On local integrability of fundamental solutions (Q1426893)

The regularity properties of fundamental solutions are studied. The author gives a sufficient condition (S) on a parametrix of partial differential equation with constant coefficients to get the character of hypoellipticity. In the two dimensional case, he also gives a necessary and sufficient condition for the parametrices to have the property (S) locally, and it is not satisfied by all hypoelliptic operators. On the other hand, he proves that for every hypoelliptic operator of order \(m\) in \(\mathbb R^{2}\) there is a parametrix in \(L^{p}\) with \(1\leq p<m+1,\) and every parametrix \(\;\)is in \(L_{\text{Loc}}^{P}\) when \(1\leq p<m+1.\) Finally he gives some examples of a hypoelliptic operator in \(\mathbb R^{n}\) where the parametrices are not locally integrable if \(n\geq 14.\)