# Symmetries on manifolds: Generalizations of the Radial Lemma of Strauss

## TL;DR

This paper extends the Strauss lemma to Riemannian manifolds with group symmetries, showing that invariance under a group action implies decay and smoothness properties, with applications to inequalities for invariant functions.

## Contribution

It generalizes the classical Strauss lemma to manifolds with symmetries, providing new decay and smoothness results for invariant functions.

## Key findings

- Invariant functions exhibit decay properties on manifolds.
- New inequalities of Caffarelli-Kohn-Nirenberg type for invariant functions.
- Extension of Strauss lemma to Riemannian manifolds with group actions.

## Abstract

For a compact subgroup $G$ of the group of isometries acting on a Riemannian manifold $M$ we investigate subspaces of Besov and Triebel-Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that $G$-invariance of functions implies certain decay properties and better local smoothness. As an application we obtain inequalities of Caffarelli-Kohn-Nirenberg type for $G$-invariant functions. Our results generalize those obtained by Skrzypczak. The main tool in our investigations are atomic decompositions adapted to the $G$-action in combination with trace theorems.