REMARK ON MEDIAN OSCILLATION DECOMPOSITION AND DYADIC POINT WISE DOMINATION

In this note, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures. In the light of the analogy between median and mean oscillation, our extension can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO. As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John-Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of zero complexity. Furthermore, by using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity.Peer reviewe