Regularity for Subelliptic PDE Through Uniform Estimates in Multi-Scale Geometries

TL;DR

This paper reviews and extends recent results on the stability of geometric and analytic estimates in subRiemannian structures, generalizing from Carnot groups to H"ormander vector fields using multi-scale geometric analysis.

Contribution

It extends stability results of doubling, Poincaré, and heat kernel estimates from Carnot groups to general H"ormander vector fields in subelliptic PDEs.

Findings

Stability of doubling properties in general subRiemannian structures

Extension of Poincaré inequalities to H"ormander vector fields

Gaussian heat kernel estimates in multi-scale geometries

Abstract

We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea [19] and Manfredini [17] concerning stability of doubling properties, Poincar\'e inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of H\"ormander vector fields.