Cancellation for the multilinear Hilbert transform

Abstract

For any natural number $k$, consider the $k$-linear Hilbert transform $$H_k(f_1,\dots ,f_k)(x):=p.v.{\int }_{\mathbf{R}}{f}_1(x+t)\dots f_k(x+kt)\frac{dt}{t}$$ for test functions $f_1,\dots ,f_k:\mathbf{R}\to \mathbf{C}$. It is conjectured that $H_k$ maps $L^{p_1}(\mathbf{R})\times \cdots \times L^{p_k}(\mathbf{R})\to L^p(\mathbf{R})$ whenever $1<p_1,\dots ,p_k,p<\infty$ and $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$. In this paper, we consider the truncated operators $$H_{k,r,R}(f_1,\dots ,f_k)(x):={\int }_{r\le |t|\le R}{f}_1(x+t)\dots f_k(x+kt)\frac{dt}{t}$$ for $R>r>0$. The above conjecture is equivalent to the uniform boundedness of $\Vert H_{k,r,R}{\Vert }_{L^{p_1}(\mathbf{R})\times \cdots \times L^{p_k}(\mathbf{R})\to L^p(\mathbf{R})}$ 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 $\Vert H_{k,r,R}{\Vert }_{L^{p_1}(\mathbf{R})\times \cdots \times L^{p_k}(\mathbf{R})\to L^p(\mathbf{R})}$ slightly to $o(\log \frac{R}{r})$ in the limit $\frac{R}{r}\to \infty$ 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.

17 pages, no figures. An error pointed out to the author by Pavel Zorin-Kranich has been corrected

Cited by in corpus (1)

• Modulation invariant operators