Unbounded harmonic functions on homogeneous manifolds of negative curvature

Richard Penney¹ ; Roman Urban²

¹ Department of Mathematics Purdue University West Lafayette, IN 47907, U.S.A.
² Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Colloquium Mathematicum, Tome 91 (2002) no. 1, pp. 99-121.

Résumé

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group $N$ and $A={\mathbb R}^+.$ We prove that if $F$ is harmonic and satisfies some growth condition then $F$ has an asymptotic expansion as $a\to 0$ with coefficients from ${\cal D}^\prime (N).$ Then we single out a set of at most two of these coefficients which determine $F.$ Then using asymptotic expansions we are able to prove some theorems answering partially the following question. Is a given harmonic function the Poisson integral of “something" from the boundary $N$?

Détail

DOI : 10.4064/cm91-1-8

Keywords: study unbounded harmonic functions second order differential operator homogeneous manifold negative curvature which semidirect product nilpotent lie group mathbb prove harmonic satisfies growth condition has asymptotic expansion coefficients cal prime single out set these coefficients which determine using asymptotic expansions able prove theorems answering partially following question given harmonic function poisson integral something boundary