An $L^{4/3}$ $SL_2$ Kakeya maximal inequality
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916095846252544 |
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| author | Harris, Terence L. J. |
| author_facet | Harris, Terence L. J. |
| contents | It is shown that $SL_2$ Besicovitch sets of measure zero exist in $\mathbb{R}^3$. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the $SL_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $SL_2$ Kakeya maximal inequality cannot hold for $p> 3/2$; even in the model case where the $δ$-tubes have $δ$-separated directions and the cardinality of the tube family is $\sim δ^{-2}$. It is then shown that, with $C_ε δ^{-ε}$ loss, the $SL_2$ Kakeya maximal inequality does hold if $p \leq 4/3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $SL_2$ case). The proof is via an $L^{4/3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4/3 < p \leq 3/2$ remains an open problem. A related $L^{6/5}$ inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in $\mathbb{R}^3$ with "light rays" and "light planes". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14667 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An $L^{4/3}$ $SL_2$ Kakeya maximal inequality Harris, Terence L. J. Classical Analysis and ODEs 28A78, 28A80 It is shown that $SL_2$ Besicovitch sets of measure zero exist in $\mathbb{R}^3$. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the $SL_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $SL_2$ Kakeya maximal inequality cannot hold for $p> 3/2$; even in the model case where the $δ$-tubes have $δ$-separated directions and the cardinality of the tube family is $\sim δ^{-2}$. It is then shown that, with $C_ε δ^{-ε}$ loss, the $SL_2$ Kakeya maximal inequality does hold if $p \leq 4/3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $SL_2$ case). The proof is via an $L^{4/3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4/3 < p \leq 3/2$ remains an open problem. A related $L^{6/5}$ inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in $\mathbb{R}^3$ with "light rays" and "light planes". |
| title | An $L^{4/3}$ $SL_2$ Kakeya maximal inequality |
| topic | Classical Analysis and ODEs 28A78, 28A80 |
| url | https://arxiv.org/abs/2311.14667 |