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)



A Liouville-type theorem on halfspaces
for the Kohn Laplacian

Author: Francesco Uguzzoni
Journal: Proc. Amer. Math. Soc. 127 (1999), 117-123
MSC (1991): Primary 31C05, 31B05, 35J15
MathSciNet review: 1458268
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Delta _{ \mathbb{H}^{n}}$ be the Kohn Laplacian on the Heisenberg group $ \mathbb{H}^{n}$ and let $\Omega$ be a halfspace of $ \mathbb{H}^{n}$ whose boundary is parallel to the center of $ \mathbb{H}^{n}$. In this paper we prove that if $u$ is a non-negative $\Delta _{ \mathbb{H}^{n}}$-superharmonic function such that

\begin{displaymath}u\in L^{1}(\Omega),\end{displaymath}

then $u\equiv 0$ in $\Omega$.

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

  • [BCDC] I. Birindelli, I. Capuzzo Dolcetta, A. Cutrì, , Ann. Inst. Henry Poincaré - Analyse non Linéaire 14 (1997), 295-308. CMP 97:13
  • [C] J. Cygan, Wiener's test for the Brownian motion on the Heisenberg group, Colloquium Math. 39 (1978), 367-373. MR 82j:60153
  • [F] G.B. Folland, , Bull. Amer. Math. Soc. 79 (1973), 373-376. MR 47:3816
  • [FS] G.B. Folland, E.M. Stein, , Comm. Pure Appl. Math. 27 (1974), 429-522. MR 51:3719
  • [G] B. Gaveau, , Acta Math. 139 (1977), 95-153. MR 57:1574
  • [Ge] D. Geller, , Comm. P.D.E. 8 (1983), 1665-1677. MR 85f:58109
  • [GL] N. Garofalo, E. Lanconelli, , Ann. Inst. Fourier Grenoble 40 (1990), 313-356. MR 91i:22014
  • [KS] A. Korányi, N.K. Stanton, , J. Funct. Anal. 60 (1985), 370-377. MR 86i:22016

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 31C05, 31B05, 35J15

Retrieve articles in all journals with MSC (1991): 31C05, 31B05, 35J15

Additional Information

Francesco Uguzzoni
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy

Keywords: Heisenberg group, Kohn Laplacian, superharmonic functions, halfspaces
Received by editor(s): February 18, 1997
Received by editor(s) in revised form: April 30, 1997
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society