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 Lp1×Lp2→Lq-boundedness of the two operators is equivalent with 1≤p1,p2<∞, and 1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.