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Optimization algorithms for Carleson and sparse collections of sets

Eline A. Honig, Emiel Lorist

TL;DR

The paper addresses turning the Carleson condition into a constructive, algorithmic framework. It develops a strongly polynomial algorithm to compute the optimal Carleson constant $\Lambda_{\mathcal{F}}$ for a finite collection $\mathcal{F}$ by formulating a submodular minimization problem using $f_{\Lambda}(\mathcal{A})=\Lambda\mu(\bigcup_{Q\in\mathcal{A}}Q)-\sum_{Q\in\mathcal{A}}\mu(Q)$ and iteratively refining $\Lambda$, with a polynomial-time bound dependent on a union-measure oracle. It then provides a constructive sparsity proof by recasting the duality between the Carleson condition and sparseness as a max-flow/min-cut problem on a carefully built weighted directed graph, allowing the explicit construction of $\varphi_Q$ (and, in the non-atomic case, $E_Q$) that realize $\frac{1}{\Lambda}$-sparseness. The approach achieves optimal constants in the finite setting and extends to general measure spaces, offering a practical, algorithmic bridge between Carleson-type control and sparse estimates in harmonic analysis. Overall, it delivers a fully constructive alternative to historical non-constructive proofs and improves the tractability of verifying and exploiting Carleson and sparse structures.

Abstract

Carleson and sparse collections of sets play a central role in dyadic harmonic analysis. We employ methods from optimization theory to study such collections. First, we present a strongly polynomial algorithm to compute the Carleson constant of a collection of sets, improving on the recent approximation algorithm of Rey. Our algorithm is based on submodular function minimization. Second, we provide an algorithm showing that any Carleson collection is sparse, achieving optimal dependence of the respective constants and thus providing a constructive proof of a result of Hänninen. Our key insight is a reformulation of the duality between the Carleson condition and sparseness in terms of the duality between the maximum flow and the minimum cut in a weighted directed graph.

Optimization algorithms for Carleson and sparse collections of sets

TL;DR

The paper addresses turning the Carleson condition into a constructive, algorithmic framework. It develops a strongly polynomial algorithm to compute the optimal Carleson constant for a finite collection by formulating a submodular minimization problem using and iteratively refining , with a polynomial-time bound dependent on a union-measure oracle. It then provides a constructive sparsity proof by recasting the duality between the Carleson condition and sparseness as a max-flow/min-cut problem on a carefully built weighted directed graph, allowing the explicit construction of (and, in the non-atomic case, ) that realize -sparseness. The approach achieves optimal constants in the finite setting and extends to general measure spaces, offering a practical, algorithmic bridge between Carleson-type control and sparse estimates in harmonic analysis. Overall, it delivers a fully constructive alternative to historical non-constructive proofs and improves the tractability of verifying and exploiting Carleson and sparse structures.

Abstract

Carleson and sparse collections of sets play a central role in dyadic harmonic analysis. We employ methods from optimization theory to study such collections. First, we present a strongly polynomial algorithm to compute the Carleson constant of a collection of sets, improving on the recent approximation algorithm of Rey. Our algorithm is based on submodular function minimization. Second, we provide an algorithm showing that any Carleson collection is sparse, achieving optimal dependence of the respective constants and thus providing a constructive proof of a result of Hänninen. Our key insight is a reformulation of the duality between the Carleson condition and sparseness in terms of the duality between the maximum flow and the minimum cut in a weighted directed graph.
Paper Structure (7 sections, 6 theorems, 40 equations, 2 figures, 3 algorithms)

This paper contains 7 sections, 6 theorems, 40 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The optimal Carleson constant
  3. The Carleson condition implies sparseness
  4. Partitioning O
  5. A weighted directed graph based on F
  6. The maximum flow problem
  7. Constructing the phiQ and EQ

Key Result

Lemma 2.2

The function $f_\Lambda\colon \mathcal{P}(\mathcal{F}) \to \mathbb{R}$ is submodular for $\Lambda \geq 1$.

Figures (2)

  • Figure 1: The collection $\mathcal{F} = \cbrace{Q_1,Q_2,Q_3}$ and the sets $E_{\mathcal{A}_j}$ for $j\in\cbrace{1,\ldots,4}$ with \mathcal{A}_1:= \{Q_3\},\mathcal{A}_3:= \{Q_2, Q_3\},\mathcal{A}_2:= \{Q_1, Q_3\},\mathcal{A}_4:= \{Q_1, Q_2, Q_3\}.
  • Figure 2: The graph $G$ based on the collection $\mathcal{F}$ in Figure \ref{['fig:rectangles']}.

Theorems & Definitions (19)

  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 9 more