# Class of hypocomplex structures on the two dimensional torus

## TL;DR

This paper investigates the solvability of certain complex vector fields on the two-dimensional torus, utilizing Theta functions and integral operators to establish a similarity principle for solutions of related equations.

## Contribution

It introduces a new approach using Theta functions to analyze hypocomplex structures and derives a similarity principle for specific complex vector field equations on the torus.

## Key findings

- Established H"{o}lder solvability conditions for complex vector fields on the torus.
- Developed a Cauchy-Pompeiu type integral operator using Theta functions.
- Proved a similarity principle for solutions of $Lu=au+bar{u}$.

## Abstract

We study the H\"{o}lder solvability of a class of complex vector fields on the torus $\mathbb{T}^2$. We make use of the Theta function to associate a Cauchy-Pompeiu type integral operator. A similarity principle for the solutions of the equation $Lu=au+b\bar{u}$ is obtained.