  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.
