Some remarks on the $n$-linear Hilbert transform for $n\geq 4$

TL;DR

This paper demonstrates that for n ≥ 4, certain n-linear Hilbert transform-type operators do not satisfy expected L^p estimates, contrasting with the bi-linear case where such estimates hold.

Contribution

It proves the failure of L^p estimates for a class of n-linear Hilbert transform operators when n ≥ 4, extending previous results and providing counterexamples.

Findings

n-linear operators with specific symbols lack L^p estimates for n ≥ 4

Counterexamples involve symbols singular along subspaces of maximal dimension

Contrasts with the bi-linear case where estimates are known to hold

Abstract

We prove that for every integer $n\geq 4$, the $n$-linear operator whose symbol is given by a product of two generic symbols of $n$-linear Hilbert transform type, does not satisfy any $L^p$ estimates similar to those in H\"{o}lder inequality. Then, we extend this result to multi-linear operators whose symbols are given by a product of an arbitrary number of generic symbols of $n$-linear Hilbert transform kind. As a consequence, under the same assumption $n\geq 4$,these immediately imply that for any $1< p_1, ..., p_n \leq \infty$ and $0<p<\infty$ with $1/p_1 + ... + 1/p_n = 1/p$, there exist non-degenerate subspaces $\Gamma\subseteq \mathbb{R}^n$ of maximal dimension $n-1$, and Mikhlin symbols $m$ singular along $\Gamma$, for which the associated $n$-linear multiplier operators $T_m$ do not map $L^{p_1}\times ... \times L^{p_n}$ into $L^p$. These counterexamples are in sharp contrast with the bi-linear case, where similar operators are known to satisfy many such $L^p$ estimates.