The multilinear Hilbert transform boundedness conjecture

For a natural number $k$ and test functions $f_1,\dots,f_k:\mathbb{R} \to \mathbb{C}$, define the $k$-linear Hilbert transform by

$$H_k(f_1,\dots,f_k)(x):=\operatorname{p.v.}\int_{\mathbb{R}}f_1(x+t)\dots f_k(x+kt)\frac{dt}{t}.$$

Multilinear Hilbert transform boundedness conjecture. For $k\geq 1$ and exponents $1<p_1,\dots,p_k,p<\infty$ satisfying $\frac{1}{p}=\frac{1}{p_1}+\dots+\frac{1}{p_k}$, the truncated transforms $H_{k,r,R}$ should satisfy

$$\lVert H_{k,r,R}(f_1,\dots,f_k)\rVert_{L^p(\mathbb{R})}\leq C_{k,p,p_1,\dots,p_k}\prod_{i=1}^k\lVert f_i\rVert_{L^{p_i}(\mathbb{R})},$$

for $f_i\in L^{p_i}(\mathbb{R})$ and $0<r<R$, with $C_{k,p,p_1,\dots,p_k}$ independent of $r,R$; consequently, the same bound holds for $H_k$ on test functions. This conjecture extends the classical and bilinear Hilbert transform estimates. It is proven for $k=1,2$, but remains open for larger $k$.