A Liouville-type theorem on halfspaces for the Kohn Laplacian
HTML articles powered by AMS MathViewer
- by Francesco Uguzzoni PDF
- Proc. Amer. Math. Soc. 127 (1999), 117-123 Request permission
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 \[ u\in L^{1}(\Omega ),\] then $u\equiv 0$ in $\Omega$.References
- I. Birindelli, I. Capuzzo Dolcetta, A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. Henry Poincaré - Analyse non Linéaire 14 (1997), 295–308.
- Jacek Cygan, Wiener’s test for the Brownian motion on the Heisenberg group, Colloq. Math. 39 (1978), no. 2, 367–373. MR 522380, DOI 10.4064/cm-39-2-367-373
- G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
- 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 367477, DOI 10.1002/cpa.3160270403
- Bernard Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95–153. MR 461589, DOI 10.1007/BF02392235
- Daryl Geller, Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), no. 15, 1665–1677. MR 729197, DOI 10.1080/03605308308820319
- 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
- Adam Korányi and Nancy K. Stanton, Liouville-type theorems for some complex hypoelliptic operators, J. Funct. Anal. 60 (1985), no. 3, 370–377. MR 780503, DOI 10.1016/0022-1236(85)90045-X
Additional Information
- Francesco Uguzzoni
- Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
- Email: uguzzoni@dm.unibo.it
- Received by editor(s): February 18, 1997
- Received by editor(s) in revised form: April 30, 1997
- Communicated by: Jeffrey B. Rauch
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 117-123
- MSC (1991): Primary 31C05, 31B05, 35J15
- DOI: https://doi.org/10.1090/S0002-9939-99-04519-0
- MathSciNet review: 1458268