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

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.