  In this paper we give sharp extension results for convoluted solutions of
abstract Cauchy problems in Banach spaces. The main technique is the use of
algebraic structure (for usual convolution product $\ast$) of these solutions
which are defined by a version of the Duhamel formula. We define algebra
homomorphisms from a new class of test-functions and apply our results to
concrete operators. Finally, we introduce the notion of $k$-distribution
semigroups to extend previous concepts of distribution semigroups.
