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

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 $(T,\varphi (T))$ where $\varphi (T)$ is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials $\varphi (T)=T^k$ 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.