Solvability near the characteristic set for a special class of complex vector fields

Abstract

This work deals with the solvability near the characteristic set Σ = {0} × S 1 of operators of the form $${L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}$$ , $${b\not\equiv0}$$ and a(0) ≠ 0, defined on $${\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}$$ , $${\epsilon >0 }$$ , where a and b are real-valued smooth functions in $${(-\epsilon,\epsilon)}$$ and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of $${C^\infty(\Omega_\epsilon)}$$ there is a solution $${u\in L^\infty}$$ of the equation Lu = f in a neighborhood of Σ; moreover, the L ∞ regularity is sharp.