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 $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ 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 $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$