Remark on dyadic pointwise domination and median oscillation decomposition

In this note, we do the following: a) 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: $$\sum_{S\in\mathcal{S}} \langle f \rangle^\mu_{S^{(k)}} 1_S\lesssim (k+1) \sum_{S'\in\mathcal{S}'} \langle f \rangle^\mu_{S'} 1_{S'}.$$ b) By following the analogue between median and mean oscillation, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures: $$\lvert f-m(f,\hat{S_0})\rvert 1_{S_0}\lesssim \sum_{S\in\mathcal{S}} (\omega_\lambda(f;S)+\lvert m(f,S)-m(f,\hat{S}) \rvert )1_S.$$ This 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 complexity zero.