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I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from this paper of Ben Green and myself) to obtain a slight amount of progress towards the open problem of obtaining ${L^p}$ bounds for the trilinear and higher Hilbert transforms (as discussed in this previous blog post). For instance, the trilinear Hilbert transform

$\displaystyle H_3( f_1, f_2, f_3 )(x) := p.v. \int_{\bf R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}$

is not known to be bounded for any ${L^{p_1}({\bf R}) \times L^{p_2}({\bf R}) \times L^{p_3}({\bf R})}$ to ${L^p({\bf R})}$, although it is conjectured to do so when ${1/p =1/p_1 +1/p_2+1/p_3}$ and ${1 < p_1,p_2,p_3,p < \infty}$. (For ${p}$ well below ${1}$, one can use additive combinatorics constructions to demonstrate unboundedness; see this paper of Demeter.) One can approach this problem by considering the truncated trilinear Hilbert transforms

$\displaystyle H_{3,r,R}( f_1, f_2, f_3 )(x) := \int_{r \leq |t| \leq R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}$

for ${0 < r < R}$. It is not difficult to show that the boundedness of ${H_3}$ is equivalent to the boundedness of ${H_{3,r,R}}$ with bounds that are uniform in ${R}$ and ${r}$. On the other hand, from Minkowski’s inequality and Hölder’s inequality one can easily obtain the non-uniform bound of ${2 \log \frac{R}{r}}$ for ${H_{3,r,R}}$. The main result of this paper is a slight improvement of this trivial bound to ${o( \log \frac{R}{r})}$ as ${R/r \rightarrow \infty}$. Roughly speaking, the way this gain is established is as follows. First there are some standard time-frequency type reductions to reduce to the task of obtaining some non-trivial cancellation on a single “tree”. Using a “generalised von Neumann theorem”, we show that such cancellation will happen if (a discretised version of) one or more of the functions ${f_1,f_2,f_3}$ (or a dual function ${f_0}$ that it is convenient to test against) is small in the Gowers ${U^3}$ norm. However, the arithmetic regularity lemma alluded to earlier allows one to represent an arbitrary function ${f_i}$, up to a small error, as the sum of such a “Gowers uniform” function, plus a structured function (or more precisely, an irrational virtual nilsequence). This effectively reduces the problem to that of establishing some cancellation in a single tree in the case when all functions ${f_0,f_1,f_2,f_3}$ involved are irrational virtual nilsequences. At this point, the contribution of each component of the tree can be estimated using the “counting lemma” from my paper with Ben. The main term in the asymptotics is a certain integral over a nilmanifold, but because the kernel ${\frac{dt}{t}}$ in the trilinear Hilbert transform is odd, it turns out that this integral vanishes, giving the required cancellation.

The same argument works for higher order Hilbert transforms (and one can also replace the coefficients in these transforms with other rational constants). However, because the quantitative bounds in the arithmetic regularity and counting lemmas are so poor, it does not seem likely that one can use these methods to remove the logarithmic growth in ${R/r}$ entirely, and some additional ideas will be needed to resolve the full conjecture.