The Bellman Function for Level Sets of Sparse Operators

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.

Figures (7)

• Figure 1: The surface $z = \mathbb{M}_r(\omega, A)$ (shown here with $r=0.8$).
• Figure 2: Regions partitioning the $\Omega_\mathbb{M}$-domain.
• Figure 3: Graph of $\mathbb{M}_r$ and the "Bellman envelope" surface $z=\Phi_r(\omega, A)$.
• Figure 4: An accordion of Bellman functions.
• Figure 5: A sparse collection $\alpha$, shown to the left spanning five dyadic generations within $\mathcal{D}(I)$, and shown to the right with its three sparse generations.

Theorems & Definitions (18)

• Proposition 1
• Corollary 1
• Theorem 1
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3: Least Supersolution Property of $\mathbb{B}$
• proof
• Lemma 2