Interior Lp-estimates and local Ap-weights

Abstract

Let Ω be a nonempty open proper and connected subset of Rn, n≥3. Consider the elliptic Schrödinger type operator LEu=AEu+Vu=−Σijaij(x)uxixj+Vu in Ω, and the linear parabolic operator LPu=APu+Vu= ut−Σaij(x,t)uxixj+Vu in ΩT=Ω×(0,T), where the coefficients aij∈VMO and the potential V satisfies a reverse Hölder condition. The aim of this paper is to obtain a priori estimates for the operators LE and LP in weighted Sobolev spaces involving the distance to the boundary and weights in a local Ap class.