Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917400598806528 |
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| author | Alpay, Natanael Ivanisvili, Paata |
| author_facet | Alpay, Natanael Ivanisvili, Paata |
| contents | 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 \[ \|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrtτ\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}}, \] where $τ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16045 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets Alpay, Natanael Ivanisvili, Paata Classical Analysis and ODEs Combinatorics 46B09, 60E15, 05C35, 65G30 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 \[ \|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrtτ\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}}, \] where $τ$ 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. |
| title | Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets |
| topic | Classical Analysis and ODEs Combinatorics 46B09, 60E15, 05C35, 65G30 |
| url | https://arxiv.org/abs/2502.16045 |