Lower Bounds for Dyadic Square Functions of indicator functions of sets

Abstract: We prove that for any Borel measurable subset $A\subset [0,1]$, the inequality $|S_{2}(\mathbbm{1}<em>{A})|</em>{1} \geq I(|A|)$ holds, where $I$ denotes the Gaussian isoperimetric profile. This improves upon the classical lower bound $ |S_{2}(\mathbbm{1}<em>{A})|</em>{1} \gtrsim |A|(1-|A|) $ by a factor of $\sqrt{\log\frac{1}{|A|(1-|A|)}}$. In addition, we study lower bounds for the $\alpha$-norm of $S_1(\mathbbm{1}<em>{A})$, and we obtain a threshold behavior around $\alpha=1$. We show that $$ |S</em>{1}(\mathbbm{1}<em>{A})|</em>{1} \geq \min{|A|, 1-|A|}\log_{2}\frac{1}{\min{|A|, 1-|A|}}, $$ and that this bound is sharp at points $|A|=2<sup>{-k}$ or $|A|=1-2<sup>{-k}$ for every nonnegative integer $k$. For each fixed $\alpha\in (0,1)$, we further establish that $|S_{1}(\mathbbm{1}<em>{A})|</em>{\alpha} \geq \min{|A|, 1-| A|},$ with the decay rate $|A|$, as $|A|\to 0$, being optimal.