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

Abdelhamid Meziani

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arXiv · 2205.00461

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

Abdelhamid Meziani

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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.

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Abdelhamid Meziani. 2022-05-01. A Lojasiewicz Inequality in Hypocomplex Structures of $\mathbb{R}^2$. https://arxiv.org/abs/2205.00461

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