Paper detail

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

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 $T_{Ω,α}^{\#}$ with homogeneous convolution kernel $Ω(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$. We show that if $0<α<\mathbb{Q}$, $Ω\in L^{1}(Σ)$ and satisfies the cancellation condition of order $[α]$, 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 $α=0$, 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 $0\leqα<\mathbb{Q}$ and $Ω$ satisfies the same cancellation condition but a stronger condition that $Ω\in L^{q}(Σ)$ for some $q>\mathbb{Q}/α$, 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*}