$p$-Adic Fourier Theory of Differentiable Functions

Abstract

Let $\mathbf{K}$ be a finite extension of ${\mathbb{Q}}_p$ of degree $d$ and ${\mathcal{O}}_{\mathbf{K}}$ its ring of integers; let ${\mathbb{C}}_p$ be the completed algebraic closure of ${\mathbb{Q}}_p$. The Fourier polynomials $P_n:{\mathcal{O}}_{\mathbf{K}}\to {\mathbb{C}}_p$ show that the topological algebra of all locally analytic distributions $\mu :{\mathcal{C}}^{\mathrm{la}}({\mathcal{O}}_{\mathbf{K}},{\mathbb{C}}_p)\to {\mathbb{C}}_p$ is, by $\mu \mapsto \sum \mu (P_n)X^n$, isomorphic to that of all power series in ${\mathbb{C}}_p[[X]]$ that converge on the open unit disc of ${\mathbb{C}}_p$. Given a real number $r\ge d$, we determine the power series that correspond under this isomorphism to all distributions $\mu :{\mathcal{C}}^r({\mathcal{O}}_{\mathbf{K}},{\mathbb{C}}_p)\to {\mathbb{C}}_p$ that extend to all $r$-times differentiable functions (as arisen in the $p$-adic Langlands program): A function $f:{\mathcal{O}}_{\mathbf{K}}\to {\mathcal{C}}_p$ is $r$-times differentiable if and only if $f(x)=\Sigma a_n{P}_n(x)$ with $|a_n|n^{r/d}\to 0$ as $n\to \infty$.