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

TL;DR

This paper studies the level-set structure of height functions arising from sparse dyadic operators by applying the Bellman function method. The authors define the exact Bellman function G_C(A,\\lambda) as the supremum of level-set measures over all binary Carleson sequences with a fixed Carleson constraint and derive a least-supersolution framework to identify it. They construct an explicit candidate \\mathbf{\\widetilde{G}}_C via Jump Inequality and A-concavity, verify it for various C (including non-integer values), and prove it satisfies the defining properties, thereby obtaining the exact Bellman function. The resulting characterization yields sharp weak-(1,1) level-set bounds for localized sparse operators on constants and advances the understanding of sparse domination in dyadic harmonic analysis.

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 $\mathcal{A}_α1\!\!1$, where $\mathcal{A}_α$ is the (localized) sparse operator associated with a binary Carleson sequence.

Figures (6)

• Figure 1: The first five dyadic generations in $\mathcal{D}(I)$.
• Figure 5: Constructing the candidate for $C=1$.
• Figure 6: Constructing the candidate for $C=2$.
• Figure 7: The candidate for $C=2$.
• Figure 8: Constructing the candidate for non-integer $C$.

Theorems & Definitions (23)

• Example 1
• Example 2
• Example 3
• Definition 1
• Example 4
• Remark 1
• Definition 2
• Proposition 1
• proof
• Remark 2