\(L^p\) bounds for singular integral operators along twisted surfaces (Q6569658)

This paper considers the following singular integrals along twisted surfaces on product domains: \N\[\N\mathcal{S}_{\Omega,h,\Lambda}(f)(x,y)=\int_{\mathbb{R}^n\times\mathbb{R}^m}f((x,y)-\Lambda(u,v))\frac{\Omega(u,v)}{|u|^n|v|^m}h(|u|,|v|)dudv,\N\]\Nwhere \(\Lambda(u,v)=\Phi(|v|)u,\Psi(|u|)v)\) with \(\Phi,\,\Psi:\,(0,\infty)\to \mathbb{R}\). For certain \(\Phi,\,\Psi\) which are the mappings more general than polynomials and convex functions, the authors obtain the \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\)-boundedness of \(\mathcal{S}_{\Omega,h,\Lambda}\) with \(1<p<\infty\), provided that the kernels \(\Omega\in L(\log L)^2(\mathbb{S}^{(n-1)\times(m-1)})\) and \(h\in L^2(\mathbb{R}^+\times\mathbb{R}^+,r^{-1}s^{-1}drds)\). Meanwhile, for the maximal function with \(\Omega\in L^1(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1})\), \N\[\N\nu_{\Omega,\Lambda}(f)(x,y)=\sup_{j,k\in \mathbb{Z}}\iint_{\substack{2^j<|u|<2^{j+1}}\\\N{2^k<|v|<2^{k+1}}}|f((x,y)-\Lambda(u,v))| \frac{|\Omega(u,v)|}{|u|^n|v|^m}dudv,\N\]\Nthe corresponding \(L^p(\mathbb{R}^n\times\mathbb{R}^m)\)-boundedness is also given.\N\NThese results can be regarded as the generalization of ones obtained by \textit{A. Al-Salman} [Commun. Pure Appl. Anal. 21, No. 1, 159--181 (2022; Zbl 1494.42012); Bull. Korean Math. Soc. 58, No. 4, 1003--1019 (2021; Zbl 1480.42023); Front. Math. China 16, No. 1, 13--28 (2021; Zbl 1468.42011)] etc.