Removable sets for homogeneous linear PDE in Carnot groups

TL;DR

This paper investigates the conditions under which sets can be removed without affecting solutions to certain homogeneous linear PDEs in Carnot groups, using Hausdorff dimension and novel tilings.

Contribution

It provides precise criteria based on Carnot--Carathéodory Hausdorff dimension for removability of sets for $ ext{L}$-solutions, including sharp conditions in some cases.

Findings

Criteria for removability depend on Hausdorff dimension.

Sharp conditions established for certain regularity hypotheses.

Introduction of novel local self-similar tilings 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.