A LIOUVILLE THEOREM FOR MATRIX-VALUED HARMONIC FUNCTIONS ON NILPOTENT GROUPS
Cho-Ho Chu, TUAN GIAO VU · Bulletin of the London Mathematical Society · 2003
Let σ be a non-degenerate positive Mn-valued measure on a locally compact group G with ‖σ‖ = 1. An Mn-valued Borel function f on G is called σ-harmonic if f ( x ) = ∫ G f ( x y − 1 ) d σ ( y ) for all x ∈ G. Given such a function f which is bounded and left uniformly continuous on G, it is shown that every central element in G is a period of f. Further, it is shown that f is constant if G is nilpotent or central. 2000 Mathematics Subject Classification 31C05, 43A05, 45E10, 46G10.