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This paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe a hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau–Hadamard–Kolmogorov inequalities on the function-theoretic side.

C 0 -Rigidity of the Double Poisson Bracket