# Laplace Transform: Are there theorems similar to the Bernstein Theorem?

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

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Actually, Bernstein's theorem says that such a function is the Laplace transform of a finite measure on $(0,\infty)$, not necessarily of a function. For example, $f(x) = e^{-x}$ is the Laplace transform of a point mass. –  Mark Meckes Dec 1 '11 at 16:24
You are right of course. Sorry for this impreciseness. My question, however, remains the same. –  florian Dec 1 '11 at 17:49