On Kakeya-Nikodym type maximal inequalities

Abstract

We show that for any dimension $d\ge 3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in ${\mathbb{R}}^d$ for $d\ge 3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional $L^2$ estimate with an auxiliary maximal function. We also prove that the same $L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $(M^d,g)$ with constant curvature, which generalizes Sogge's results for $d=3$ to any $d\ge 3$. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function.

arXiv comment21 pages, minor corrections. To appear in Transaction of the American Mathematical Society

References in corpus (2)

• An improved bound on the Minkowski dimension of Besicovitch sets in R^3
• On Wolff's $L^{\frac{5}{2}}-$Kakeya maximal inequality in ${\mathbb{R}}^3$

Cited by in corpus (1)

• On Instability of the Nikodym Maximal Function bounds over Riemannian Manifolds