Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

In this paper, we determine the $L^p($\mathbb{R}$)\times L^q($\mathbb{R}$)\rightarrow L^r($\mathbb{R}$)$ boundedness of the bilinear Hilbert transform $H_{\gamma }(f,g)$ along a convex curve $\gamma$ $$H_{\gamma }(f,g)(x):=\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x-t)g(x-\gamma (t))\frac{\text{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, and $r>\frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p($\mathbb{R}$)\times L^q($\mathbb{R}$)\rightarrow L^r($\mathbb{R}$)$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{\gamma }(f,g)$ along a convex curve $\gamma$ $$M_{\gamma }(f,g)(x):=\underset{\epsilon >0}{\sup }\frac{1}{2\epsilon }{\int }_{-\epsilon }^{\epsilon }\mathrm{\mid }f(x-t)g(x-\gamma (t))\mathrm{\mid }\text{d}t\mathrm{.}$$