Abstract
AbstractWe obtain a sparse domination principle for an arbitrary family of functions$f(x,Q)$, where$x\in {\mathbb R}^{n}$andQis a cube in${\mathbb R}^{n}$. 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 vector-valued square functions.
Highlights
Let us consider a particular line of research in this direction, for which the starting point is the so-called local mean oscillation estimate
The local mean oscillation estimate can be regarded as the first operator-free sparse domination result, but its main application was to operators
For some cd ≥ 1 and a Borel measure μ that satisfies the doubling ball property μ B(s, 2ρ) ≤ cμ μ B(s, ρ), s ∈ S, ρ > 0 for some cμ ≥ 1. It was shown by Anderson and Vagharshakyan [1] that the sparse domination principle based on the median oscillation estimate (1.1) could be generalised from the Euclidean space Rn equipped with the Lebesgue measure to a space of homogeneous type
Summary
Sparse domination is a recent technique allowing one to estimate (in norm, pointwise or dually) many operators in harmonic analysis by simple expressions of the form f p,Q χQ,. Starting from [30], the local mean oscillation estimate (1.1) has not played a role in the obtained sparse domination results This development can be viewed as an evolution from sparse domination for arbitrary functions (expressed in (1.1)) to sparse domination for operators.
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