$${{\varvec{W}}}^{{\varvec{1,p}}}_{\varvec{\varphi }}$$-estimates for Green’s functions of the linearized Monge–Ampère operator

Abstract

It is proved that Green’s functions associated to the linearized Monge–Ampere operator satisfy certain Sobolev-type estimates within the natural first-order calculus. Our main result extends the classical Sobolev estimates for Green’s functions due to Gruter and Widman (Manuscr Math 37(3):303–342, 1982) in the uniformly elliptic case and it addresses a question posed by Le (Manuscr Math 149:45–62, 2016) in the degenerate and/or singular Monge–Ampere setting.