Let $O\subset\mathbb{R}^d$
 be a bounded domain of the class $C^{1,1}$ 
 (or $C^2$
 for simplicity). Let the operator $A_D$
 be formally given by the differential expression $A=-\operatorname{div}g(x)\nabla$ 
 with the Dirichlet boundary condition. ($A$
 means the expression, and $A_D$
 is the corresponding operator.) Here $g(x)>0$, $g,g^{−1}\in L_\infty$. 
The precise definition is given via the quadratic form on $H^1_0(O)$. 


If the matrix of coefficients $g$
 and the boundary are smooth enough, we can define $A_D$
 by the differential expression $-\operatorname{div}g(x)\nabla$ 
 on $H^2(O)\cap H^1_0(O)=\operatorname{Ran}A_D^{-1}$, so for the expression $A$
 acting on the resolvent $A^{−1}_D$
 we have
$$\Vert A A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}=\Vert A_D A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}=1.$$

**Question:**

Is there are any hope to prove the estimate 
$$\Vert A A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \mathrm{const}$$
for the operator with non-smooth coefficients in the domain of class $C^{1,1}$
 (or $C^2$
)? (Here the expression $A$ is initially considered as operator from $H^1$
 to $H^{−1}$
.) E. g., by using of mollification of coefficients.

(I am also interested in this question for strongly elliptic systems in a divergent form: $A=b(D)^∗g(x)b(D)$
, $b=\sum _{l=1}^d b_lD_l$
, $b_l$
 are constant matrices, the symbol of $b(D)$
 has maximal rank.)

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I am trying to do the following. Write $A=D^*gD$, $D=-i\nabla$. Let $f\in H^1_0(O)$, $\phi\in L_2(O)$. We have $(AA_D^{-1}\phi ,f)_{L_2}=(g^{1/2}D A_D^{-1}\phi,g^{1/2}Df)_{L_2}$. Functions $A_D^{-1}\phi$ and $f$ are in $H^1_0(O)$ that is in the domain of the quadratic form of $A_D$. So,  we can continue our equality as  $(AA_D^{-1}\phi ,f)_{L_2}=(A_D^{1/2}A_D^{-1}\phi,A_D^{1/2}f)_{L_2}=(A_D^{-1/2}\phi,A_D^{1/2}f)_{L_2}=(\phi,f)_{L_2}$. So, $(AA_D^{-1})^*=A_D^{-1}A$ coincides with the identity operator $I$ on $H^1_0(O)\subset L_2(O)$. And I can extend it by continuity to the whole space $L_2(O)$, right?