Rough Singular Integrals and Marcinkiewicz Integrals on Product Spaces, with $H^1$ Characterizations on Product Spheres

We prove the $L^p(\mathbb R^n\times\mathbb R^m)$ boundedness of the rough product singular integral $T_Ω$, its maximal truncation $T_Ω^*$, and the product Marcinkiewicz integral $μ_Ω$ for $1<p<\infty$, assuming that $Ω\in H^1(S^{n-1}\times S^{m-1})$ satisfies separate cancellation. These res