Left $\theta$-derivations on weighted convolution algebras

Let $\theta$ be a homomorphism on $L_0^\infty({\Bbb R}^+, \omega)^*$. In this paper, we study left $\theta$-derivations on $L_0^\infty({\Bbb R}^+, \omega)^*$. We show that every left $\theta$-derivation on $L_0^\infty({\Bbb R}^+, \omega)^*$ is always a $\theta$-derivation, and if $\theta$ is isomorphism, then $L_0^\infty({\Bbb R}^+, \omega)^*$ has no non-zero left $\theta$-derivation. We also investigate automatic continuity, Singer-Wermer's conjecture and Posner's first theorem for left $\theta-$derivations on $L_0^\infty({\Bbb R}^+, \omega)^*$.