Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves

Junfeng Li and Haixia Yu

Vol. 310 (2021), No. 2, 375–446

DOI: 10.2140/pjm.2021.310.375

Abstract

We determine the $L^p\left(ℝ\right)\times L^q\left(ℝ\right)\to L^r\left(ℝ\right)$ boundedness of the bilinear Hilbert transform $H_{\gamma }\left(f,g\right)$ along a convex curve $\gamma$:

$$H_{\gamma }\left(f,g\right)\left(x\right):=p.v.{\int \; <!--nolimits-->}_{-\infty }^{\infty }f\left(x-t\right)g\left(x-\gamma \left(t\right)\right)\frac{dt}{t},$$

where $p$, $q$, and $r$ satisfy $1∕p+1∕q=1∕r$, and $r>\frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p\left(ℝ\right)\times L^q\left(ℝ\right)\to L^r\left(ℝ\right)$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{\gamma }\left(f,g\right)$ along a convex curve $\gamma$

$$M_{\gamma }\left(f,g\right)\left(x\right):=\underset{𝜀>0}{sup}\frac{1}{2𝜀}{\int \; <!--nolimits-->}_{-𝜀}^{𝜀}\left|f\left(x-t\right)g\left(x-\gamma \left(t\right)\right)\right|dt.$$

Keywords

bilinear Hilbert transform, (sub)bilinear maximal function, convex curve, time frequency analysis

Mathematical Subject Classification

Primary: 42B20

Secondary: 47B38

Milestones

Received: 7 June 2020

Revised: 2 September 2020

Accepted: 4 September 2020

Published: 8 March 2021