Weak amenability of commutative Beurling algebras

Abstract

For a locally compact Abelian group $G$ and a continuous weight function $\omega$ on $G$ we show that the Beurling algebra $L^1(G, \omega )$ is weakly amenable if and only if there is no nontrivial continuous group homomorphism $\phi$: $G\to \mathbb {C}$ such that $\sup _{t\in G}\frac {|\phi (t)|}{\omega (t)\omega (t^{-1})} < \infty$. Let $\widehat \omega (t) = \limsup _{s\to \infty }\omega (ts)/\omega (s)$ ($t\in G$). Then $L^1(G, \omega )$ is $2$-weakly amenable if there is a constant $m> 0$ such that $\liminf _{n\to \infty }\frac {\omega (t^n)\widehat \omega (t^{-n})}{n} \leq m$ for all $t\in G$.