$L^p$ 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 ($1<p<\infty$ and $w$ a Muckenhoupt $A_p$ weight) of fractional maximal singular integral operators $T_{\hat{I}\text{c◯},\hat{I}\pm }^{\#}$ with homogeneous convolution kernel $\hat{I}\text{c◯}(x)$ on an arbitrary homogeneous group $\mathbb{H}$ of dimension $\mathbb{Q}$. We show that if $0<\hat{I}\pm <\mathbb{Q}$, $\hat{I}\text{c◯}\in L^1(\hat{I}\pounds )$ and satisfies the cancellation condition of order $[\hat{I}\pm ]$, then for any $1<p<\infty$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(\mathbb{H})}\lesssim\|Ω\|_{L^{1}(Σ)}\|f\|_{L_α^{p}(\mathbb{H})}, \end{align*} where for the case $\hat{I}\pm =0$, the $L^p$ boundedness of rough singular integral operator and its maximal operator were studied by Tao (\cite{Tao}) and Sato (\cite{sato}), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if $0\le \hat{I}\pm <\mathbb{Q}$ and $\hat{I}\text{c◯}$ satisfies the same cancellation condition but a stronger condition that $\hat{I}\text{c◯}\in L^q(\hat{I}\pounds )$ for some $q>\mathbb{Q}/\hat{I}\pm$, then for any $1<p<\infty$ and $w\in A_p$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(w)}\lesssim\|Ω\|_{L^{q}(Σ)}\{w\}_{A_p}(w)_{A_p}\|f\|_{L_α^{p}(w)},\ \ 1<p<\infty. \end{align*}

40 pages, Shorten the proof in the previous version, To appear in J. Geom. Anal