On Kakeya maps with regularity assumptions

Abstract

For a $n-$dimensional Kakeya set $(n\geq 3)$ we may define a Kakeya map associated to it which parametrizes the Kakeya set by $[0,1]\times S^{n-1},$ where $S^{n-1}$ is thought of as the space of unit directions. We show that if the Kakeya map is either $\alpha-$H\"{o}lder continuous with $\alpha>\frac{(n-2)n}{(n-1)^2},$ or continuous and in the Sobolev space $ H^{s} $ for some $s>(n-1)/2,$ then the Kakeya set has positive Lebesgue measure.