Carleson families of cubes related to porous sets

TL;DR

This paper refines Carleson packing and sparseness conditions for dyadic covers of porous sets and investigates when such families generate the original porous set.

Contribution

It introduces refined conditions for Carleson families related to porous sets and explores the inverse problem of generating porous sets from these families.

Findings

Refined Carleson packing condition for dyadic covers of porous sets

Characterization of when a Carleson family generates a porous set

Analysis of the inverse problem linking Carleson families to porous sets

Abstract

Given a porous set $E\in \mathbb{R}^d$ and a dyadic lattice $\mathcal{D}$, we refine the Carleson packing condition and the sparseness property for the dyadic cover $\mathcal{D}_E=\{Q \in \mathcal{D}: \: Q \cap E \neq \varnothing\}$. We study the inverse problem, when a Carleson family $\mathcal{S} \subset \mathcal{D}$ generates the porous set $E$ such that $\mathcal{S} \subset \mathcal{D}_E$.