Compactness and weak-star continuity of derivations on weighted convolution algebras

Let $\omega$ be a continuous weight on $\mathbb R^+$ and let $L^1(\omega)$ be the corresponding convolution algebra. By results of Gr{\o}nb{\ae}k and Bade & Dales the continuous derivations from $L^1(\omega)$ to its dual space $L^{\infty}(1/\omega)$ are exactly the maps of the form $$(D_{\phi}f)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,\phi(t+s)\,ds\qquad\text{($t\in\mathbb R^+$ and $f\in L^1(\omega)$)}$$ for some $\phi\in L^{\infty}(1/\omega)$. Also, every $D_{\phi}$ has a unique extension to a continuous derivation $\bar{D}_{\phi}:M(\omega)\to L^{\infty}(1/\omega)$ from the corresponding measure algebra. We show that a certain condition on $\phi$ implies that $\bar{D}_{\phi}$ is weak-star continuous. The condition holds for instance if $\phi\in L_0^{\infty}(1/\omega)$. We also provide examples of functions $\phi$ for which $\bar{D}_{\phi}$ is not weak-star continuous. Similarly, we show that $D_{\phi}$ and $\bar{D}_{\phi}$ are compact under certain conditions on $\phi$. For instance this holds if $\phi\in C_0(1/\omega)$ with $\phi(0)=0$. Finally, we give various examples of functions $\phi$ for which $D_{\phi}$ and $\bar{D}_{\phi}$ are not compact.