  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.
