×

On the analytic singular support for the solutions of a class of degenerate elliptic operators. (English) Zbl 1484.35158

Summary: 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 \(u, \operatorname{sing}\operatorname{supp} u\), is a “negligible” set. In particular, we provide (optimal) upper estimates for the Hausdorff dimension of \(\operatorname{sing}\operatorname{supp} u\). Finally, we show that in dimension \(n\geq 4\), there exists an operator in such a class and a distribution \(u\) such that \(\operatorname{sing}\operatorname{supp} u\) is of dimension \(n\).

MSC:

35H10 Hypoelliptic equations
35H20 Subelliptic equations
35A20 Analyticity in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 10.1090/S0065-9266-2012-00663-0 · Zbl 1293.35005 · doi:10.1090/S0065-9266-2012-00663-0
[2] 10.1016/j.jfa.2018.03.009 · Zbl 1404.35112 · doi:10.1016/j.jfa.2018.03.009
[3] 10.1090/S0002-9904-1972-12955-0 · Zbl 0276.35023 · doi:10.1090/S0002-9904-1972-12955-0
[4] ; Bers, Partial differential equations, 131 (1964)
[5] 10.1007/s40627-020-00055-8 · Zbl 1462.35152 · doi:10.1007/s40627-020-00055-8
[6] 10.5802/aif.395 · Zbl 0215.45405 · doi:10.5802/aif.395
[7] 10.1007/BF02392081 · Zbl 0156.10701 · doi:10.1007/BF02392081
[8] 10.1002/cpa.3160240505 · Zbl 0226.35019 · doi:10.1002/cpa.3160240505
[9] ; Métivier, C. R. Acad. Sci. Paris Sér. I Math., 292, 401 (1981) · Zbl 0481.35033
[10] 10.14492/hokmj/1525852965 · Zbl 0531.35022 · doi:10.14492/hokmj/1525852965
[11] 10.1007/BF02414189 · Zbl 0456.35019 · doi:10.1007/BF02414189
[12] 10.1080/03605307808820074 · Zbl 0384.35055 · doi:10.1080/03605307808820074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.