C k -solvability near the characteristic set for a class of elliptic vector fields with degeneracies

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This paper deals with the solvability near the characteristic set \({\Sigma = \{0\} \times S^{1}}\) of operators of the form \({L= \partial / \partial t + (x^{n}a(x) + ixb(x))\partial / \partial x, b(0) \neq 0 \,\,{\rm and}\,\, n \geq 2, \,\,{\rm defined \,\, on}\,\, \Omega_{\epsilon} = (-\epsilon,\epsilon) \times S^{1}, \epsilon > 0,}\) where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon). \,\,{\rm For \,\, fixed}\,\, k \geq 1}\), it is shown that given f belonging to a subspace of finite codimension (depending on k) of \({C^{\infty}(\Omega_{\epsilon})}\) there is a solution \({u \in C^{k}}\) of the equation Lu = f in a neighborhood of \({\Sigma}\).