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

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.