Estimates of solutions of elliptic differential-difference equations with degeneration

We consider a second-order differential-difference equation in a bounded domain $Q\subset\mathbb R^n$. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the Gårding inequality.
We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well.
It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space $W^1_2(Q)$. However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain $Q$, but inside some subdomains $Q_r$ generated by the shifts of the boundary, where $\bigcup_r\overline{Q_r}=\overline Q$.