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- W2330956782 abstract "We prove that for every integer $ngeq 4$, 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 $ngeq 4$,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 $Gammasubseteq mathbb{R}^n$ 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." @default.
- W2330956782 created "2016-06-24" @default.
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- W2330956782 date "2014-01-01" @default.
- W2330956782 modified "2023-09-28" @default.
- W2330956782 title "Some remarks on the $n$-linear Hilbert transform for $n geq 4$" @default.
- W2330956782 doi "https://doi.org/10.4310/mrl.2014.v21.n5.a11" @default.
- W2330956782 hasPublicationYear "2014" @default.
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