Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

We prove that in arbitrary Carnot groups G of step 2, with a splitting G = W · L with L one-dimensional, the intrinsic graph of a continuous function ϕ : U ⊆ W → L is C1H -regular precisely when ϕ satisfies, in the distributional sense, a Burgers’ type system D ϕ ϕ = 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. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution ϕ to a Burgers’ type system D ϕ ϕ = ω, with ω continuous, is actually a broad solution to D ϕ ϕ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to D ϕ ϕ = ω, with ω continuous, enjoy 1/2-little H¨older regularity along vertical directions.