Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b)

Are there any theorems of the same flavor, which characterize the inverse Laplace transform of functions, which are locally bounded or locally $L^p$?

More precisely, for \begin{equation} f(x)=\int_0^\infty dt \; e^{-xt} \; g(t) \; , \end{equation} what do I have to prove for $f$ to be able to conclude, that $g \in L_{\textrm{loc}}^p$, for some $1\le p < \infty$?