Lusin area functions on local fields

Abstract

1* By a local field, we mean a locally compact, nondiscrete, totally disconnected, (complete) field. Various aspects of harmonic analysis on local fields have been studied. A list of references can be found in [4]. We also refer to [4] for notation and preliminaries. Let if be a fixed local field with the ring of integers £?. &\& = GF(q) where & is the maximal ideal in & and q is a prime power. For keZ, let ^~k = {xe if: £ qk), {έ? = ^°). ^*r* = V + ^~h are spheres. The Haar measure on K has been normalized so that |<^| = dx = 1 and |^V| = qk for all k. The theory of regular functions which are the local field analogue of harmonic functions is studied in [10] and [4]. In particular, distributions on K have been identified with regular functions on if x Z and the regularization kernel Rk(x) = q~kΦ-k{x)y where Φ_fe is the characteristic function of ^~k, serves as the Poisson kernel. Write (&*y k) = {{x, k)eK x Z:xe ^~1}. For a nonnegative integer ί and ^e K, let Γι{z) = {(x, k)e K x Z: \x - z ^ qk+ι} = U/fc (^~(*+l), A;). For a distribution / on if or a regular function /(a?, k) on if x Z, denote dkf(x) — f(x, k) — f(x, k + 1). The Lusin area function of / with respect to Γt is given by