Certain square functions on product spaces (Q2762269)

Let NEWLINE\[NEWLINEP_{\mathcal K}(u)=\sum^{\mathcal K}_{k=1}a_ku^k\quad\text{ and}\quad Q_{\mathcal J}(v)=\sum^{\mathcal J}_{j=1}b_jv^jNEWLINE\]NEWLINE be two real polynomials with \(P_{\mathcal K}(0)=Q_{\mathcal J}(0)=0.\) For nonzero \(x\in {\mathbb{R}}^n\), let \(x'=x/|x|\). For \(n\geq 2\) and \(m\geq 3\), let \(\Omega(x',y')\in L^1(S^{n-1}\times S^{m-1})\) satisfy NEWLINE\[NEWLINE\int_{S^{n-1}}\Omega(x',y') d\sigma(x')= \int_{S^{m-1}}\Omega(x',y') d\sigma(y')=0.NEWLINE\]NEWLINE For a bounded function \(B(u,v)\) supported in \([0,1]^2\), define NEWLINE\[NEWLINE\Phi(x,y)=B(|x|,|y|)|x|^{-n+1}|y|^{-m+1}\Omega(x',y')NEWLINE\]NEWLINE and \(\Phi_{t,s}(x,y)=2^{-nt-ms}\Phi(x/2^t,y/2^s)\). The Littlewood-Paley square function and the singular integral operator are, respectively, defined by NEWLINE\[NEWLINET_\Phi(f)(\xi,\eta)=\left(\int_{{\mathbb{R}}^n} |\Phi_{t,s}\#f(\xi,\eta)|^2 dt ds\right)^{1/2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINES_\Phi(f)(\xi,\eta)=\int_{{\mathbb{R}}^n} \Phi_{t,s}\#f(\xi,\eta) dt ds,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\Phi_{t,s}\#f(\xi,\eta)=\int_{{\mathbb{R}}^n\times {\mathbb{R}}^m} \Phi_{t,s}(x,y)f(\xi-P_{\mathcal K}(|x|)x', \eta-Q_{\mathcal J}(|y|)y') dx dy.NEWLINE\]NEWLINE The authors prove that if \(\Omega\in L^q(S^{n-1}\times S^{m-1})\) for some \(q>1\), then \(T_\Phi\) and \(S_\Phi\) are bounded in \(L^p({\mathbb{R}}^n\times{\mathbb{R}}^m)\) for \(p\in (1,\infty)\) and the bound is independent of the coefficients of \(P_{\mathcal K}\) and \(Q_{\mathcal J}\).