Vol. 3, No. 3, 2021

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

Paolo Albano and Marco Mughetti

Vol. 3 (2021), No. 3, 473–486
DOI: 10.2140/paa.2021.3.473
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 $n$.

Keywords
sums of squares, analytic hypoellipticity, analytic singular support, analytic stratifications
Mathematical Subject Classification
Primary: 35H10, 35H20
Secondary: 35A20, 35A27, 35B65