A Łojasiewicz inequality in hypocomplex structures of ℝ²

Abstract

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