Remarks on the construction of $K_σ$ sets associated to trees not satisfying a separation condition
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| Format: | Preprint |
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2024
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| _version_ | 1866915382420307968 |
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| author | Hagelstein, Paul Radillo-Murguia, Blanca Stokolos, Alex |
| author_facet | Hagelstein, Paul Radillo-Murguia, Blanca Stokolos, Alex |
| contents | $K_σ$ sets involving sticky maps $σ$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p(\mathbb{R}^2)$ for all $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $ε> 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, ε}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $σ: \mathcal{B}^{h(\mathcal{T}_{N, ε})} \rightarrow \mathcal{T}_{N, ε}$, one has $|K_σ \cap ((1,2) \times \mathbb{R})| \geq 1 - ε$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on the construction of $K_σ$ sets associated to trees not satisfying a separation condition Hagelstein, Paul Radillo-Murguia, Blanca Stokolos, Alex Classical Analysis and ODEs 42B25 $K_σ$ sets involving sticky maps $σ$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p(\mathbb{R}^2)$ for all $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $ε> 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, ε}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $σ: \mathcal{B}^{h(\mathcal{T}_{N, ε})} \rightarrow \mathcal{T}_{N, ε}$, one has $|K_σ \cap ((1,2) \times \mathbb{R})| \geq 1 - ε$. |
| title | Remarks on the construction of $K_σ$ sets associated to trees not satisfying a separation condition |
| topic | Classical Analysis and ODEs 42B25 |
| url | https://arxiv.org/abs/2409.12280 |