Rough bilinear singular integrals (Q1694299)

Given \(\Omega\in L^\infty(\mathbb S^{2n-1})\), with mean value zero, set \[ K(y,z)=\frac{\Omega((y,z)')}{|(y,z)|^{2n}}, \] where \(x'=x/|x|\). Then, the authors prove the boundedness of the rough bilinear operator \[ T_\Omega(f,g)(x)=\text{p.v.}\int_{\mathbb R^n}\int_{\mathbb R^n} K(x-y,x-z)f(y)g(z)\,dydz, \] on \(L^p(\mathbb R^n)\), for all \(p>1/2\) and all dimensions: \[ T_\Omega:L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)\rightarrow L^{p}(\mathbb R^n), \] \(1 < p_1,p_2<\infty\) and \(1/p=1/p_1+1/p_2\). \(T_\Omega\) was first introduced in [\textit{R. R. Coifman} and \textit{Y. Meyer}, Trans. Am. Math. Soc. 212, 315--331 (1975; Zbl 0324.44005)], where they proved a similar result, in dimension 1, under the more restrictive smoothness condition that \(\Omega\) is a function of bounded variation on the circle. The extension to higher dimensions, for Lipschitz functions \(\Omega\) in \(\mathbb S^{2n-1}\), was obtained in [\textit{L. Grafakos} and \textit{R. H. Torres}, Adv. Math. 165, No. 1, 124--164 (2002; Zbl 1032.42020)]. If \(\Omega\in L^q(\mathbb S^{2n-1})\), \(2\leq q<\infty\), a similar results holds, for \((1/p_1,1/p_2,1/p)\) on a suitable neighborhood of \((1/2,1/2,1)\). The idea of the proof is to write the kernel \(K=\sum_jK_j\), using a Littlewood-Paley decomposition, and then to obtain an exponential decay of the norm of \(T_j\), based on a particular compactly supported wavelet expansion, combined with combinatorial arguments and exploiting orthogonality. Some motivations for these ideas can be found in [\textit{J. Duoandikoetxea} and \textit{J. L. Rubio de Francia}, Invent. Math. 84, 541--561 (1986; Zbl 0568.42012)]. A couple of interesting open problems are stated at the end of this work: if \(\Omega\in L^q(\mathbb S^{2n-1})\), \(2\leq q<\infty\), find the optimal range of indices \(p_1,p_2,p\), for which \(T_\Omega\) is bounded, and study whether the operator is bounded when \(\Omega\in L^q(\mathbb S^{2n-1})\), for \(q<2\).