Rough maximal bilinear singular integrals

Abstract

We study the rough maximal bilinear singular integral

$$\begin{aligned} T^{*}_\varOmega (f,g)(x)=\! \sup _{\varepsilon >0}\left| \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\! \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\!\frac{ \varOmega ((y,z)/|(y,z)|)}{ |(y,z)|^{2n}}f(x-y)g(x-z) dydz\right| , \end{aligned}$$

where $\varOmega$ is a function in $L^\infty (\mathbb S^{2n-1})$ with vanishing integral. We prove it is bounded from $L^p\times L^q\rightarrow L^r,$ where $1<p,q<\infty$ and $1/r=1/p+1/q.$ We also discuss results for $\varOmega \in L^s(\mathbb S^{2n-1}),$$1<s<\infty$.