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Multi-parameter singular integral operators and representation theorem. (English) Zbl 1366.42016
The author formulates a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. The main result obtained for this class of operators is a multi-parameter representation theorem stating that a generic operator in this class can be represented as an average of sums of dyadic shifts, which implies a new multi-parameter \(T1\) theorem as a byproduct. This extends the representation principles of Hytönen’s and Martikainen’s to the multi-parameter setting. Furthermore, equivalence between the studied class and Journé’s class of multi-parameter operators is established, whose proof requires the multiparameter \(T1\) theorem.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30H35 BMO-spaces
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