Rough singular integrals on product spaces (Q1777881)

Let \(d\geq 2\) (\(d=n\) or \(d=m\)) and \(S^{d-1}\) be the unit sphere in \(\mathbb{R}^d\) equipped with the normalized Lebesgue measure \(d\sigma\). Suppose that \(\Omega(x,y)\) is a homogeneous function of degree zero in both variables \(x\) and \(y\) and it satisfies \(\Omega\in L(S^{n-1}\times S^{m-1})\) and \[ \int_{S^{n-1}}\Omega(x,y)\,d\sigma(x)= \int_{S^{m-1}}\Omega(x,y)\,d\sigma(y)= 0,\quad\text{for all }x\text{ and }y. \] For the Calderón-Zygmund type kernel \(K(x,y)= \Omega(x,y)|x|^{-n}|y|^{-m}\), and suitable \(C^\infty\)-mappings \(\Phi: B_n(0,r)\to \mathbb{R}^N\), \(\Psi: B_m(0,r)\to \mathbb{R}^M\), the authors study the singular integral operator \[ Tf(x,y)= \text{p.v. }\int_{B_n(0,1)\times B_m(0,1)} f(x- \phi(u), y-\Psi(v))\,K(u,v)\,du\,dv. \] They obtain the following main theorem in the paper. Theorem. Assume that \(\Omega\) is in certain block space \(B^{0,1}_q(S^{n-1}\times S^{m-1})\), \(q> 1\). If \(\Phi\) and \(\Psi\) are of finite type at \(0\), then the operator \(T\) is bounded on \(L^p(\mathbb{R}^N\times \mathbb{R}^M)\) for \(1< p<\infty\). The authors also establish a similar boundedness theorem for the corresponding truncated maximal operator \(T^*\).