Sharp restricted weak-type estimates for sparse operators
Irina Holmes Fay, Guillermo Rey, Kristina Ana Škreb
Abstract
We find the exact Bellman function associated to the level-sets of sparse operators acting on characteristic functions.
Table of Contents
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Sharp restricted weak-type estimates for sparse operators
Abstract
We find the exact Bellman function associated to the level-sets of sparse operators acting on characteristic functions.
Paper Structure (7 sections, 20 theorems, 194 equations, 19 figures)
This paper contains 7 sections, 20 theorems, 194 equations, 19 figures.
Table of Contents
Key Result
Corollary 1
If $\mathcal{S}$ is sparse, $E \subseteq [0,1)$, $|E| = 2^{-n}$, and $\lambda \in \mathbb{N} - 2^{-n}$ is at least $2$, then and equality is attained for certain pairs $(E, \mathcal{S})$.
Figures (19)
- Figure 1: $(x, A)$-domain regions for $0<\lambda\leq 1$
- Figure 2: Diagram of $\mathbf{f}$. Here $C$ and $D$ represent the two possible positions of the point $(x_\ast, \lambda_\ast)$.
- Figure 3: Concatenation of $\alpha$ and $\beta$, assigning $\gamma$ to the top interval. Here $I = [0,1)$ and the suffix $+$ or $-$ denotes the right or left child respectively.
- Figure 4: Inductive application of \ref{['construction:x=1']} to obtain \ref{['construction:x=1:full']}.
- Figure 5: Lower bounds for $\mathbb{F}$. The green line is given by Proposition \ref{['construction:first_nontrivial']}, and the red is obtained by applying \ref{['construction:C']} and \ref{['bellman:jump']}.
- ...and 14 more figures
Theorems & Definitions (37)
- Corollary
- Theorem A
- Theorem 2.1
- proof
- Definition 2.1
- Definition 2.2
- Theorem 2.2
- proof
- Theorem 2.3
- proof
- ...and 27 more