Operator-valued dyadic harmonic analysis beyond doubling measures
Jos\'e M. Conde-Alonso, Luis Daniel L\'opez-S\'anchez
TL;DR
This paper characterizes the weak-type (1,1) bounds for Haar shift operators in the operator-valued setting with respect to general measures, using a noncommutative Calderón-Zygmund decomposition.
Contribution
It provides a complete characterization of weak-type (1,1) bounds for Haar shift operators beyond doubling measures in the operator-valued context.
Findings
Characterization of weak-type (1,1) for Haar shift operators
Development of a noncommutative Calderón-Zygmund decomposition for arbitrary measures
Extension of harmonic analysis tools to operator-valued measures
Abstract
We obtain a complete characterization of the weak-type for Haar shift operators in terms of generalized Haar systems adapted to a Borel measure in the operator-valued setting. The main technical tool in our method is a noncommutative Calder\'on-Zygmund decomposition valid for arbitrary Borel measures.
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