Normalization of complex-valued planar vector fields which degenerate along a real curve

Abstract

Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.

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

“…As mentioned in [10], we have no example showing that regularity results of normalization given by Theorem 4 are optimal. 2.…”

“…It is worth mentioning that [10] proved that, for each λ ∈ C with (λ) > 0, there exists a complex vector field L, given by (4), such that the function a + ib satisfies (5), which is not equivalent to a multiple of T λ by any C ∞ -diffeomorphism. Moreover, the optimal regularity of the class of the diffeomorphism given by Theorem 4 is unknown.…”

On first and second order planar elliptic equations with degeneracies

“…When λ ∈ R, only a C 1 -diffeomorphism Φ is achieved in [7]. A generalization is obtained by Cordaro and Gong in [4] to include C k -smoothness of Φ when λ ∈ R\Q. It is also proved in [4], that, in general, a C ∞ -normalization cannot be achieved.…”

A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies

“…The C w -solvability of the model operator T λ was studied in [8] when λ ∈ C \ R, and in [5] when λ ∈ R (see also [6]). Moreover, if λ ∈ R \ Q, the following was proved in [10]. Proposition 1.1 Let L be given by (1.4) with m = 1 and irrational λ.…”