Removable sets for homogeneous linear PDE in Carnot groups

Abstract

Let $\cL$ be a homogeneous left invariant differential operator on a Carnot group. Assume that both $\cL$ and $\cL^t$ are hypoelliptic. We study the removable sets for $\cL$-solutions. We give precise conditions in terms of the Carnot--Carath\'eodory Hausdorff dimension for the removability for $\cL$-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self similar tilings in Carnot groups.