Properties of derivations on some convolution algebras

For all the convolution algebras $L^1[0,1),\ L^1_{\text{loc}}$ and $A(\omega)=\bigcap_n L^1(\omega_n)$, the derivations are of the form $D_{\mu} f=Xf*\mu$ for suitable measures $\mu$, where $(Xf)(t)=tf(t)$. We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure $\mu$. Moreover, for all these algebras we show that the extension of $D_{\mu}$ to a natural dual space is weak-star continuous.