Abstract
In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈ℝn. We also apply this result to obtain the fundamental solutions for the Grushin operator in ℝ2 and the sub-Laplacian in the Heisenberg group H n .
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Beals, R.: A note on fundamental solutions. Comm. PDE, 24(1 & 2), 369–376 (1999)
Glimm, J., Jaffe, A.: Quantum physics: A functional integral point of view, 2nd edition, Spring–Verlag, Berlin–New York–Heidelberg, 1987
Berenstein, C., Chang, D. C., Tie, T.: Laguerre Calculus and Its Applications on the Heisenberg Group, AMS/IP series in advanced mathematics #22, International Press, Cambridge, Massachusetts, 2001
Chang, D. C., Greiner, P.: Analysis and geometry on Heisenberg groups. to appear Proceedings of Second International Congress of Chinese Mathematicians (C. S. Lin and S. T. Yau ed.), AMS/IP series in advanced mathematics, International Press, Cambridge, Massachusetts, 2002
Magnus, W., Oberhettinger, F., Soni, R. P.: Formulas and Theorems for the Special Functions of Mathematical Physics, Springer–Verlag, Berlin–New York–Heidelberg, 1964
Hörmander, L.: Hypoelliptic second–order differential equations. Acta Math., 119, 147–171 (1967)
Calin, O., Chang, D. C., Tie, J.: Local Nonsolvability for Gruhsin type operators, preprint, (2003)
] Andrews, G., Askey, R., Roy, R.: Special Functions, Encyclopedia of mathematics and its applications #71, Cambridge University Press, Cambridge, United Kingdom, 1999
Beals, R., Gaveau, B., Greiner, P. C.: Complex Hamiltonian mechanics and parametrics for subelliptic Laplacians, I,II,III. Bull. Sci. Math., 121, 1–36, 97–149, 195–259 (1997)
Beals, R., Greiner, P. C.: Calculus on Heisenberg manifolds, Ann. Math. Studies #119, Princeton University Press, Princeton, New Jersey, 1988
Chang, D. C., Tie, J.: Estimates for spectral projection operators of the sub–Laplacian on the Heisenberg group. J. Analyse Math., 71, 315–347 (1997)
Chang, D. C., Tie, J.: Estimates for powers of sub–Laplacian on the non–isotropic Heisenberg group. J. Geom. Anal., 10, 653–678 (2000)
Folland, G. B., Stein, E. M.: Estimates for the ¯∂b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 27, 429–522 (1974)
Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents. Acta Math., 139, 95–153 (1977)
Greiner, P. C., Stein, E. M.: Estimates for the ¯∂–Neumann problem, Math. Notes #19, Princeton University Press, Princeton, New Jersey, 1977
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Dedicated to Professor Silei Wang on the occasion of his 70th brithday
The research is partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University
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Chang, D.C., Tie, J.Z. A Note on Hermite and Subelliptic Operators. Acta Math Sinica 21, 803–818 (2005). https://doi.org/10.1007/s10114-004-0336-0
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DOI: https://doi.org/10.1007/s10114-004-0336-0