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

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Keywords
sums of squares, analytic hypoellipticity, analytic singular support, analytic stratifications
Mathematical Subject Classification
Primary: 35H10, 35H20
Secondary: 35A20, 35A27, 35B65
Milestones
Received: 17 April 2020
Revised: 8 May 2021
Accepted: 21 June 2021
Published: 12 September 2021
Authors
Paolo Albano
Dipartimento di Matematica
Università di Bologna
Bologna
Italy
Marco Mughetti
Dipartimento di Matematica
Università di Bologna
Bologna
Italy