Maximal operators given by Fourier multipliers with dilation of fractional dimensions

In this paper, we investigate Lᵖ 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 Lᵖ bounds of the maximal operators for each p. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.