Estimates for maximal Fourier multiplier operators on ${\mathbb{R}}^2$ via square functions

We consider certain Littlewood-Paley square functions on ${\mathbb{R}}^2$ and prove sharp estimates for them, from which we can deduce $L^p$ boundedness of maximal functions defined by Fourier multipliers of Bochner-Riesz type on ${\mathbb{R}}^2$. This is a generalization of a result due to A. Carbery 1983.