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The Newtonian potential of an Euclidean ball $B$ of $\mathbb{R}^n$ centered at $x_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in $\mathbb{R}^n$, containing $x_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x_0$, then $D$ is an Euclidean ball with center $x_0$. In this paper we generalize this last result to more general measures and domains.

We construct the fundamental solutionsΓ\Gammaandγ\gammafor the non-divergence form operators∑i,jai,j(x,t)XiXj−∂t\,{\textstyle \sum _{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial _t\,and∑i,jai,j(x)XiXj{\,\textstyle \sum _{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j, where theXiX_i’s are Hörmander vector fields generating a stratified groupG\mathbb {G}and(ai,j)i,j(a_{i,j})_{i,j}is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates ofΓ\Gammaand its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates ofΓ\Gammaallow us to constructγ\gammausing bothtt-saturation and approximation arguments.

Many results of classical Potential Theory are extended to sub-Laplacians ▵𝔾 on Carnot groups 𝔾. Some characterizations of ▵𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵𝔾-subharmonic functions in ℝ N .

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T_A(x), whose differential preserves the first layer of the stratification of Lie(G).