Let $(X_t)$ be a subordinator (not killed). Since $(X_t)$ is a non-decreasing Levy process, we have the corresponding infinitesimal generator: \begin{equation} Af(x)=\delta f'(x)+\int_{0}^{\infty}(f(x+y)-f(x))\prod(dy),\;\;\;(1) \end{equation} where $\delta\ge0$ is the drift constant and $\prod$ is the jump measure.
Because I got this by applying the general form of infinitesimal generator for Levy processes, the domain is $C^{2}_{\infty}(\mathbb{R})$, that is, the function $f$ should satisfy $\lim_{|x|\to\infty}f(x)=0$.
Is it possible that (1) holds under the only conditions that $f\ge0$, $f$ is continuously differentiable and $f\in L^{p}(\mathbb{R})$ for $p\ge1$ (without vanishing at infinity)? Any help, advice or reference would be of great help. Thank you very much.