On the bilinear Hilbert transform along two polynomials

Abstract:

We prove that the bilinear Hilbert transform along two polynomials $B_{P,Q}(f,g)(x)=\int _{\mathbb {R}}f(x-P(t))g(x-Q(t))\frac {dt}{t}$ is bounded from $L^p \times L^q$ 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 \[ \mathcal {M}_{P,Q}(f,g)(x)=\sup _{\epsilon >0}\frac {1}{2\epsilon }\int _{-\epsilon }^{\epsilon } |f(x-P(t))g(x-Q(t))|dt.\]