Let $\Omega \in \mathbb{R}^d$ be a Lipschitz domain. Let $A$ be a linear second-order differential operator, not necessarily self-adjoint. Let $\Omega$ and the coefficients of $A$ satisfy assumptions (e.g. "Partial Differential Equations" by Evans, Section 6.3.2), such that one gets that the differential equation \begin{align*} A u &= f \text{ in } \Omega, \\ u(x) &= 0 \text{ on } \partial \Omega \end{align*} gives a solution $u \in H^2$ for any $f \in L^2$, i.e., $A^{-1}$ is a bounded linear operator from $L^2$ to $H^2$.
Consider the operator $B = A^{-1} D^\alpha$ for $\alpha$ being an order two multi-index and $D^\alpha$ being the according differential operator. Can one show that the operator $B$ is bounded from $L^2$ to $L^2$, meaning that it is bounded for a dense subset? I know that it is true for $D^{\alpha} A^{-1}$, but I am not sure about this case. If it is not true, can one find a counter-example?