Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Kac's principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group


Authors: Michael Ruzhansky and Durvudkhan Suragan
Journal: Proc. Amer. Math. Soc. 144 (2016), 709-721
MSC (2010): Primary 35R03, 35S15
DOI: https://doi.org/10.1090/proc/12792
Published electronically: June 26, 2015
MathSciNet review: 3430847
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we construct an integral boundary condition for the Kohn Laplacian in a given domain on the Heisenberg group extending to the setting of the Heisenberg group M. Kac's ``principle of not feeling the boundary''. This also amounts to finding the trace on smooth surfaces of the Newton potential associated to the Kohn Laplacian. We also obtain similar results for higher powers of the Kohn Laplacian.


References [Enhancements On Off] (What's this?)

  • [1] Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277-304 xii (French, with English summary). MR 0262881 (41 #7486)
  • [2] V. Fischer and M. Ruzhansky, Quantization on nilpotent Lie groups, to appear in Progress in Mathematics, Birkhäuser, 2015.
  • [3] G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. MR 0461588 (57 #1573)
  • [4] G. B. Folland and E. M. Stein, Estimates for the $ \bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. MR 0367477 (51 #3719)
  • [5] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161-207. MR 0494315 (58 #13215)
  • [6] Daryl Geller, Liouville's theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), no. 15, 1665-1677. MR 729197 (85f:58109), https://doi.org/10.1080/03605308308820319
  • [7] Daryl Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, Mathematical Notes, vol. 37, Princeton University Press, Princeton, NJ, 1990. MR 1030277 (91d:58243)
  • [8] P. C. Greiner and E. M. Stein, Estimates for the $ \overline \partial $-Neumann problem, Mathematical Notes, No. 19, Princeton University Press, Princeton, N.J., 1977. MR 0499319 (58 #17218)
  • [9] David S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I, J. Funct. Anal. 43 (1981), no. 1, 97-142. MR 639800 (83c:58081a), https://doi.org/10.1016/0022-1236(81)90040-9
  • [10] M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 189-215. MR 0045333 (13,568b)
  • [11] M. Kac, Integration in function spaces and some of its applications, Accademia Nazionale dei Lincei, Pisa, 1980. Lezioni Fermiane. [Fermi Lectures]. MR 660839 (83g:60096)
  • [12] T. Sh. Kal'menov and D. Suragan, On spectral problems for the volume potential, Dokl. Akad. Nauk 428 (2009), no. 1, 16-19 (Russian); English transl., Dokl. Math. 80 (2009), no. 2, 646-649. MR 2596645 (2011b:31009), https://doi.org/10.1134/S1064562409050032
  • [13] T. Sh. Kal'menov and D. Suragan, Boundary conditions for the volume potential for the polyharmonic equation, Differ. Equ. 48 (2012), no. 4, 604-608. Translation of Differ. Uravn. 48 (2012), no. 4, 595-599. MR 3177192, https://doi.org/10.1134/S0012266112040155
  • [14] T. Sh. Kalmenov and D. Suragan, A boundary condition and spectral problems for the Newton potential, Modern aspects of the theory of partial differential equations, Oper. Theory Adv. Appl., vol. 216, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 187-210. MR 2848241 (2012j:35056), https://doi.org/10.1007/978-3-0348-0069-3_11
  • [15] William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312 (2001a:35051)
  • [16] Cristina Romero, Potential theory for the Kohn Laplacian on the Heisenberg group, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)-University of Minnesota. MR 2686874
  • [17] Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247-320. MR 0436223 (55 #9171)
  • [18] Michael Ruzhansky and Ville Turunen, Pseudo-differential operators and symmetries, Background analysis and advanced topics, Pseudo-Differential Operators. Theory and Applications, vol. 2, Birkhäuser Verlag, Basel, 2010. MR 2567604 (2011b:35003)
  • [19] Naoki Saito, Data analysis and representation on a general domain using eigenfunctions of Laplacian, Appl. Comput. Harmon. Anal. 25 (2008), no. 1, 68-97. MR 2419705 (2009d:35038), https://doi.org/10.1016/j.acha.2007.09.005
  • [20] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35R03, 35S15

Retrieve articles in all journals with MSC (2010): 35R03, 35S15


Additional Information

Michael Ruzhansky
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: m.ruzhansky@imperial.ac.uk

Durvudkhan Suragan
Affiliation: Institute of Mathematics and Mathematical Modelling, 125 Pushkin Str., 050010 Almaty, Kazakhstan
Email: suragan@math.kz

DOI: https://doi.org/10.1090/proc/12792
Keywords: Sub-Laplacian, Kohn Laplacian, integral boundary conditions, Heisenberg group, Newton potential
Received by editor(s): January 26, 2015
Published electronically: June 26, 2015
Additional Notes: The authors were supported in part by EPSRC grant EP/K039407/1 and by the Leverhulme grant RPG-2014-02, as well as by MESRK grant 5127/GF4.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society