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Let $Sf$ be a discrete martingale square function. Then, for any set $V$ ofpositive probability, we have $\mathbb{E} S(\mathbf{1}_V)^2 \geq \eta\mathbb{P}(V)$ for an absolute constant $\eta >0$. We extend this to waveletsquare functions, and discuss some related open questions.

Lower Estimate on Square Function of an Indicator Set