# Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$

## TL;DR

This paper investigates critical and subcritical Trudinger-Moser inequalities within fractional Sobolev spaces on the real line, establishing their relationships and exploring whether the supremum values are attained.

## Contribution

It provides new insights into the connection between critical and subcritical inequalities and analyzes the attainability of supremum in fractional Sobolev spaces.

## Key findings

- Established the relation between critical and subcritical inequalities.
- Proved results on the attainability of supremum values.
- Extended the understanding of Trudinger-Moser inequalities in fractional settings.

## Abstract

In this paper, we are concerned with the critical and subcritical Trudinger-Moser type inequalities for functions in a fractional Sobolev space $H^{1/2,2}$ on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.