Simplex Calderón–Zygmund form conjecture

Let $\mathbb{V}=\mathbb{R}^{\mathscr{d}}$, and let $F_{0},\dotsc,F_{n}$ be functions of $n$ variables. For $x=(x_{0},\dotsc,x_{n})\in\mathbb{V}^{n+1}$, write $x_{(i)}=(x_{0},\dotsc,x_{i-1},x_{i+1},\dotsc,x_{n})$. Define

$$\Lambda_{K}(F_{0},\dotsc,F_{n})=\int_{\mathbb{V}^{n+1}}\prod_{i=0}^{n}F_{i}(x_{(i)})K\left(\sum_{i=0}^{n}x_{i}\right)\,\mathrm{d}x.$$

Suppose that $K$ is a sufficiently smooth Calderón–Zygmund kernel.

Simplex Calderón–Zygmund form conjecture. Then

$$\left|\Lambda_{K}(F_{0},\dotsc,F_{n})\right|\lesssim\prod_{i=0}^{n}\|F_{i}\|_{p_{i}},\qquad \sum_{i=0}^{n}\frac{1}{p_{i}}=1,\qquad n<p_{i}<\infty.$$

This is presented as the most optimistic conjecture for multidimensional modulation-invariant operators consistent with the negative results discussed in the paper. Its status is open in the stated generality.