The 2/3-Convergence Rate for the Poisson Bracket

In this paper we introduce a new method for approaching the C ⁰-rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C ⁰ perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous results of Cardin–Viterbo, Zapolsky, Entov and Polterovich. Using our method, we prove a C ⁰-rigidity result in the spirit of the work of Humilière. We also discuss a general question of the C ⁰-rigidity for multilinear differential operators.