Properties of solutions of a class of hypocomplex vector fields

Abstract

A Cauchy type integral operator is associated to a class of integrable vector fields with complex coefficients. Properties of the integral operator are used to deduce Hölder solvability of semilinear equations Lu = F (x, y, u) and a strong similarity principle between the solutions of the equation Lu = au + bu and those of the equation Lu = 0.

Class of hypocomplex structures on the two-dimensional torus

“…This paper deals with the solvability of a class of complex vector fields on the two-dimensional torus T 2 . The main results are generalizations of those contained in the recent papers [5] and [6] where the focus was on solvability in domains of the plane R 2 . Study of complex vector fields in T 2 , or on compact manifolds, was considered in many papers (see for instance [2], [3], [4], [7], [8], [11]) under the assumption of separation of variables: the coefficients of the induced equations are independent on certain variables.…”

“…In this section we give a necessary and sufficient condition for the solvability of the equation y)) . It follows from [5] and property (3.3) (3.7). We restrict the action of P k to the subspace C(R).…”

“…As in [5], we can assume that there exist local coordinates (s, t) near points p ∈ Σ i in which the differential form ω is a multiple of (2.2) ds + i|t| σi dt and first integral ζ i…”

“…where dµ s is the density measure in R 2 . A simple version of this operator was considered in [9] and [10] for other classes of vector fields, and more recently in [5] and [6]. Let T ω g ∈ C α (R), with α = 2 − p − µ p , p = q q − 1 , and µ = σ σ + 1 .…”

“…To prove the necessity of (4.3), suppose equation (4.2) has a solution in T 2 (note that in this case the solution is necessarily Hölder continuous by results contained in [5]). Then the function V (x, y) = Π * u(x, y)e −TωΠ * A(x,y) satisfies LV = 0 in R 2 .…”