Let $\mu$ be a measure concentrated on a domain $ D \subset \mathbb {R}^N$ , and let $ x_0 \in D$. Denote by $ \Gamma$ the fundamental solution of the Laplacian, and by $\Gamma_{\mu}$ the Newtonian potential of $\mu$. We say that $\mu$ is a polarity measure for $D$ at $x_0$ if $\Gamma_{\mu} = \Gamma (x_0 - x)$ for every $x$ in the complementary of $D$. If we also have $\Gamma_{\mu} < \Gamma (x_0 - x)$ for every $x \in D$ then we say that $\mu$ is a strong polarity measure for $D$ at $x_0$. In the present talk we first recall the following results: A. Every sufficiently smooth domain supports a polarity measure at an arbitrarily given point. B. Every strong polarity measure characterizes its supporting domain. Then we show how to extend A and B to the general context of the hypoellipitic semi-elliptic linear second order PDEs. All the results we present have been obtained in collaboration with Giovanni Cupini.