An L^{3} bound for the trilinear Hilbert transform

Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L^3(\mathbb R)^3 to L^1(\mathbb R). This resolves the L³ exponent case of the standard conjecture for slopes 1, 2, 3.

Manuscript

An L³ bound for the trilinear Hilbert transform

October 5, 2026 · source · cite as \cite{OAI:An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026}

We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R) to L^1(\mathbb R), resolving this case of the trilinear Hilbert transform conjecture.

Result 086 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.