Seminario de análisis y aplicaciones

Viernes, 7 de marzo de 2025

11:30 - 12:30, Aula 420, departamento de Matemáticas

Emiel Lorist
TU Delft

Quantifying the overlap of a collection of sets

Resumen:
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, H¨ anninen 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–H ¨ anninen–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.
ICMAT CSIC-UAM-UC3M-UCM
Departamento de Matemáticas. U.A.M.
Proyecto CEX2019-000904-S financiado por MCIN/ AEI/10.13039/501100011033.