface by F. Dyson. MR 2766495, DOI 10.1007/b138553
Makoto Matsumoto, A slope of a mountain is a Finsler surface with respect to a time measure, J. Math. Kyoto Univ. 29 (1989), no. 1, 17–25. MR 988060, DOI 10.1215/kjm/1250520303
Xiaohuan Mo and Libing Hang, On curvature decreasing property of a class of navigation problems, Publ. Math. Debrecen 71 (2007), no. 1-2, 141–163. MR 2340039
Shin-ichi Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 211–249. MR 2546027, DOI 10.1007/s00526-009-0227-4
Shin-Ichi Ohta and Karl-Theodor Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), no. 10, 1386–1433. MR 2547978, DOI 10.1002/cpa.20273
B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
Hans-Bert Rademacher, Nonreversible Finsler metrics of positive flag curvature, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 261–302. MR 2132661
Michael Ruzhansky and Durvudkhan Suragan, Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math. 317 (2017), 799–822. MR 3682685, DOI 10.1016/j.aim.2017.07.020
Michael Ruzhansky and Durvudkhan Suragan, Uncertainty relations on nilpotent Lie groups, Proc. A. 473 (2017), no. 2201, 20170082, 12. MR 3668138, DOI 10.1098/rspa.2017.0082
Michael Ruzhansky and Durvudkhan Suragan, Layer potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups, Adv. Math. 308 (2017), 483–528. MR 3600064, DOI 10.1016/j.aim.2016.12.013
Bin Shen, Strongly and weakly affine vector fields on Finsler manifolds, Differential Geom. Appl. 62 (2019), 190–211. MR 3881649, DOI 10.1016/j.difgeo.2018.11.006
Zhongmin Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128 (1997), no. 2, 306–328. MR 1454401, DOI 10.1006/aima.1997.1630
Zhongmin Shen, Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001. MR 1967666, DOI 10.1007/978-94-015-9727-2
Zhongmin Shen, Finsler metrics with $\mathbf K=0$ and $\mathbf S=0$, Canad. J. Math. 55 (2003), no. 1, 112–132. MR 1952328, DOI 10.4153/CJM-2003-005-6
Zhongmin Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. MR 1845637, DOI 10.1142/9789812811622
Zhi-Qiang Wang and Michel Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203 (2003), no. 2, 550–568. MR 2003359, DOI 10.1016/S0022-1236(03)00017-X
Changyu Xia, The Caffarelli-Kohn-Nirenberg inequalities on complete manifolds, Math. Res. Lett. 14 (2007), no. 5, 875–885. MR 2350131, DOI 10.4310/MRL.2007.v14.n5.a14
Qiaohua Yang, Dan Su, and Yinying Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), no. 2, 1350043, 24. MR 3195155, DOI 10.1142/S0219199713500430
Lixia Yuan, Wei Zhao, and Yibing Shen, Improved Hardy and Rellich inequalities on nonreversible Finsler manifolds, J. Math. Anal. Appl. 458 (2018), no. 2, 1512–1545. MR 3724742, DOI 10.1016/j.jmaa.2017.10.036
Wei Zhao and Yibing Shen, A universal volume comparison theorem for Finsler manifolds and related results, Canad. J. Math. 65 (2013), no. 6, 1401–1435. MR 3121676, DOI 10.4153/CJM-2012-053-4