Calderón–Zygmund Decomposition for Sobolev Functions

Calderón–Zygmund decompositions are a classical tool in harmonic analysis. The idea is to decompose a function on all scales into good and bad functions, where good means small in a certain sense and the bad functions can be large but are at least localized in a good sense. The classical Calderón–Zygmund decomposition happens at the level of Lebesgue spaces. Here, we show a decomposition in \(\mathbb {W}^{1,p}_{\mathbb {D}}\) -spaces.