Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincar\'e 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 $(M,\mathbf{d},\mu)$ of topological dimension $d$ and analytic dimension $2$ that satisfies a Poincar\'e inequality. For every $Q>\frac{d(d-1)+2}{2}$, the construction yields a metric measure manifold $(M,\mathbf{d},\mu)$ that is doubling, satisfies a Poincar\'e inequality, and for which $M$ has $\mu$-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 H\"older 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 Poincar\'e inequality: weighted and deformed Carnot groups and smooth sub-Riemannian spaces. The construction is modeled on a singular variant of a filiform Carnot group.