Premium
A Liouville Theorem for Matrix‐Valued Harmonic Functions on Nilpotent Groups
Author(s) -
Chu ChoHo,
Vu Tuan Giao
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303002169
Subject(s) - mathematics , nilpotent , bounded function , pure mathematics , harmonic function , degenerate energy levels , matrix (chemical analysis) , mathematical analysis , physics , quantum mechanics , materials science , composite material
Let σ be a non‐degenerate positive M n ‐valued measure on a locally compact group G with ‖σ‖ = 1. An M n ‐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.