Nearness and solvability of non-invariant equations on stratified groups

TL;DR

This paper establishes the well-posedness of certain non-invariant, possibly nonlinear second-order differential equations on stratified groups, including the Heisenberg group and Euclidean space, using a nearness approach that requires minimal regularity.

Contribution

It extends the theory of differential equations on stratified groups by handling non-invariant, nonlinear operators with bounded coefficients, broadening the scope of solvability results.

Findings

Proves well-posedness of $Au=f$ on stratified groups.

Extends results to unbounded domains in Euclidean space.

Provides explicit analysis for the Heisenberg group.

Abstract

We prove the well-posedness of the differential equation $Au=f$ in the setting of a stratified group $\mathbb{G}$ when the considered second-order differential operator $A$ can be non-invariant and non-linear. Our approach follows the Campanato theory of nearness of operators, allowing one to treat equations with only bounded coefficients, without any regularity assumptions. Our analysis becomes explicit in the particular case of the Heisenberg group $\mathbb{H}^n$ of any dimension and on the Euclidean case $\mathbb{R}^n$, where in the latter case our results also extend the known results by treating the unbounded domain setting.