A Lojasiewicz Inequality in Hypocomplex Structures of $\mathbb{R}^2$

TL;DR

This paper establishes a Lojasiewicz inequality for hypocomplex structures in ^2, linking the solvability of certain PDEs with integrability conditions and revealing a similarity principle with holomorphic functions.

Contribution

It introduces a new Lojasiewicz inequality for hypocomplex structures and connects bounded solutions of PDEs to holomorphic functions through a similarity principle.

Findings

Bounded solutions exist for ^2 equations with data in L^p for p>1+ inequality

A specific  inequality number   attached to the structure

Similarity principle between solutions of PDEs and holomorphic functions

Abstract

For a real analytic complex vector field $L$ in an open set of $\mathbb{R}^2$, with local first integrals that are open maps, we attach a number $\mu \ge 1$ (obtained through Lojasiewicz inequalities) and show that the equation $Lu=f$ has bounded solutions when $f\in L^p$ with $p>1+\mu$. We also establish a similarity principle between the bounded solutions of the equation $Lu=Au+B\overline{u}$ (with $A,B\in L^p$) and holomorphic functions.