Estimates of Oscillations of the Hardy Transform

For the Hardy transform of a nonincreasing function we obtain a sharp two-sided estimate of the $\operatorname {BLO}$-norm and sharp inequalities between the $\operatorname {BMO}$- and the $\operatorname {BLO}$-norms of a nonincreasing function. A well-known lower bound for the $\operatorname {BMO}$-norm of the Hardy transform is improved on the basis of these inequalities.