Analysis & Applications Seminar
Quantifying the overlap of a collection of sets
Speaker: Emiel Lorist (TU Delft)Date: Friday, 07 March 2025 - 11:30Place: Aula 420, Módulo 17, Departamento de Matemáticas, UAM
Abstract:
Quantifying the overlap or almost pairwise disjointness among a collection of sets in a measure space plays an important role in harmonic analysis. Two key notions for quantifying this are sparseness and the so-called Carleson condition. For dyadic cubes, these notions were shown to be equivalent by Verbitsky, drawing on ideas from Dor. More recently, Hänninen generalized this result to general sets in Rd. This equivalence is very useful in harmonic analysis: the Carleson condition is often straightforward to verify, while sparseness proves incredibly useful for obtaining (sharp) estimates.
The remarkably elegant Dor–Hänninen–Verbitsky proof is based on duality and the Hahn–Banach separation theorem and is therefore non-constructive, even for finite collections. In this talk, we will reformulate the equivalence between sparseness and the Carleson condition as a graph theory problem—more specifically, as a max-flow problem on a weighted directed graph. Borrowing some methods from optimization theory, we will deduce a constructive proof of the equivalence between the Carleson condition and sparseness.
This talk is based on joint work with Eline Honig.
