Partial answer. If $g$ is known to be a non-negative function, Karameta Tauberian theorem relates the behavior of $G = \int0^\cdot$ at infinity (resp. at 0) to the behavior of $\mathcal{L}g$ at $0$ (resp. at infinity). Reference: William Feller's book {\it Probability}.
In particular, $G(t) \sim 2ct^{1/2} as t \to \infty$ if and only if $\lc{g}(t) \sim \Gamma(1/2)c s^{-1/2}$ as $s \to 0$.
From $G(t) \sim 2ct^{1/2} as t \to \infty$, one needs regularity assumptions on $g$ to derive $g(t) \sim ct^{-1/2} as t \to \infty$. Assuming th