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

${H}_{\gamma }\left(f,g\right)\left(x\right):=p.v.{\int }_{-\infty }^{\infty }f\left(x-t\right)g\left(x-\gamma \left(t\right)\right)\phantom{\rule{0.3em}{0ex}}\frac{dt}{t},$

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 ${L}^{p}\left(ℝ\right)×{L}^{q}\left(ℝ\right)\to {L}^{r}\left(ℝ\right)$ boundedness property holds for the corresponding (sub)bilinear maximal function ${M}_{\gamma }\left(f,g\right)$ along a convex curve $\gamma$

${M}_{\gamma }\left(f,g\right)\left(x\right):=\underset{𝜀>0}{sup}\frac{1}{2𝜀}{\int }_{-𝜀}^{𝜀}|f\left(x-t\right)g\left(x-\gamma \left(t\right)\right)|\phantom{\rule{0.3em}{0ex}}dt.$

##### Keywords
bilinear Hilbert transform, (sub)bilinear maximal function, convex curve, time frequency analysis
Primary: 42B20
Secondary: 47B38
##### Milestones
Received: 7 June 2020
Revised: 2 September 2020
Accepted: 4 September 2020
Published: 8 March 2021
##### Authors
 Junfeng Li School of Mathematical Sciences Dalian University of Technology Dalian China Haixia Yu Department of Mathematics Shantou University Shantou China