Let
\(\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}\) be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in
\(\mathbb{R}^{n}\). We show that the
L p norm, 1<
p<∞, of the related maximal operator
$$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$
is at most
C(log(
N+2))
n/2. We show that this bound is sharp.