On maximal functions for Mikhlin–Hörmander multipliers

Given Mikhlin–Hörmander multipliers , with uniform estimates we prove an optimal bound in Lp for the maximal function and related bounds for maximal functions generated by dilations. These improve the results in [M. Christ, L. Grafakos, P. Honzík, A. Seeger, Maximal functions associated with multipliers of Mikhlin–Hörmander type, Math. Z. 249 (2005) 223–240].