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Solvability near the characteristic set for a class of planar vector fields of infinite type

Alberto P. Bergamasco; Abdelhamid Meziani — 2005

Annales de l’institut Fourier

We study the solvability of equations associated with a complex vector field $L$ in $ℝ^{2}$ with $C^{\infty}$ or $C^{ω}$ coefficients. We assume that $L$ is elliptic everywhere except on a simple and closed curve $Σ$ . We assume that, on $Σ$ , $L$ is of infinite type and that $L \land \bar{L}$ vanishes to a constant order. The equations considered are of the form $L u = p u + f$ , with $f$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $L \land \bar{L}$ is $> 1$ , the equation $L u = f$ is solvable in the $C^{\infty}$ category but not in the $C^{ω}$ category....

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