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 $L^{\mathrm{\infty }}(1\mathrm{/}\omega )$ are exactly the maps of the form $$(D_{\varphi }f)(t)={\int }_0^{\mathrm{\infty }}f(s)\frac{s}{t+s}\varphi (t+s)ds\text{(}t\in {\mathbb{R}}^{+}\text{ and }f\in L^1(\omega )\text{)}$$ for some $\varphi \in L^{\mathrm{\infty }}(1\mathrm{/}\omega )$. Also, every $D_{\varphi }$ has a unique extension to a continuous derivation ${\bar{D}}_{\varphi }:M(\omega )\to L^{\mathrm{\infty }}(1\mathrm{/}\omega )$ from the corresponding measure algebra. We show that a certain condition on $\varphi$ implies that ${\bar{D}}_{\varphi }$ is weak-star continuous. The condition holds for instance if $\varphi \in L_0^{\mathrm{\infty }}(1\mathrm{/}\omega )$. We also provide examples of functions $\varphi$ for which ${\bar{D}}_{\varphi }$ is not weak-star continuous. Similarly, we show that $D_{\varphi }$ and ${\bar{D}}_{\varphi }$ are compact under certain conditions on $\varphi$. For instance this holds if $\varphi \in C_0(1\mathrm{/}\omega )$ with $\varphi (0)=0$. Finally, we give various examples of functions $\varphi$ for which $D_{\varphi }$ and ${\bar{D}}_{\varphi }$ are not compact.