A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one

TL;DR

This paper extends recent results on a Moser-Trudinger inequality within Sobolev-Slobodeckij spaces in one dimension, focusing on the entire real line and addressing an open question from prior research.

Contribution

It advances the analysis of the inequality on \\mathbb{R} and provides an answer to an open problem posed by Parini and Ruf.

Findings

Extended the inequality to the whole real line.

Provided an answer to an open question in the field.

Enhanced understanding of Sobolev-Slobodeckij spaces in one dimension.

Abstract

We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole $\mathbb{R}$ and we give an answer to one of their open questions.