Comptes Rendus
Functional Analysis
Counterexamples for multi-parameter weighted paraproducts
Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 529-534.

We build the plethora of counterexamples to bi-parameter two weight embedding theorems. Two weight one parameter embedding results (which is the same as results of boundedness of two weight classical paraproducts, or two weight Carleson embedding theorems) are well known since the works of Sawyer in the 80’s. Bi-parameter case was considered by S. Y. A. Chang and R. Fefferman but only when underlying measure is Lebesgue measure. The embedding of holomorphic functions on bi-disc requires general input measure. In [9] we classified such embeddings if the output measure has tensor structure. In this note we give examples that without tensor structure requirement all results break down.

Dans le présent article, nous construisons une pléthore de contre-exemples aux théorèmes de plongements à deux poids et à deux paramètres. Les résultats de plongement à un paramètre et à deux poids (qui sont la même chose que les résultats de paraproduits bornés classiques à deux poids) sont bien connus depuis les travaux de Sawyer dans les années 80. S. Y. A. Chang et R. Fefferman ont examiné le cas des deux paramètres, mais uniquement lorsque la mesure sous-jacente est la mesure de Lebesgue. Le plongement de fonctions holomorphes sur le bi-disque nécessite une mesure générale en entrée. Dans [9], nous avons classé ces plongements lorsque la mesure obtenu en sortie a une structure tensorielle. Dans cette note, nous donnons des contre-exemples d’après lesquels tous les résultats deviennent faux en l’absence d’hypothèse d’une structure tensorielle.

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DOI: 10.5802/crmath.52

Pavel Mozolyako 1; Georgios Psaromiligkos 1; Alexander Volberg 1

1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg. Counterexamples for multi-parameter weighted paraproducts. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 529-534. doi : 10.5802/crmath.52. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.52/

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