arXiv subjects
Publications and source records attributed to Marco Di Marco.
We prove that the intrinsic jump set of every locally integrable function on an equiregular Carnot-Carathéodory space is intrinsically countably rectifiable. A key new ingredient is a local scalar Whitney extension theorem for intrinsic $C^1$ functions.
By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincaré inequality for special functions with bounded variation in equiregular Carnot-Carathéodory spaces which satisfy an additional natural assumption, called property $\mathcal R$.
We study optimal quantization of probability measures on Carnot groups equipped with a left-invariant homogeneous distance. We prove two main results. First, we establish a Zador-type asymptotic formula for the quantization error: after the natural rescaling, the error converges as the number of centers tends to infinity, and the exponent is determined by the homogeneous dimension of the group. The limiting constant is a Carnot cell constant, defined through the quantization problem on a reference cell. Second, we prove weak convergence of the empirical measures associated with optimal centers, and describe the limit in terms of the density of the absolutely continuous part of the measure. The proof combines a tiling of Carnot groups by exponential cubes with a sub-Riemannian version of Pierce's lemma, which allows us to treat measures with non-compact support.
We observe that the diameter of small (in a locally uniform sense) balls in $C^{1,1}$ sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to $C^0$, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.
We introduce the space SBV$_X$ of special functions with bounded $X$-variation in Carnot-Carathéodory spaces and study its main properties. Our main outcome is an approximation result, with respect to the BV$_X$ topology, for SBV$_X$ functions.
We discuss the notion of submanifolds with boundary with intrinsic $C^1$ regularity in sub-Riemannian Heisenberg groups and we provide some examples. Eventually, we present a Stokes' Theorem for such submanifolds involving the integration of Rumin's differential forms in Heisenberg groups.
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
We introduce and study the notion of $C^1_\mathbb{H}$-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for $C^1_\mathbb{H}$-regular submanifolds with boundary that takes into account Rumin's complex of differential forms in Heisenberg groups.