$L^{p}$ estimates and weighted estimates of fractional maximal rough singular integrals on homogeneous groups

Abstract

In this paper, we study the $L^{p}$ boundedness and $L^{p}(w)$ boundedness ($1 \mathbb{Q}/\alpha$, then for any $1<p<\infty$ and $w\in A_{p}$, \begin{align*} \|T_{\Omega,\alpha}^{\#}f\|_{L^{p}(w)}\lesssim\|\Omega\|_{L^{q}(\Sigma)}\{w\}_{A_p}(w)_{A_p}\|f\|_{L_{\alpha}^{p}(w)},\ \ 1<p<\infty. \end{align*}