  In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier
operators with dilation of fractional dimensions. For the Fourier multipliers,
we suggest a criterion related to dimensions of dilation sets which guarantees
$L^p$ bounds of the maximal operators for each $p$. Our criterion covers
Mikhlin-type multipliers, multipliers with limited decay, and multipliers with
slow decay.
