Cancellation for the multilinear Hilbert transform

Abstract

For any natural number k, consider the k-linear Hilbert transform

$$\begin{aligned} H_k( f_1,\dots ,f_k )(x) := {\text {p.v.}} \int _\mathbb {R}f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned}$$

for test functions $f_1,\dots ,f_k: \mathbb {R}\rightarrow \mathbb {C}$. It is conjectured that $H_k$ maps $L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})$ whenever $1 < p_1,\dots ,p_k,p < \infty$ and $\frac{1}{p} = \frac{1}{p_1} + \dots + \frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger k. In this paper, we consider the truncated operators

$$\begin{aligned} H_{k,r,R}( f_1,\dots ,f_k )(x) := \int _{r \leqslant |t| \leqslant R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned}$$

for $R > r > 0$. The above conjecture is equivalent to the uniform boundedness of $\Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})}$ in r, R, whereas the Minkowski and Hölder inequalities give the trivial upper bound of $2 \log \frac{R}{r}$ for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on $\Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})}$ slightly to $o( \log \frac{R}{r} )$ in the limit $\frac{R}{r} \rightarrow \infty$ for any admissible choice of k and $p_1,\dots ,p_k,p$. This establishes some cancellation in the k-linear Hilbert transform $H_k$, but not enough to establish its boundedness in $L^p$ spaces.