Hardy type inequalities related to Carnot-Carathéodory distance on the Heisenberg group

Yang, Qiao-Hua

doi:10.1090/S0002-9939-2012-11322-X

Hardy type inequalities related to Carnot-Carathéodory distance on the Heisenberg group
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Proc. Amer. Math. Soc. 141 (2013), 351-362 Request permission

Abstract:

Being motivated by a representation formula associated with the Korányi-Folland nonisotropic gauge proved by Cohn and Lu, we prove an analogous representation formula related to the Carnot-Carathéodory distance on the Heisenberg group. Using this formula, we obtain some Hardy inequalities associated with the Carnot-Carathéodory distance on such groups.

References

Richard Beals, Bernard Gaveau, and Peter C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl. (9) 79 (2000), no. 7, 633–689. MR 1776501, DOI 10.1016/S0021-7824(00)00169-0
William S. Cohn and Guozhen Lu, Best constants for Moser-Trudinger inequalities on the Heisenberg group, Indiana Univ. Math. J. 50 (2001), no. 4, 1567–1591. MR 1889071, DOI 10.1512/iumj.2001.50.2138
L. D’Ambrozio, Some Hardy inequalities on the Heisenberg group, Differ. Uravn. 40 (2004), no. 4, 509–521, 575 (Russian, with Russian summary); English transl., Differ. Equ. 40 (2004), no. 4, 552–564. MR 2153649, DOI 10.1023/B:DIEQ.0000035792.47401.2a
Lorenzo D’Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 725–734. MR 2019949, DOI 10.1090/S0002-9939-03-07232-0
Lorenzo D’Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 3, 451–486. MR 2185865
Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc, Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces, Sobolev spaces in mathematics. I, Int. Math. Ser. (N. Y.), vol. 8, Springer, New York, 2009, pp. 117–151. MR 2508841, DOI 10.1007/978-0-387-85648-3_{5}
Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc, Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces, Potential Anal. 34 (2011), no. 3, 223–242. MR 2782971, DOI 10.1007/s11118-010-9190-0
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
N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313–356 (English, with French summary). MR 1070830
Jerome A. Goldstein and Qi S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal. 186 (2001), no. 2, 342–359. MR 1864826, DOI 10.1006/jfan.2001.3792
Roberto Monti, Some properties of Carnot-Carathéodory balls in the Heisenberg group, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 3, 155–167 (2001) (English, with English and Italian summaries). MR 1841689
Pengcheng Niu, Huiqing Zhang, and Yong Wang, Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3623–3630. MR 1860496, DOI 10.1090/S0002-9939-01-06011-7
Simone Secchi, Didier Smets, and Michel Willem, Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris 336 (2003), no. 10, 811–815 (English, with English and French summaries). MR 1990020, DOI 10.1016/S1631-073X(03)00202-4
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