Marcinkiewicz integrals along subvarieties on product domains (Q1777921)

Let \(\Omega: \mathbb R^{n}\times\mathbb R^{m}\to \mathbb C\) be a component-wise homogeneous function of degree zero with cancellation properties \(\int_{S^{n-1}}\Omega(x',y')\,d\sigma(x')=0,\) \(\int_{S^{m-1}}\Omega(x',y')\,d\sigma(y')=0.\) The Marcinkiewicz integral in a product domain is defined by \[ \mu_\Omega f(x,y)=\bigl(\int_{0}^{\infty}\int_{0}^{\infty} | F_{s,t}f(x,y)| ^2\frac{dsdt}{s^3t^3} \bigr)^{\frac12}, \] where \[ F_{s,t}f(x,y)=\int_{| x-u| \leq t}\int_{| y-v| \leq t} \frac{\Omega(x-u,y-v)} {| x-u| ^{n-1}| y-v| ^{m-1}}f(u,v)dudv. \] It is known that the Marcinkiewicz integral operator \(\mu_\Omega\) is bounded on \(L^p(\mathbb R^{n}\times\mathbb R^{m})\) under the assumption \(\Omega| _{S^{n-1}\times S^{m-1}}\in L(\log^+L)(S^{n-1}\times S^{m-1})\), by \textit{Y. Choi} [J. Math. Anal. Appl. 261, No. 1, 53--60 (2001; Zbl 0993.42009)] and \textit{H. Al-Qassem, A. Al-Salman, L. Cheng} and \textit{Y. Pan} [Stud. Math. 167, No. 3, 227--234 (2005; Zbl 1084.42008)]. On the other hand, in the usual case (i.e., non-product type) \textit{J. Chen, D. Fan} and \textit{Y. Pan} [Math. Nachr. 227, 33--42 (2001; Zbl 0994.42012)] gave \(L^p\)-boundedness of the Marcinkiewicz integral \(((2+\alpha)/(1+\alpha)<p<2+\alpha)\) under the condition \(\sup_{\xi\in S^{n-1}}\int_{S^{n-1}} | \Omega(y')| (\log 1/| \xi\cdot y'| )^{1+\alpha}d\sigma(y')<0\) \(\alpha>0\). (This condition was introduced by \textit{L. Grafakos} and \textit{A. Stepanov} [Indiana Univ. Math. J. 47, No. 2, 455--469 (1998; Zbl 0913.42014)].) The author gives similar results under the same but more complicated assumption.