Lower Estimate on Square Function of an Indicator Set

2023/07/11 by Christina Giannitsi, Michael T. Lacey, Giannitsi, Christina +1
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2307.05692

openalex publication_date 2023/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Sf be a discrete martingale square function. Then, for any set V of positive probability, we have 𝔼 S(1_V)² ≥ ηℙ(V) for an absolute constant η>0. We extend this to wavelet square functions, and discuss some related open questions.

Related

• On the failure of lower square function estimates in the non-homogeneous weighted setting
• A Gaussian upper bound for martingale small-ball probabilities
• Lower Bounds for Dyadic Square Functions of indicator functions of sets
• Martingale transforms of bounded random variables and indicator functions of events
• On a Bellman function associated with the Chang--Wilson--Wolff theorem: a case study