Weak amenability of l(1) (G, omega) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of l(1) (G, for two important non-commutative locally compact groups G: the free group F-2 which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of l(1) (G, omega) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for l(1) (F-2, omega) nor for l(1) ((ax + b), omega) to be weakly amenable. We prove that for several important classes of weights omega the algebra l(1) (F-2, omega) is weakly amenable if and only if the weight w is diagonally bounded. In particular, the polynomial weight omega(alpha) (x) = (1 + vertical bar x vertical bar)(alpha), where vertical bar x vertical bar denotes the length of the element x is an element of F-2 and alpha > 0, never makes l(1) (F-2, omega(alpha)) weakly amenable. We also study weak amenability of an Abelian algebra l(1) (Z(2), omega).
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