Removable singularities and Harnack inequality for nonlinear Hörmander degenerate subelliptic equations
Jiayi Qiang, Yawei Wei, Mengnan Zhang
TL;DR
The paper develops removability criteria and Harnack-type regularity for nonlinear degenerate subelliptic equations driven by general Hörmander vector fields, under weaker coefficient integrability than previously assumed. By leveraging sharp Sobolev inequalities tied to the generalized Métivier index $\tilde{v}$ and capacities cap$_s$, the authors extend prior Serrin–Meier–Capogna–Danielli–Garofalo results to broader Hörmander geometries and remove reliance on the (H) hypothesis. The main contributions are: (i) removable singularity results for weak solutions across sets of capacity zero, (ii) a Harnack inequality for nonnegative bounded weak solutions, and (iii) Hölder continuity in equiregular domains, with quantified dependence on the data. An application to higher-step Grushin-type operators demonstrates practical reach, including explicit corollaries for $P_{k'}(u)=f$ and removability at singular points, thereby broadening the scope of regularity theory in degenerate subelliptic PDEs.
Abstract
This paper concerns the quasilinear subelliptic function derived from Hörmander vector fields. Based on the significant work of J. Serrin in \cite{SER}, M. Meier in \cite{MM1}, and L. Capogna, D. Danielli and N. Garofalo in \cite{LC1,LDN}, we obtain the removable singularities and Harnack inequality by a sharp Sobolev inequalities under weaker integrability of coefficients in structure conditions. Furthermore, we get the Hölder continuity when domain $Ω$ is equiregular.