A decomposition of Calder\'on--Zygmund type and some observations on differentiation of integrals on the infinite-dimensional torus

In this note we will show a Calder'on–Zygmund decomposition associated with a function $f\in L^1({\mathbb{T}}^{\omega })$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.