Rigidity of the L^p norm of the Poisson bracket on surfaces

For a symplectic manifold $M$ let $\{\cdot ,\cdot \}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto |{F,G}|_{L^p(M)}$islower-semicontinuouswithrespecttothe$C^0$-normon$C^\infty_c(M)$when$\dim M = 2$and$p < \infty$, extending previous rigidity results for $p=\mathrm{\infty }$ in arbitrary dimension.