Criterion on Lp1×Lp2→Lq-Boundedness for Oscillatory Bilinear Hilbert Transform

Abstract

We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that the $L^{p_1}\times L^{p_2}\to L^q$-boundedness of the two operators is equivalent with $1\le p_1,p_2$ , and $1/q=1/p_1+1/p_2$. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.