Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

Abstract

We prove that in arbitrary Carnot groups $\mathbb{G}$ of step 2, with a splitting $\mathbb{G}=\mathbb{W}\cdot \mathbb{L}$ with $\mathbb{L}$ one-dimensional, the graph of a continuous function $\phi :U\subseteq \mathbb{W}\to \mathbb{L}$ is $C_{\mathrm{H}}^1$-regular precisely when $\phi$ satisfies, in the distributional sense, a Burgers' type system $D^{\phi }\phi =\omega$, with a continuous $\omega$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $\phi$ to a Burgers' type system $D^{\phi }\phi =\omega$, with $\omega$ continuous, is actually a broad solution to $D^{\phi }\phi =\omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{\phi }\phi =\omega$, with $\omega$ continuous, enjoy $1/2$-little Hölder regularity along vertical directions.