Chen, Jiecheng; Ding, Yong; Fan, Dashan Certain square functions on product spaces. (English) Zbl 0991.42006 Math. Nachr. 230, 5-18 (2001). Let \[ P_{\mathcal K}(u)=\sum^{\mathcal K}_{k=1}a_ku^k\quad\text{ and}\quad Q_{\mathcal J}(v)=\sum^{\mathcal J}_{j=1}b_jv^j \] 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 \[ \int_{S^{n-1}}\Omega(x',y') d\sigma(x')= \int_{S^{m-1}}\Omega(x',y') d\sigma(y')=0. \] For a bounded function \(B(u,v)\) supported in \([0,1]^2\), define \[ \Phi(x,y)=B(|x|,|y|)|x|^{-n+1}|y|^{-m+1}\Omega(x',y') \] 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 \[ T_\Phi(f)(\xi,\eta)=\left(\int_{{\mathbb{R}}^n} |\Phi_{t,s}\#f(\xi,\eta)|^2 dt ds\right)^{1/2} \] and \[ S_\Phi(f)(\xi,\eta)=\int_{{\mathbb{R}}^n} \Phi_{t,s}\#f(\xi,\eta) dt ds, \] where \[ \Phi_{t,s}\#f(\xi,\eta)=\int_{{\mathbb{R}}^n\times {\mathbb{R}}^m} \Phi_{t,s}(x,y)f(\xi-P_{\mathcal K}(|x|)x', \eta-Q_{\mathcal J}(|y|)y') dx dy. \] 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 \(Q_{\mathcal J}\). Reviewer: Yang Dachun (Beijing) Cited in 8 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory Keywords:product space; square function; singular integral; rough kernel; boundedness PDFBibTeX XMLCite \textit{J. Chen} et al., Math. Nachr. 230, 5--18 (2001; Zbl 0991.42006) Full Text: DOI References: [1] Chen, Chinese Annals of Mathematics pp 39– (2000) [2] Duoandikoetxea, Ann. Inst. Fourier 36 pp 185– (1986) · Zbl 0568.42011 · doi:10.5802/aif.1073 [3] : Multiparameter Fourier Analysis, Beijing Lectures in Harmonic Analysis, 47-130, Edited by E. M. Stein, Annals of Math. Study #112, Princeton Univ. Press, Princeton, NJ, 1986 [4] Fan, Amer. Jour. Math. 119 pp 799– (1997) · Zbl 0899.42002 · doi:10.1353/ajm.1997.0024 [5] Stein, Trans. Amer. Math. Soc. 88 pp 430– (1958) · doi:10.1090/S0002-9947-1958-0112932-2 [6] : Harmonic Analysis Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.