# A Hardy inequality for ultraspherical expansions with an application to the sphere

## TL;DR

This paper establishes a Hardy inequality for ultraspherical expansions using ground state methods, leading to uncertainty principles and a Hardy inequality on spheres with double singularities.

## Contribution

It introduces a Hardy inequality for ultraspherical expansions and applies it to derive uncertainty principles and sphere inequalities with singular potentials.

## Key findings

- Hardy inequality for ultraspherical expansions proved
- Uncertainty principles derived from the inequality
- Hardy inequality on spheres with double singularity

## Abstract

We prove a Hardy inequality for ultraspherical expansions by using a proper ground state representation. From this result we deduce some uncertainty principles for this kind of expansions. Our result also implies a Hardy inequality on spheres with a potential having a double singularity.