Rough singular integrals and maximal operator with radial-angular integrability

Abstract:

In this paper, we study the rough singular integral operator \begin{equation*} T_\Omega f(x)=\text {p.v.}\int _{\mathbb {R}^n}f(x-y)\frac {\Omega (y’)}{|y|^n}dy, \end{equation*} and the corresponding maximal singular integral operator \begin{equation*} T^*_\Omega f(x)=\sup _{\varepsilon >0}\Big |\int _{|y|\geq \varepsilon }f(x-y)\frac {\Omega (y’)}{|y|^n}dy\Big |, \end{equation*} where the kernel $\Omega \in H^1(\mathrm {S}^{n-1})$ has zero mean value and $n\geq 2$. We prove that $T_\Omega$ and $T^*_\Omega$ are bounded on the mixed radial-angular spaces $L_{|x|}^pL_{\theta }^{\tilde {p}}(\mathbb {R}^n)$ for some suitable indexes $1<p, \tilde {p}<\infty$. The corresponding vector-valued versions are also established.