On Kac’s principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group
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- by Michael Ruzhansky and Durvudkhan Suragan
- Proc. Amer. Math. Soc. 144 (2016), 709-721
- DOI: https://doi.org/10.1090/proc/12792
- Published electronically: June 26, 2015
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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
- 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 262881
- V. Fischer and M. Ruzhansky, Quantization on nilpotent Lie groups, to appear in Progress in Mathematics, Birkhäuser, 2015.
- G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
- 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 367477, DOI 10.1002/cpa.3160270403
- 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
- Daryl Geller, Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), no. 15, 1665–1677. MR 729197, DOI 10.1080/03605308308820319
- Daryl Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, Mathematical Notes, vol. 37, Princeton University Press, Princeton, NJ, 1990. MR 1030277, DOI 10.1515/9781400860739
- 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
- David S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I, J. Functional Analysis 43 (1981), no. 1, 97–142. MR 639800, DOI 10.1016/0022-1236(81)90040-9
- 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-Los Angeles, Calif., 1951, pp. 189–215. MR 0045333
- M. Kac, Integration in function spaces and some of its applications, Accademia Nazionale dei Lincei, Pisa, 1980. Lezioni Fermiane. [Fermi Lectures]. MR 660839
- 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, DOI 10.1134/S1064562409050032
- 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, DOI 10.1134/S0012266112040155
- 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, DOI 10.1007/978-3-0348-0069-3_{1}1
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
- 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
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
- Michael Ruzhansky and Ville Turunen, Pseudo-differential operators and symmetries, Pseudo-Differential Operators. Theory and Applications, vol. 2, Birkhäuser Verlag, Basel, 2010. Background analysis and advanced topics. MR 2567604, DOI 10.1007/978-3-7643-8514-9
- 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, DOI 10.1016/j.acha.2007.09.005
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Bibliographic Information
- Michael Ruzhansky
- Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- MR Author ID: 611131
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 709-721
- MSC (2010): Primary 35R03, 35S15
- DOI: https://doi.org/10.1090/proc/12792
- MathSciNet review: 3430847