Journal of the
Korean Mathematical Society JKMS

In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels $h$ and sphere kernels $\Omega$ by assuming $h\in{\triangle}_{\gamma}(\mathbb{R}_+)$ and $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ for some $\gamma>1$ and $\beta>1$. Here $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

Keywords: Singular integrals, maximal operators, rough kernels, weighted norm inequalities

MSC numbers: Primary 42B20; Secondary 42B15, 42B25

Supported by: The research was supported by NNSF of China (Nos. 11771358, 11871101)