Properties of derivations on some convolution algebras

For all the convolution algebras $L^1[0,1),L_{\text{loc}}^1$ and $A(\omega )={\bigcap }_n{L}^1({\omega }_n)$, the derivations are of the form $D_{\mu} f=Xf*\mu$forsuitablemeasures$\mu$,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.