Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality

Abstract

For every integer $d\geq 3$ and real number $Q\in \left(d,\frac{d(d-1)+2}{2}\right]$, we construct a $Q$-Ahlfors regular metric measure manifold $(M,\mathbf{d},μ)$ of topological dimension $d$ and analytic dimension $2$ that satisfies a Poincaré inequality. For every $Q>\frac{d(d-1)+2}{2}$, the construction yields a metric measure manifold $(M,\mathbf{d},μ)$ that is doubling, satisfies a Poincaré inequality, and for which $M$ has $μ$-essential Hausdorff dimension $Q$. The manifold is $\mathbb{R}^d$ equipped with Lebesgue measure and a sub-Riemannian distance given by vector fields that are locally Hölder continuous but not Lipschitz continuous. These metrics give an uncountable family of new models that are genuinely distinct from the only known families of metric manifolds satisfying a Poincaré inequality: weighted and deformed Carnot groups and smooth sub-Riemannian spaces. The construction is modeled on a singular variant of a filiform Carnot group.