On Kakeya-Nikodym type maximal inequalities

Abstract

We show that for any dimension d ≥ 3 d\ge 3 , one can obtain Wolff’s L ( d + 2 ) / 2 L^{(d+2)/2} bound on Kakeya-Nikodym maximal function in R d \mathbb R^d for d ≥ 3 d\ge 3 without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional L 2 L^2 estimate with an auxiliary maximal function. We also prove that the same L ( d + 2 ) / 2 L^{(d+2)/2} bound holds for Nikodym maximal function for any manifold ( M d , g ) (M^d,g) with constant curvature, which generalizes Sogge’s results for d = 3 d=3 to any d ≥ 3 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.