Reinforcing a Philosophy: A counting approach to square functions over local fields

Abstract

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\ddot{I}•(T))$ where $\ddot{I}•(T)$ is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials $\ddot{I}•(T)=T^k$ for $k\ge 3$ which are uniform over all local fields whose characteristic is coprime to \(k\). Key to our approach is a systematic analysis of the second order differencing polynomial and its geometry in local fields.

19 pages, no figures, comments welcome!