Weak amenability and 2-weak amenability of Beurling algebras

Let L1 ω(G) be a Beurling algebra on a locally compact abelian group G. We look for general conditions on the weight which allows the vanishing of continuous derivations of L1 ω(G). This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of the derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2- weakly amenable.