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 Unboundedness of the trilinear Hilbert transform under the critical index J. Korean Math. Soc.Published online May 14, 2021 Yasuo Komori-Furuya Tokai University Abstract : Demeter (2008) and Kuk and Lee (2012) proved the unboundedness of the trilinear Hilbert transforms $H_{a,b,c}$ under the critical index $1/2$ for some parameters $a,b$ and $c$. We show the unboundedness of $H_{a,b,c}$ for any parameters. Keywords : bilinear Hilbert transform, trilinear Hilbert transform, trilinear fractional integral operator MSC numbers : 42B20 Full-Text :