Non-Archimedean Kelvin Transformation

TL;DR

The paper develops a non-Archimedean analogue of the Kelvin transformation associated with the Vladimirov-Taibleson operator $D^{\alpha,n}$ on $K^n$, exploiting unramified extensions to define an inversion map. It establishes a fundamental identity $(D^{\alpha,n}u)(x)=\|x\|_{K^n}^{\alpha+n}(D^{\alpha,n}(\mathcal{K}u))(x)$ linking the operator applied to a function and to its Kelvin transform, thereby translating $\alpha$-harmonicity across inversion. The analysis is embedded in a Fourier/Sobolev framework with an isomorphism between $K^n$ and an unramified extension $L$ (where $\gamma=\alpha/n$), enabling reduction to a 1D model on $L$. The work provides a non-Archimedean analogue of Kelvin invariance and a p-adic conformal viewpoint, facilitating study of $\alpha$-harmonic functions on $K^n$ via inversion and transform techniques.

Abstract

We introduce and study an analog of the Kelvin transformation connected with the Vladimirov-Taibleson operator acting on real- or complex-valued functions on a space $K^n$ over a non-Archimedean local field $K$.

Theorems & Definitions (2)

• Theorem 4.1
• Corollary 4.1