Global solvability for a class of variable-coefficient Vekua-type operators on compact lie groups

Abstract We investigate the global solvability of a class of Vekua-type operators in the form: $$\begin{aligned} Pu = Lu - (s(t) + i\delta q(t)) \cdot u - \alpha q(t) \overline{u}, \end{aligned}$$ P u = L u - ( s ( t ) + i δ q ( t ) ) · u - α q ( t ) u ¯ , which act on smooth functions defined on $$\mathbb {T}^1 \times G$$ T 1 × G , where $$G$$ G is a compact Lie group. The operator $$L$$ L is given by $$\begin{aligned} L = \partial _t - (p_0 + i\lambda q(t)) \cdot D, \end{aligned}$$ L = ∂ t - ( p 0 + i λ q ( t ) ) · D , with $$D$$ D being a diagonal operator on $$G$$ G . The term $$\overline{u}$$ u ¯ makes $$P$$ P real-linear but not complex-linear, leading to significant differences in the analysis of global solvability. We establish that $$P$$ P is globally solvable in a smooth sense under specific non-resonance assumptions.