Global Orlicz regularity for sub-Laplace equations on homogeneous groups

Abstract

Let G be a homogeneous group, X 1 , X 2 , … , X p 0 be left invariant real vector fields of homogeneous of degree one and generate the Lie algebra on G . We consider the following sub-Laplace equation: − ∑ j = 1 p 0 X j 2 u ( x ) = f ( x ) , x ∈ G . By establishing the higher order derivatives estimates for the strong solutions of the above equation in Sobolev and Hölder spaces and using high order Poincaré inequalities proved by W.S. Cohn, G. Lu and P. Wang (2007) [5] , we obtain the global higher order Orlicz estimates.