Consider a differential operator of the form
$$Af(x) = a(x) f''(x) + b(x) f'(x) $$
for smooth, bounded functions $a,b : \mathbb R \rightarrow \mathbb R$. Let $P_t = e^{A t}$ be the associated semigroup, so that for a bounded continuous function $f : \mathbb R \rightarrow \mathbb R$, the function $P_t$ solves $\frac{d}{dt} P_t f(x) = A P_t f(x)$.
I would like to say that if $f$ is bounded and twice continuously differentiable, with bounded first and second derivatives, then $P_t f(x)$ is also twice continuously differentiable. Is there a simple proof or a reference for this fact?