A domain description and Green's function estimates up to the boundary for elliptic operator with singular potential

We consider the Friedrichs extension of the operator A=A0+q(x), defined on a bounded domain Ω in Rn, n⩾1. For n=1, we assume that Ω=]a,b[. Here A0=A0(x,D) is an elliptic operator of order 2m with bounded smooth coefficients and q a function in Lp(Ω). Under some assumptions for q we obtain the uniform up to the boundary estimates for the Green's function of the Friedrichs extension of the operator A+λI, for λ sufficiently large. Under some stronger assumptions for q we give a description for the domain of the Friedrichs extension of A.