The trilinear Hilbert transform conjecture at the L3 x L3 x L3 to L1 point (slopes 1, 2, 3)

Theorem 1.1: there is a finite $C$ with $\|T(f_1,f_2,f_3)\|_{L^1}\le C\prod_{j}\|f_j\|_{L^3}$ for Schwartz functions, so $T$ extends uniquely to a bounded trilinear map $L^3\times L^3\times L^3\to L^1$. The proof combines the Leng-Sah-Sawhney inverse theorem with new compression, curvature-localization and sparse counting lemmas that sum the scales without loss. It concerns only the fixed slopes 1, 2, 3 and the single exponent point $(3,3,3;1)$: the rest of the conjectured range, other slopes, maximal or truncation-uniform versions and higher-degree multilinear Hilbert transforms are not treated, as the manuscript itself says.

Theorem 1.1: there is a finite $C$ with $\|T(f_1,f_2,f_3)\|_{L^1}\le C\prod_{j}\|f_j\|_{L^3}$ for Schwartz functions, so $T$ extends uniquely to a bounded trilinear map $L^3\times L^3\times L^3\to L^1$. The proof combines the Leng-Sah-Sawhney inverse theorem with new compression, curvature-localization and sparse counting lemmas that sum the scales without loss. It concerns only the fixed slopes 1, 2, 3 and the single exponent point $(3,3,3;1)$: the rest of the conjectured range, other slopes, maximal or truncation-uniform versions and higher-degree multilinear Hilbert transforms are not treated, as the manuscript itself says.