The boundedness of the triangular Hilbert transforms along curves

Abstract

In this paper, for general plane curves γ satisfying some suitable smoothness and curvature conditions, we establish the $L^p\left(R^2\right)\times L^q\left(R^2\right)\to L^r\left(R^2\right)$ boundedness of the triangular Hilbert transform along curves $(t,\gamma (t\left)\right)$ with $p,q\in (1,\infty ),r\in [1,2)$ satisfying $1/p+1/q=1/r$, and the $L^p\left(R^2\right)\times L^q\left(R^2\right)\to L^r\left(R^2\right)$ boundedness for the associated maximal function, where $p,q$ as above and $r\in [1,\infty )$.