WEAK AMENABILITY OF COMMUTATIVE BEURLING ALGEBRAS

For a locally compact Abelian group G and a continuous weight function w on G we show that the Beurling algebra L-1(G, w) is weakly amenable if and only if there is no nontrivial continuous group homomorphism phi: G -> C such that sup t is an element of G vertical bar phi(t)vertical bar/w(t)w(t(-1)) < infinity. Let <(w)over cap>(t) = lim sup(s ->infinity) w(ts)/w(s) (t is an element of G). Then L-1 (G,w) is 2-weakly amenable if there is a constant m > 0 such that lim inf(n ->infinity) w(t(n))(w) over cap (t(-n))/n <= m for all t is an element of G.