# Extremals for fractional Moser-Trudinger inequalities in dimension 1 via harmonic extensions and commutator estimates

## TL;DR

This paper establishes the existence of extremal functions for fractional Moser-Trudinger inequalities in one dimension using blow-up analysis, harmonic extensions, and commutator estimates, advancing understanding in fractional Sobolev spaces.

## Contribution

It introduces new sharp commutator estimates and applies blow-up analysis to prove extremal existence for fractional Moser-Trudinger inequalities in 1D.

## Key findings

- Existence of extremals in interval and on the real line.
- Development of new commutator estimates for fractional operators.
- Application of blow-up analysis to fractional inequalities.

## Abstract

We prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp estimates obtained via commutator techniques.