Some remarks on the n-linear Hilbert transform for n≥ 4

2012/09/27 by Camil Muscalu, Muscalu, Camil
Mathematics · #42 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Partial Differential Equations #math.CA #msc:42

paper · pdf · doi:10.48550/arxiv.1209.6391

15 pages, one figure

openalex publication_date 2012/09/27 · arxiv created 2013/01/28 · arxiv updated 2013/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for every integer n≥ 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≥ 4,these immediately imply that for any 1< p₁, ..., p_n ≤ ∞ and 0

Citations

• Calderón commutators and the Cauchy integral on Lipschitz curves revisited II. Cauchy integral and generalizations
• Calderón commutators and the Cauchy integral on Lipschitz curves revisited I. First commutator and generalizations
• Calderón commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions

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