Rellich inequalities for sub-Laplacians with drift

Ruzhansky, Michael; Yessirkegenov, Nurgissa

doi:10.1090/proc/14312

Rellich inequalities for sub-Laplacians with drift
HTML articles powered by AMS MathViewer

by Michael Ruzhansky and Nurgissa Yessirkegenov

Proc. Amer. Math. Soc. 147 (2019), 1335-1349

DOI: https://doi.org/10.1090/proc/14312

Published electronically: December 6, 2018

PDF | Request permission

Abstract:

In this note we prove horizontal weighted Rellich inequalities for the sub-Laplacian and for sub-Laplacians with drift on general stratified groups. We show how the presence of a drift improves the known inequalities. Moreover, we obtain several versions of weighted Rellich inequalities for the sub-Laplacian with drift on the polarizable Carnot groups, also with the weights associated with the fundamental solution of the sub-Laplacian. The obtained results are already new for the Laplacian in the usual Euclidean setting of ${\mathbb R}^n$, embedding the classical Rellich inequality into a family of Rellich inequalities with parameter dependent drifts.

References

Adimurthi, Massimo Grossi, and Sanjiban Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem, J. Funct. Anal. 240 (2006), no. 1, 36–83. MR 2259892, DOI 10.1016/j.jfa.2006.07.011
G. Barbatis, Best constants for higher-order Rellich inequalities in $L^p(\Omega )$, Math. Z. 255 (2007), no. 4, 877–896. MR 2274540, DOI 10.1007/s00209-006-0056-5
Alexander A. Balinsky, W. Desmond Evans, and Roger T. Lewis, The analysis and geometry of Hardy’s inequality, Universitext, Springer, Cham, 2015. MR 3408787, DOI 10.1007/978-3-319-22870-9
Zoltán M. Balogh and Jeremy T. Tyson, Polar coordinates in Carnot groups, Math. Z. 241 (2002), no. 4, 697–730. MR 1942237, DOI 10.1007/s00209-002-0441-7
G. Barbatis and A. Tertikas, On a class of Rellich inequalities, J. Comput. Appl. Math. 194 (2006), no. 1, 156–172. MR 2230976, DOI 10.1016/j.cam.2005.06.020
E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_p(\Omega )$, Math. Z. 227 (1998), no. 3, 511–523. MR 1612685, DOI 10.1007/PL00004389
D. E. Edmunds and W. D. Evans, The Rellich inequality, Rev. Mat. Complut. 29 (2016), no. 3, 511–530. MR 3538645, DOI 10.1007/s13163-016-0200-7
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
Veronique Fischer and Michael Ruzhansky, Quantization on nilpotent Lie groups, Progress in Mathematics, vol. 314, Birkhäuser/Springer, [Cham], 2016. MR 3469687, DOI 10.1007/978-3-319-29558-9
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Jerome A. Goldstein, Ismail Kombe, and Abdullah Yener, A general approach to weighted Rellich type inequalities on Carnot groups, Monatsh. Math. 186 (2018), no. 1, 49–72. MR 3788226, DOI 10.1007/s00605-017-1060-z
Waldemar Hebisch, Giancarlo Mauceri, and Stefano Meda, Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator, J. Funct. Anal. 210 (2004), no. 1, 101–124. MR 2052115, DOI 10.1016/j.jfa.2003.08.010
Waldemar Hebisch, Giancarlo Mauceri, and Stefano Meda, Spectral multipliers for sub-Laplacians with drift on Lie groups, Math. Z. 251 (2005), no. 4, 899–927. MR 2190149, DOI 10.1007/s00209-005-0839-0
Yongyang Jin and Shoufeng Shen, Weighted Hardy and Rellich inequality on Carnot groups, Arch. Math. (Basel) 96 (2011), no. 3, 263–271. MR 2784910, DOI 10.1007/s00013-011-0220-y
I. Kombe, Hardy, Rellich and Uncertainty principle inequalities on Carnot Groups, arXiv:math/0611850, (2006).
Ismail Kombe, Sharp weighted Rellich and uncertainty principle inequalities on Carnot groups, Commun. Appl. Anal. 14 (2010), no. 2, 251–271. MR 2667055
Baosheng Lian, Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B (Engl. Ed.) 33 (2013), no. 1, 59–74. MR 3003743, DOI 10.1016/S0252-9602(12)60194-5
A. Martini, A. Ottazzi, and M. Vallarino, A multiplier theorem for sub-Laplacians with drift on Lie groups, arXiv:1705.04752, (2017).
Franz Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 243–250 (German). MR 0088624
Michael Ruzhansky and Durvudkhan Suragan, On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups, J. Differential Equations 262 (2017), no. 3, 1799–1821. MR 3582213, DOI 10.1016/j.jde.2016.10.028
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
Michael Ruzhansky and Durvudkhan Suragan, Local Hardy and Rellich inequalities for sums of squares of vector fields, Adv. Differential Equations 22 (2017), no. 7-8, 505–540. MR 3646469
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, Durvudkhan Suragan, and Nurgissa Yessirkegenov, Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups, Integral Equations Operator Theory 90 (2018), no. 1, Paper No. 10, 33. MR 3769720, DOI 10.1007/s00020-018-2437-7
Bolys Sabitbek and Durvudkhan Suragan, Horizontal weighted Hardy-Rellich type inequalities on stratified Lie groups, Complex Anal. Oper. Theory 12 (2018), no. 6, 1469–1480. MR 3829602, DOI 10.1007/s11785-017-0650-z
A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math. 209 (2007), no. 2, 407–459. MR 2296305, DOI 10.1016/j.aim.2006.05.011
Qiaohua Yang, Best constants in the Hardy-Rellich type inequalities on the Heisenberg group, J. Math. Anal. Appl. 342 (2008), no. 1, 423–431. MR 2440809, DOI 10.1016/j.jmaa.2007.12.014
Yongyang Jin and Yazhou Han, Weighted Rellich inequality on H-type groups and nonisotropic Heisenberg groups, J. Inequal. Appl. , posted on (2010), Art. ID 158281, 17. MR 2671023, DOI 10.1155/2010/158281