Regularization of $p$-Adic Distributions Associated to Functions on $p$-Adic Fields With Moderate Variation

Abstract

The $p$-adic distributions attached to ordinary functions defined on $p$-adic fields with moderate variation are studied. We first give a sufficient growth condition on ordinary functions to construct $p$-adic distributions. Then a moderate variation condition on functions for regularization of these $p$-adic distributions is imposed which provides a general method to construct $p$-adic measures. The $p$-adic integrals against these measures are also explicitly transformed to integrals against Bernoulli measures.