  In this paper, we introduce a criterion for maximal operators associated with
Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples
satisfying the criterion are multipliers of the Mikhlin type or limited decay
which are not necessarily radial. To do so, we make use of modified square
function estimates and bilinear interpolation. In result, we obtain convergence
results for fractional half-wave equations and surface averages as well as the
$L^p$ boundedness for the maximal operators.
