The height function of a sparse collection: a Bellman function approach

2025/09/30 by Aggarwal, Shivam, Hernandez, Samuel, Fay, Irina Holmes +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.00247

Abstract

Sparse operators have emerged as a powerful method to extract sharp constants in harmonic analysis inequalities, for example in the context of bounding singular integral operators. We investigate the level sets of height functions for sparse collections, or, in other words, weak-type (1,1) inequalities for sparse operators applied to constant functions. We use another notable method from dyadic harmonic analysis, also famous for its ability to produce sharp constants, the Bellman function method. Specifically, we find the exact Bellman function maximizing level sets of A_α1 1, where A_α is the (localized) sparse operator associated with a binary Carleson sequence.

Citations

• Sharp restricted weak-type estimates for sparse operators
• Paraproducts, Bloom BMO and Sparse BMO Functions
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Intuitive dyadic calculus: the basics
• Bellman functions and two weight inequalities for Haar multipliers

Related

• The Bellman Function for Level Sets of Sparse Operators
• Sharp restricted weak-type estimates for sparse operators
• Properties of the Bellman function related to the Carleson Imbedding theorem for the dyadic maximal operator
• A Bellman function counterexample to the A₁ conjecture: the blow-up of the weak norm estimates of weighted singular operators
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution