On the analytic singular support for the solutions of a class of degenerate elliptic operators

We study a class of degenerate elliptic operators (which is a slight extension of the sums of squares of real-analytic vector fields satisfying the Hörmander condition). We show that, in dimensions 2 and 3, for every operator L in such a class and for every distribution u such that Lu is real-analytic, the analytic singular support of u, singsuppu, is a “negligible” set. In particular, we provide (optimal) upper estimates for the Hausdorff dimension of singsuppu. Finally, we show that in dimension n≥4, there exists an operator in such a class and a distribution u such that singsuppu is of dimension n.