# Limit of Fractional Power Sobolev Inequalities

## TL;DR

This paper establishes the Moser-Trudinger-Onofri inequalities on 2- and 4-spheres as limits of fractional Sobolev inequalities, using a dimensional continuation approach inspired by Branson.

## Contribution

It introduces a novel approach linking fractional Sobolev inequalities to classical inequalities via dimensional continuation.

## Key findings

- Moser-Trudinger-Onofri inequalities derived as limits of fractional Sobolev inequalities
- Dimensional continuation method justified for spheres
- Connections established between fractional and classical inequalities

## Abstract

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.