Flag multiresolutions: T1, paraproducts and commutators

@misc{foster2026flag,
  title = {{Flag multiresolutions: T1, paraproducts and commutators}},
  author = {Ben Foster and Kangwei Li and Henri Martikainen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38092},
  eprint = {2609.38092},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38092},
}

TY  - UNPB
TI  - Flag multiresolutions: T1, paraproducts and commutators
AU  - Ben Foster
AU  - Kangwei Li
AU  - Henri Martikainen
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38092
UR  - https://arxiv.org/abs/2609.38092
AB  - 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 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.
ER  -