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: $$\sum _{S\in \mathcal{S}}⟨f{⟩}_{S^{(k)}}^{\mu }{1}_S\lesssim (k+1)\sum _{S^{\mathrm{\prime }}\in {\mathcal{S}}^{\mathrm{\prime }}}⟨f{⟩}_{S^{\mathrm{\prime }}}^{\mu }{1}_{S^{\mathrm{\prime }}}\mathrm{.}$$ b) By following the analogue between median and mean oscillation, we extend Lerner’s local median oscillation decomposition to arbitrary (possibly non-doubling) measures: $$\mid f-m(f,\widehat{S_0})\mid 1_{S_0}\lesssim \sum _{S\in \mathcal{S}}({\omega }_{\lambda }(f;S)+\mid m(f,S)-m(f,\widehat{S})\mid )1_S\mathrm{.}$$ 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.