The Bellman Function for Level Sets of Sparse Operators

TL;DR

This paper determines the exact weak-$(1,1)$ norm of sparse operators by constructing a Bellman function that maximizes their level sets, providing precise bounds for these operators in harmonic analysis.

Contribution

The paper introduces a Bellman function approach to exactly compute the weak-$(1,1)$ bounds of sparse operators, advancing understanding of their boundedness properties.

Findings

Exact weak-$(1,1)$ norm of sparse operators determined

Bellman function constructed for level set maximization

Provides precise bounds for sparse operator boundedness

Abstract

We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the exact weak-$(1,1)$ norm of these sparse operators.