A bound for the (n, k) maximal function via incidence combinatorics.

摘要

In this dissertation, we discuss a Kakeya-type operator known as the (n, k) maximal function in the setting of a vector space over a finite field F. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from Lp to Lq when 2 ≤ k ≤ n - 2, 1 ≤ p ≤ kn+k+1kk+1 and 1 ≤ q ≤ (n - k) p'. Very briefly, we arrive at this estimate by first showing that estimates for the maximal function in question are equivalent, to estimates for the number of incidences between families of points P ⊂ Fn and k-planes pi. Then, we arrive at a non-trivial incidence bound by estimating the number of (k + 1)-simplices with vertices from P and faces from pi. We also motivate a few open problems related to the main result, and consider an alternate proof of the main result which utilizes more traditional analytic techniques, and may prove useful in trying to reproduce the result in Euclidean space.