Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets

TL;DR

This paper establishes sharp lower bounds for dyadic square functions of indicator functions, improving classical bounds with logarithmic factors and connecting to Brownian motion and Takagi functions.

Contribution

It provides the first sharp lower bounds for dyadic square functions of indicator functions, revealing new logarithmic improvements and connections to stochastic processes and fractal functions.

Findings

Sharp lower bound for $S_2$ involving Brownian motion exit time.

Logarithmic improvement over classical bounds.

Sharp inequality for $S_1$ involving the Takagi function.

Abstract

We study lower bounds for dyadic square functions of indicator functions. In the case of the dyadic square function $S_{2}$ we obtain a sharp lower bound: for every measurable $A \subset {[0,1)}$, we have \[ \|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrt{\tau}\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}}, \] where $\tau$ is the first exit time from $(0,1)$ of a standard Brownian motion started at $|A|$, and $|A|^{*}:=\min\{|A|,1-|A|\}$. This estimate gives logarithmic improvement over the classical Burkholder--Davis--Gundy lower bound $|A|^{*}$. In addition, we show a sharp inequality \[ \|S_{1}(\mathbbm{1}_{A})\|_{1} \ge T(|A|)\asymp |A|^{*}\log_{2}\frac{1}{|A|^{*}}, \] where $T(x)=\sum_{k=0}^{\infty}\frac{\operatorname{dist}(2^{k}x,\mathbb{Z})}{2^{k}}$ is the Takagi function.