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

We consider the Friedrichs extension of the operator A = A 0 + q ( x ) , defined on a bounded domain Ω in R n , n ⩾ 1 . For n = 1 , we assume that Ω = ] a , b [ . Here A 0 = A 0 ( x , D ) is an elliptic operator of order 2m with bounded smooth coefficients and q a function in L p ( Ω ) . Under some assumptions for q we obtain the uniform up to the boundary estimates for the Greenʹs function of the Friedrichs extension of the operator A + λ I , for λ sufficiently large. Under some stronger assumptions for q we give a description for the domain of the Friedrichs extension of A.