On Schatten p-norms of commutators

For any odd integer d≥3d≥3, we determine the sharpest constant Cp,q,rCp,q,r such that‖XY−YX‖p≤Cp,q,r‖X‖q‖Y‖r for all X,Y∈Md, where MdMd denotes the set of all d×dd×d complex matrices, ‖⋅‖p‖⋅‖p, 1≤p≤∞1≤p≤∞, denotes the Schatten p  -norm on MdMd, and 1≤p,q,r≤∞1≤p,q,r≤∞ satisfy 1p>1q+1r. This is a continuation of the study of the problem considered in Wenzel and Audenaert (2010) [8].