A constructive algorithm to prove the equivalence of the Carleson and sparse condition

Abstract

Let (S, Σ, μ) be a divisible measure space. Let F be a collection of subsets of S that are in Σ. For some applications it can be useful to describe the overlap between the sets in F. The sparse and Carleson constant both describe this overlap in a different way. The closer both of these constants are to 1, the closer the sets in F are to being pairwise disjoint. It has been shown that the sparse and Carleson condition are actually equivalent: we always have that F is Λ-Carleson if and only if F is Λ^-1-sparse. Proving that a Λ^-1-sparse collection is Λ-Carleson is quite simple, but proving that every Λ-Carleson collection also is Λ^-1-sparse turns out to be much harder. Previous proofs of the fact that Λ-Carleson are Λ^-1-sparse, such as the one by Hänninen [14] and Rey [25], have all relied on difficult theory. There is also no known method to exactly find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition. Rey is able to approximate these sets, but his algorithm has a logarithmic loss that can only be removed when imposing geometric restrictions.

In this paper I will give a proof of the equivalence of the sparse and Carleson condition for any finite collection F that relies only on basic set and optimisation theory. This proof can be extended to infinite collections F with only a minimal restriction. Asides from proving the equivalence, I also describe an algorithm that can find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition if F is finite and Carleson with respect to a divisible measure μ. Finally, I will describe an algorithm that we can use to find the Carleson constant of a finite collection F if it is unknown.