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

J. P. Velasquez-Rodriguez

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 $L^r_α(\mathbb{G})$, $1<r<\infty$, $α>0$, 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.

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

Abstract

Let be a prime number, and let be a compact -adic Lie group. This work provides multiplier theorems for invariant operators on acting on , , , 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 -boundedness of bounded functions of the Vladimirov-Taibleson operator on compact Vilenkin groups.
Paper Structure (29 sections, 21 theorems, 279 equations, 1 figure)

This paper contains 29 sections, 21 theorems, 279 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Multipliers on Abelian Vilenkin Groups
  3. Organization of the Paper
  4. Compact Vilenkin groups
  5. The compact Heisenberg group
  6. Dual of a compact $p$-adic Lie group
  7. Equivalent norms on $L^2_\alpha (\mathbb{G})$
  8. Difference operators on $\mathbb{G}$
  9. The symbol of the Vladimirov-Taibleson operator
  10. Main results
  11. Multipliers on $L^2_\alpha(\mathbb{G})$
  12. Multipliers on $L^r_\alpha (\mathbb{G})$, $r \neq 2$
  13. Preliminaries
  14. Fourier analysis on compact groups
  15. Dual of a compact Vilenkin group
  16. ...and 14 more sections

Key Result

Theorem 1.4

Let $G$ be a locally compact abelian Vilenkin group with an order bounded sequence of compact open subgroups $\{G_n\}_{n \in \mathbb{Z}}$. Let $\widehat{G}$ be its unitary dual endowed with the sequence of compact open subgroups $\{\widehat{G}_n\}_{n \in \mathbb{Z}}$ associated to $\{G_n\}_{n \in \m Then $T_\sigma$ extends to a bounded operator on $L^r(G)$ for $1 < r < \infty$.

Figures (1)

  • Figure 1: The dual group $\widehat{\mathbb{Z}}_p$, $p=3$, as an infinite tree. Here we can see the first 3 ultrameric balls, which produce finite trees.

Theorems & Definitions (69)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: Saloff-Coste, SaloffCoste1986
  • Remark 1.5
  • Theorem 1.6: Onneweer
  • Theorem 1.7: Quek
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 59 more