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

Abstract

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

References in corpus (1)

• Uniform estimates for some paraproducts

Cited by in corpus (1)

• $L^p({\mathbb{R}}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients