Narrow Results

The notion of horizontal energy minimizers between C-C spaces is introduced. We prove the existence of such energy minimizers when the domain is a C², noncharacteristic bounded open set in a C-C space and the target is a C-C space of Carnot type.

This paper deals with a notion of Sobolev space W^{1, p} introduced by Bourgain, Brezis and Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by Ponce to obtain a Poincaré-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressed in terms of the intrinsic distance is equivalent to the L^p-norm of the intrinsic gradient, and provide a Poincaré-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{\text{MCP}}(K,N)$ if and only if $K\le 0$ and $N\ge k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

In the setting of Carnot groups, we prove the $q$-logarithmic Sobolev inequality for probability measures as a function of the Carnot–Carathéodory distance. As an application, we use the Hamilton–Jacobi equation in the setting of Carnot groups to prove the $p$-Talagrand inequality and hypercontractivity.