Asymptotic average solutions for second order hypoelliptic PDEs

The Newtonian potential of a merely continuous function f does not have, in general, continuous second order derivatives. As a consequence, the Poisson equation Δ ⁢ u = - f {Δ u=-f} is not, in general, solvable in the pointwise classical sense. However, a 1909 result by Pizzetti shows that the previous Poisson equation is pointwise solvable in a suitable generalized sense for every continuous function f . We show how Pizzetti’s approach works for wide classes of hypoelliptic linear second order PDEs.