Weighted $L^p$ boundedness of Carleson type maximal operators

In 2001, E. M. Stein and S. Wainger gave the $L^p$ boundedness of the Carleson type maximal operator $\mathcal {T}^\ast$, which is defined by

$$\mathcal {T}^\ast f(x)=\sup _\lambda \bigg \vert\int _{{\mathbb{R}}^n}e^{iP_\lambda (y)}K(y)f(x-y)dy\bigg \vert.$$

In this paper, the authors show that if $K$ is a homogeneous kernel, i.e. $K(y)=\Omega (y')\vert y\vert^{-n}$, then Stein-Wainger's result still holds on the weighted $L^p$ spaces when $\Omega$ satisfies only an $L^q$-Dini condition for some $1<q\le \infty$.