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 $H_γ(f,g)$ along a convex curve $γ$ $H_γ(f,g)(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x-t)g(x-γ(t)) \,\frac{\textrm{d}t}{t},$ 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 $M_γ(f,g)$ along a convex curve $γ$ $M_γ(f,g)(x):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}|f(x-t)g(x-γ(t))| \,\textrm{d}t.$

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