Rough Bilinear Hypersingular Integrals

We study the rough bilinear hypersingular integral operator Ts(f,g)(x)=p.v.∫ℝ2nΩ((y,z)′)|(y,z)|2n+sf(x−y)g(x−z)dydz,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_{s}(f,g)(x)=\textrm{p.v.}{\int}_{\mathbb{R}^{2n}}\frac{\Omega((y,z)^{\prime})}{|(y,z)|^{2n+s}}f(x-y)g(x-z)dydz, $$\end{document} defined on all test functions f,g, where s ≥ 0, Ω is a function in Lq(S2n− 1) satisfying certain cancellation condition. For s ≥ 0, we obtain boundedness for Ts with Ω in L∞(S2n−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty }(\mathbf {S}^{2n-1})$\end{document}. The result extends some known results on bilinear singular integrals and linear hypersingular integrals.