An $L^{4/3}$ $S{L}_2$ Kakeya maximal inequality

Abstract

It is shown that $S{L}_2$ 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 $S{L}_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $S{L}_2$ Kakeya maximal inequality cannot hold for $p>3/2$; even in the model case where the $\delta$-tubes have $\delta$-separated directions and the cardinality of the tube family is $\sim {\delta }^{-2}$. It is then shown that, with $C_{\epsilon }{\delta }^{-\epsilon }$ loss, the $S{L}_2$ Kakeya maximal inequality does hold if $p\le 4/3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $S{L}_2$ case). The proof is via an $L^{4/3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4/3<p\le 3/2$ 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".

This will not be submitted for publication as it has been superseded by a joint work, available at https://hal.science/hal-04386684

References in corpus (6)

• A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in ${\mathbb{R}}^3$
• On the Dimension of Kakeya Sets in the First Heisenberg Group
• On restricted projections to planes in ${\mathbb{R}}^3$
• A note on Kakeya sets of horizontal and $SL(2)$ lines
• Kakeya maximal inequality in the Heisenberg group
• Length of sets under restricted families of projections onto lines