C⁰-rigidity of the double Poisson bracket

The 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 an 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.