On the bilinear Hilbert transform along two polynomials

We prove that the bilinear Hilbert transform along two polynomials B_P,Q(f, g)(x) = ^∫ _R f(x − P(t))g(x − Q(t)) ^dt _t is bounded from L^p × Lq to L^r for a large range of (p, q, r), as long as the polynomials P and Q have distinct leading and trailing degrees. The same boundedness property holds for the corresponding bilinear maximal function M_P,Q(f, g)(x) = sup _{ε> 0 2} ¹ _ε ^∫ ₋ ^ε _ε |f(x − P(t))g(x − Q(t))|dt.