Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

Abstract

It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field K. This includes the case where K is a p-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjolin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of th...