$L^r$-Multipliers on compact $p$-adic Lie groups

TL;DR

This paper establishes multiplier theorems for invariant operators on compact p-adic Lie groups, utilizing difference operators and conditions from harmonic analysis, and applies these results to Littlewood-Paley theory and boundedness of Vladimirov-Taibleson operators.

Contribution

It introduces new multiplier theorems for invariant operators on compact p-adic Lie groups using difference operators and extends harmonic analysis tools to this setting.

Findings

Proved multiplier theorems for invariant operators on compact p-adic Lie groups.

Established Littlewood-Paley decomposition in this context.

Demonstrated L^r-boundedness of functions of Vladimirov-Taibleson operators.

Abstract

Let $p$ be a prime number, and let $\mathbb{G}$ be a compact $p$-adic Lie group. This work provides multiplier theorems for invariant operators on $\mathbb{G}$ acting on $L^r_\alpha(\mathbb{G})$, $1<r<\infty$, $\alpha>0$, in terms of the Ruzhansky-Turunen difference operators and Saloff-Coste's condition. As an application, a Littlewood-Paley decomposition is proven, together with the $L^r$-boundedness of bounded functions of the Vladimirov-Taibleson operator on compact Vilenkin groups.