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We prove that in arbitrary Carnot groups $mathbb G$ of step 2, with a splitting $mathbb G=mathbb Wcdotmathbb L$ with $mathbb L$ one-dimensional, the graph of a continuous function $varphicolon Usubseteq mathbb Wto mathbb L$ is $C^1_{mathrm{H}}$-regular precisely when $varphi$ satisfies, in the distributional sense, a Burgers type system $D^{varphi}varphi=omega$, 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 $varphi$ to a Burgers type system $D^{varphi}varphi=omega$, with $omega$ continuous, is actually a broad solution to $D^{varphi}varphi=omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{varphi}varphi=omega$, with $omega$ continuous, enjoy $1/2$-little Holder regularity along vertical directions.
In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differen
In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-$n$ Car
We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space,
We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in Euclidean spaces,