A Łojasiewicz inequality in hypocomplex structures of ℝ²

Meziani, Abdelhamid

doi:10.1090/proc/16019

A Łojasiewicz inequality in hypocomplex structures of $\mathbb {R}^2$
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Proc. Amer. Math. Soc. 150 (2022), 3875-3887 Request permission

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 Łojasiewicz inequalities) and show that the equation $Lu=f$ has bounded solution 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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