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

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 Hölder 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 $Γ\subseteq \mathbb{R}^n$ of maximal dimension $n-1$, and Mikhlin symbols $m$ singular along $Γ$, 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.