On a Local Mean Oscillation Decomposition

In this note we generate two local median oscillation decompositions of an arbitrary measurable function and discuss some applications to Calder'{o}n-Zygmund singular integral operators $T$. These applications rely on the inequality $M_{0,s}^{\mathrm{♯}}(Tf)(x)\le cMf(x)$, and we complete the results given here with a discussion of a local version of this estimate.