Rigidity versus flexibility of the Poisson bracket with respect to the $L_p$-norm

TL;DR

This paper investigates how the rigidity of the Poisson bracket, well-known in the uniform norm case, extends or varies when considering $L_p$ norms, revealing dimension-dependent behavior.

Contribution

It demonstrates rigidity of $L_p$-Poisson bracket invariants in two dimensions and suggests flexibility in higher dimensions.

Findings

Rigidity in dimension two for $L_p$-Poisson invariants.

Evidence of flexibility in higher dimensions.

Contrast with uniform norm case.

Abstract

Rigidity of the Poisson bracket with respect to the uniform norm is one of the central phenomena discovered within function theory on symplectic manifolds. In the present work we examine the case of $L_p$ norms with $p < \infty$. We show that $L_p $ - Poisson bracket invariants exhibit rigid behavior in dimension two, and we provide an evidence for their flexibility in higher dimensions.