Revista de la
Unión Matemática Argentina
Interior $L^p$-estimates and local $A_p$-weights
Isolda Cardoso, Pablo Viola, Beatriz Viviani
Volume 59, no. 1 (2018), pp. 73–98

DOI: https://doi.org/10.33044/revuma.v59n1a04

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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.