A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

Abstract

In this article we consider parabolic systems and $L_p$ regularityof the solutions. With zero boundary condition the solutionsexperience bad regularity near the boundary. This article addressesa possible way of describing the regularity nature. Our space domainis a half space and we adapt an appropriate weight into our functionspaces. In this weighted Sobolev space setting we develop aFefferman-Stein theorem, a Hardy-Littlewood theorem and sharpfunction estimations. Using these, we prove uniqueness and existenceresults for second-order elliptic and parabolic partial differentialsystems in weighed Sobolev spaces.