Abstract

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 $n\ge 4$, there exists an operator in such a class and a distribution $u$ such that $singsuppu$ is of dimension $n$.

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