The general problem under consideration is the determination of the best (i.e., smallest) constant C-p,C-q,C-r such that

parallel to XY - YX parallel to(p) <= C-p,C-q,C-q parallel to X parallel to(q) parallel to Y parallel to(r)

for all n x n matrices X and Y,

where parallel to . parallel to(p) is the Schatten p-norm. Among the open situations, the problem is solved when

(i) 2 < p < infinity, q = r =1;

(ii) 2 < p < infinity, q = 1, 1 < r < 2, and X, Y are 2 x 2 real matrices.

The result for (i) is obtained via the study of the region (recently obtained in [5]) of singular values of the commutator XY - YX , where X and Y are normalized rank one matrices. As a result of independent interest, for 2 x 2 real matrices, the region of singular values of the commutator XY - YX, where X is rank one and normalized, and parallel to Y parallel to(r) = 1, is determined.

The result for (ii) is a consequence of the case when r lies between 1 and 2.