This work deals with the solvability near the characteristic set Σ = {0}×S 1 of operators of the form L = ∂/∂t+(x n a(x)+ix m b(x))∂/∂x, b ≡ 0 and a(0) = 0, defined on Ω = (− , ) × S 1 , > 0, where a and b are real-valued smooth functions in (− , ) and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of C ∞ (Ω ) there is a solution u ∈ L ∞ of the equation Lu = f in a neighborhood of Σ; moreover, the L ∞ regularity is sharp. Mathematics Subject Classification (2010). Primary 35A01; Secondary 58Jxx.

This paper deals with the solvability near the characteristic set Σ = {0} × S 1 of operators of the form L = ∂/∂t + (x n a(x) + ixb(x))∂/∂x, b(0) = 0 and n ≥ 2, defined on Ω = (− , ) × S 1 , > 0, where a and b are real-valued smooth functions in (− , ). For fixed k ≥ 1, it is shown that given f belonging to a subspace of finite codimension (depending on k) of C ∞ (Ω ) there is a solution u ∈ C k of the equation Lu = f in a neighborhood of Σ. Mathematics Subject Classification. Primary 35A01; Secondary 35F05.

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