Multi-parameter singular integral operators and representation theorem

Abstract

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.