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.

Locations

• Collectanea mathematica - View
• arXiv (Cornell University) - View - PDF
• Collectanea mathematica - View
• arXiv (Cornell University) - View - PDF
• Collectanea mathematica - View
• arXiv (Cornell University) - View - PDF