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Given Mikhlin-Hörmander multipliers mi, i = 1, . . . , N , with uniform estimates we prove an optimal log(N + 1) bound in L p for the maximal function sup i |F −1 [mi f ]| and related bounds for maximal functions generated by dilations. These improve results in [7].

We show that maximal operators formed by dilations of MikhlinHörmander multipliers are typically not bounded on L p (R d ). We also give rather weak conditions in terms of the decay of such multipliers under which L p boundedness of the maximal operators holds.

We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.