Operator-free sparse domination

Abstract

We obtain a sparse domination principle for an arbitrary family of functions 𝑓 (𝑥, 𝑄), where 𝑥 ∈ R𝑛 and Q is a cube in R𝑛. When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré–Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vectorvalued square functions.