For a symplectic manifold $(M,ω)$, let $\{·,·\}$ be the corresponding Poisson bracket. In this note we prove that the functional $ (F,G) \mapsto \|\{F,G\}\|_{L^p(M)} $ is lower-semicontinuous with respect to the $C^0$-norm on $C^∞_c(M)$ when $\dim M = 2$ and $p < ∞$, extending previous rigidity results for $p = ∞$ in arbitrary dimension.
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