C k -solvability near the characteristic set for a class of planar complex vector fields of infinite type

Abstract

This paper deals with semi-global C k -solvability of complex vector fields of the form $${\mathsf{L}=\partial/\partial t+x^r(a(x)+ib(x))\partial/\partial x,}$$ , r ≥ 1, defined on $${\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}$$ , $${\epsilon >0 }$$ , where a and b are C ∞ real-valued functions in $${(-\epsilon,\epsilon)}$$ . It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C k -solvability at Σ = {0} × S 1. When r = 1, it is permitted that the functions a and b of $${\mathsf L}$$ depend on the x and t variables, that is, $${\mathsf{L}=\partial/\partial t+x(a(x,t)+ib(x,t))\partial/\partial x,}$$ where $${(x, t)\in\Omega_\epsilon}$$ .