Global properties of a class of planar vector fields of infinite type

A Rudin–Carleson theorem for planar vector fields

“…The following example shows that if L has a compact orbit, L may not have any non-constant solution on D. More examples can be found in [5].…”

“…It was shown in [5] (Theorem 1.1, and part (3) of Remark 1.1) that if f is a continuous solution of L in a neighborhood of any annulus of the form A δ = {z ∈ C : r 0 − δ < |z| < r 0 + δ}, then it is constant. It follows that if u ∈ C(D) and Lu = 0 in the unit disc D, then u is constant on D since L is elliptic away from .…”

Nonexistence of global solutions for a class of complex vector fields on two-torus

“…It was proved in [4] that there no is solution u ∈ C ∞ for the equation Lu = f in any neighborhood of Σ (for related questions see also [7]). …”

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

“…For instance, it was shown in [8] that if a + ib = ix r , r 2, then L is solvable in C ∞ but there exists f ∈ G 1 (Ω ), satisfying the compatibility conditions, such that the equation Lu = f does not have any solution u ∈ G 1 (Ω ); moreover, it follows from our Example 2.3 at the end of Section 2 that there may be no solution in G s , if 1 s < r r−1 . Further references dealing with related questions about the existence and regularity of global and semiglobal solutions are [1][2][3]7,9,10,[12][13][14][15][16]25,30].…”