### Abstract:

Let G be a locally compact group, w be a continuous weight function on G, and L^1(G,w) be the corresponding Beurling algebra. In this thesis, we study weak amenability of L^1(G,w) and of its centre ZL^1(G,w) for non-commutative locally compact groups G.
We first give examples to show that the condition that characterizes weak amenability of L^1(G,w) for commutative groups G is no longer sufficient for the non-commutative case. However, we prove that this condition remains necessary for all [IN] groups G. We also provide a necessary condition for weak amenability of L^1(G,w) of a different nature, which, among other things, allows us to obtain a number of significant results on weak amenability of l^1(F_2,w) and l^1((ax+b),w).
We then study the relation between weak amenability of the algebra L^1(G,w) on a locally compact group G and the algebra L^1(G/H,^w) on the quotient group G/H of G over a closed normal subgroup H with an appropriate weight ^w induced from w. We give an example showing that L^1(G,w) may not be weakly amenable even if both L^1(G/H,^w) and L^1(H,w|_H) are weakly amenable. On the other hand, by means of constructing a generalized Bruhat function on G, we establish a sufficient condition under which weak amenability of L^1(G,w) implies that of L^1(G/H,^w). In particular, with this approach, we prove that weak amenability of the tensor product of L^1(G_1,w_1) and L^1(G_2,w_2) implies weak amenability of both Beurling algebras L^1(G_1,w_1) and L^1(G_2,w_2), provided the weights w_1, w_2 are bounded away from zero. However, given a general weight on the direct product G of G_1 and G_2, weak amenability of L^1(G,w) usually does not imply that of L^1(G_1,w|_{G_1}), even if both G_1, G_2 are commutative. We provide an example to illustrate this.
While studying the centres ZL^1(G,w) of L^1(G,w), we characterize weak amenability of ZL^1(G,w) for connected [SIN] groups G, establish a necessary condition for weak amenability of ZL^1(G,w) in the case when G is an [FC] group, and give a sufficient condition for the case when G is an [FD] group. In particular, we obtain some positive results on weak amenability of ZL^1(G,w) for a compactly generated [FC] group G with a polynomial weight w.
Finally, we briefly discuss the derivation problem for weighted group algebras and present a partial solution to it.