ON KAKEYA-NIKODYM TYPE MAXIMAL INEQUALITIES.

We show that for any dimension d ≥ 3, one can obtain Wolff's L(d+2)/2 bound on Kakeya-Nikodym maximal function in Rd for d ≥ 3 without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional L² 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 (Md, g) with constant curvature, which generalizes Sogge's results for d = 3 to any d ≥ 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. [ABSTRACT FROM AUTHOR]