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

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 $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $φ$ satisfies, in the distributional sense, a Burgers' type system $D^φφ=ω$, with a continuous $ω$. 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 $φ$ to a Burgers' type system $D^φφ=ω$, with $ω$ continuous, is actually a broad solution to $D^φφ=ω$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^φφ=ω$, with $ω$ continuous, enjoy $1/2$-little Hölder regularity along vertical directions.

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