Sharpness of Zapolsky inequality for quasi-states and Poisson brackets

TL;DR

This paper proves that the lower bound provided by Zapolsky inequality for the Poisson bracket's L1 norm in terms of quasi-states on the sphere is optimal, confirming its sharpness.

Contribution

It demonstrates that the Zapolsky inequality's lower bound for the Poisson bracket norm is sharp, establishing the precise limit of this inequality.

Findings

Zapolsky inequality's lower bound is sharp.

The Poisson bracket's L1 norm attains the lower bound.

Confirmation of the inequality's optimality.

Abstract

Zapolsky inequality gives a lower bound for the L1 norm of the Poisson bracket of a pair of C1 functions on the two-dimensional sphere by means of quasi-states. Here we show that this lower bound is sharp.