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

Let ω be a continuous weight on R + and let L 1 ( ω ) be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from L 1 ( ω ) to its dual space L ∞ ( 1 / ω ) are exactly the maps of the form ( D φ f ) ( t ) = ∫ 0 ∞ f ( s ) s t + s φ ( t + s ) d s ( t ∈ R +  and  f ∈ L 1 ( ω ) ) for some φ ∈ L ∞ ( 1 / ω ) . Also, every D φ has a unique extension to a continuous derivation D ¯ φ : M ( ω ) → L ∞ ( 1 / ω ) from the corresponding measure algebra. We show that a certain condition on φ implies that D ¯ φ is weak-star continuous. The condition holds for instance if φ ∈ L 0 ∞ ( 1 / ω ) . We also provide examples of functions φ for which D ¯ φ is not weak-star continuous. Similarly, we show that D φ and D ¯ φ are compact under certain conditions on φ . For instance this holds if φ ∈ C 0 ( 1 / ω ) with φ ( 0 ) = 0 . Finally, we give various examples of functions φ for which D φ and D ¯ φ are not compact.