Lower Bounds for Dyadic Square Functions of indicator functions of sets
Natanael Alpay, Paata Ivanisvili
TL;DR
This work delivers sharp, end-point lower bounds for dyadic square functions of indicator functions of sets. By employing a Bellman-function framework and a carefully constructed limit function on dyadic rationals, the authors connect $\|S_{2}(\mathbf{1}_{A})\|_{1}$ to the Gaussian isoperimetric profile $I(|A|)$, surpassing the classical $|A|^{*}\sqrt{\log(1/|A|^{*})}$ rate. They also develop a general method for $\|S_{\beta}(\mathbf{1}_{A})\|_{\alpha}$ bounds, revealing a threshold behavior around $\alpha=1$ and proving sharpness in the $\alpha\in(0,1)$ regime, with explicit constructions and a fractal-like extremal structure for $S_{1}$-bounds. The results strengthen connections between dyadic martingale square functions, edge isoperimetric phenomena, and Gaussian isoperimetry, with potential implications for sharp endpoint inequalities in discrete and probabilistic settings.
Abstract
We prove that for any Borel measurable subset $A\subset [0,1]$, the inequality $\|S_{2}(\mathbbm{1}_{A})\|_{1} \geq I(|A|)$ holds, where $I$ denotes the Gaussian isoperimetric profile. This improves upon the classical lower bound $ \|S_{2}(\mathbbm{1}_{A})\|_{1} \gtrsim |A|(1-|A|) $ by a factor of $\sqrt{\log\frac{1}{|A|(1-|A|)}}$. In addition, we study lower bounds for the $α$-norm of $S_1(\mathbbm{1}_{A})$, and we obtain a threshold behavior around $α=1$. We show that $$ \|S_{1}(\mathbbm{1}_{A})\|_{1} \geq \min\{|A|, 1-|A|\}\log_{2}\frac{1}{\min\{|A|, 1-|A|\}}, $$ and that this bound is sharp at points $|A|=2^{-k}$ or $|A|=1-2^{-k}$ for every nonnegative integer $k$. For each fixed $α\in (0,1)$, we further establish that $\|S_{1}(\mathbbm{1}_{A})\|_α \geq \min\{|A|, 1-| A|\},$ with the decay rate $|A|$, as $|A|\to 0$, being optimal.