$$L^{p}$$ 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 \(T_{\Omega ,\alpha }^{\#}\) with homogeneous convolution kernel \(\Omega (x)\) on an arbitrary homogeneous group \({\mathbb {H}}\) of dimension \({\mathbb {Q}}\). We show that if \(0<\alpha <{\mathbb {Q}}\), \(\Omega \in L^{1}(\Sigma )\) and satisfies the cancellation condition of order \([\alpha ]\), then for any \(1<p<\infty \), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}({\mathbb {H}})}\lesssim \Vert \Omega \Vert _{L^{1}(\Sigma )}\Vert f\Vert _{L_{\alpha }^{p}({\mathbb {H}})}. \end{aligned}$$where for the case \(\alpha =0\), the \(L^p\) boundedness of rough singular integral operator and its maximal operator were studied by Tao (Indiana Univ Math J 48:1547–1584, 1999) and Sato (J Math Anal Appl 400:311–330, 2013), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if \(0\le \alpha <{\mathbb {Q}}\) and \(\Omega \) satisfies the same cancellation condition but a stronger condition that \(\Omega \in L^{q}(\Sigma )\) for some \(q>{\mathbb {Q}}/\alpha \), then for any \(1<p<\infty \) and \(w\in A_{p}\), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}(w)}\lesssim \Vert \Omega \Vert _{L^{q}(\Sigma )}\{w\}_{A_p}(w)_{A_p}\Vert f\Vert _{L_{\alpha }^{p}(w)},\ \ 1<p<\infty . \end{aligned}$$