The 2/3 - convergence rate for the Poisson bracket

TL;DR

This paper introduces a novel approach to C^0 - rigidity of the Poisson bracket, establishing the precise convergence rate and extending previous results, with implications for multilinear differential operators.

Contribution

It provides a new method for proving C^0 - rigidity of the Poisson bracket and determines the exact semi-continuity rate, extending prior work by several researchers.

Findings

Established the lower semi-continuity under C^0 perturbations.

Determined the exact convergence rate for the Poisson bracket.

Extended C^0 - rigidity results to multilinear differential operators.

Abstract

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 Humiliere. We also discuss a general question of the C^0 - rigidity for multilinear differential operators.