Interior $L^p$-estimates and local $A_p$-weights

Abstract

Let $ \Omega$ be a nonempty open proper and connected subset of $\mathbb R^ {n}$, $n \geq 3$. Consider the elliptic Schrödinger type operator $L_ {E} u= A_ {E} u+Vu= - \Sigma_{ij} a_ {ij} (x) u_ {x_i x_j} +Vu$ in $\Omega$, and the linear parabolic operator $L_ {P} u=A_ {P} u+Vu = u_ {t} - \Sigma a_ {ij} (x,t)u_ {x_{i}x_{j}} +Vu$ in $\Omega_{T} = \Omega \times (0,T)$, where the coefficients $a_ {ij} \in \mathrm{VMO}$ and the potential $V$ satisfies a reverse Hölder condition. The aim of this paper is to obtain a priori estimates for the operators $L_ {E}$ and $L_ {P}$ in weighted Sobolev spaces involving the distance to the boundary and weights in a local $A_ {p}$ class.