Weak amenability of weighted group algebras

In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G,\omega )$ is weakly amenable if and only if every non-inner quasi-additive function in $L^{\mathrm{\infty }}(G,1\mathrm{/}\omega )$ is unbounded. This provides an answer to the question concerning weak amenability of $L^1(G,\omega )$ and improve some known results in connection with it.