Differential structure on the dual of a compact lie group

TL;DR

This paper develops a differential structure on the dual of a compact Lie group using difference operators and Sobolev spaces, proving key multiplier theorems and establishing sharpness of Sobolev exponents.

Contribution

It introduces a new differential framework on the dual of compact Lie groups and proves classical multiplier theorems within this setting.

Findings

Established H"ormander, Mihlin, and Marcinkiewicz multiplier theorems.

Proved sharpness of Sobolev exponent for H"ormander-type multipliers.

Defined Sobolev-type spaces and difference operators on the dual group.

Abstract

In this paper we define difference operators and homogeneous Sobolev-type spaces on the dual of a compact Lie group. As an application and to show that this defines a relevant differential structure, we state and prove multiplier theorems of H\"ormander, Mihlin and Marcinkiewicz types together with the sharpness in the Sobolev exponent for the one of H\"ormander type.