Flag multiresolutions: T1, paraproducts and commutators

Abstract: We develop a rich theory of dyadic-probabilistic multiresolution methods in the context of flag type singular integrals. We apply these methods to prove a full $T1$ theorem, even a representation theorem, for bi-parameter flag singular integrals. This is the first time a complete $T1$ theorem appears in a multi-parameter setting with an entangled nature; even the related recent advances with Zygmund dilations have been in the cancellative $T1 = 0$ or convolution situations. This theorem comes with a non-trivial necessity argument related to the flag type product $\mathrm{BMO}$ assumption on $T1$. We also present a version of our theorem in the cancellative tri-parameter flag setting showing that our methods are perfectly adaptable to general multi-parameter flag settings. Importantly, our $T1$ representation theorem implies weighted estimates for every flag $A_p$ weight, showing that our dyadic multiresolutions carefully retain the flag invariance of the original singular integrals. Our multiresolution methods also yield flag style paraproduct decompositions of products of functions and lead to a general and efficient method for proving commutator estimates in various flag settings.