Revista Matemática Iberoamericana

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Volume 33, Issue 1, 2017, pp. 325–350
DOI: 10.4171/RMI/939

Published online: 2017-02-22

Multi-parameter singular integral operators and representation theorem

Yumeng Ou

We formulate a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. The main result we prove for this class of operators is a multi-parameter representation theorem stating that a generic operator in our class can be represented as an average of sums of dyadic shifts, which implies a new multi-parameter $T$1 theorem as a byproduct. This extends the representation principles of Hytönen’s and Martikainen’s to the multi-parameter setting. Furthermore, equivalence between ours and Journé’s class of multi-parameter operators is established, whose proof requires the multiparameter $T$1 theorem.

Keywords: Multi-parameter singular integral operators, representation theorem, dyadic shift, product BMO

Ou Yumeng: Multi-parameter singular integral operators and representation theorem. Rev. Mat. Iberoam. 33 (2017), 325-350. doi: 10.4171/RMI/939