Abstract Cauchy problems of the form $$ u' =Au, \qquad u(0)=x, $$ often may have solutions which are not exponentially bounded or even not defined on the whole half-line $[0,\infty)$. In such cases, the powerful tool of the Laplace transform is not available to link the evolution operator and the generator $A$. Several, to some extent equivalent, approaches to such problems have been developed, ranging from distribution and ultra-distribution semigroups, through locally integrated semigroups, to local convoluted semigroups. This paper deals with local convoluted semigroups. By definition, a closed linear operator on a Banach space $X$ is the generator of a local $F$-convoluted semigroup on $[0,T)$, if the Cauchy problem $$ u'(t) =Au(t)+F^{[1]}(t)x, \quad 0<t<T,\quad u(0)=0, $$ where $F^{[1]}(t) =\int_0^tF(s)ds$, has a unique classical solution for any $x \in X$. The family $(S(t))_{t\geq 0}$ defined by $S(t)x =u'(t)$ is called the local $F$-convoluted semigroup generated by $A$. This definition yields $C_0$-semigroups for the choice $F^{[1]}(t)=1$ for all $t$, whereas, e.g., distribution semigroups correspond to $F^{[1]}(t) = t^\alpha/\Gamma(\alpha+1), \alpha\geq 0$.
The main result of this paper is a Hille-Yosida type theorem characterising generators of local convoluted semigroups. Without going into details, it states that under certain technical assumptions, $(S(t))_{t\geq 0}$ is the local $F$-convoluted semigroup on $[0,T)$ generated by $A$ if and only if for some $\omega\in \Bbb{R}$, $(\omega,\infty)\subset \rho(A)$ and $(\omega,\infty)\ni \lambda \to \int_0^\tau e^{-\lambda t}dF(t)\cdot R(\lambda,A)$ is the local Laplace-Stieltjes transform on $[0,\tau]$, $\tau<T$, of $(S(t))_{t\geq 0}$.
The paper is concluded by specializing this result to the exponentially bounded case.