A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

TL;DR

This paper proves a refined Trudinger-Moser inequality with a Tintarev-type constraint in fractional-dimensional spaces and demonstrates the existence of extremal functions in the critical case.

Contribution

It extends previous inequalities to fractional dimensions with new constraints and establishes the existence of maximizers in this setting.

Findings

Established a new Trudinger-Moser inequality in fractional-dimensional spaces.

Proved the existence of extremal functions in the critical regime.

Refined previous inequalities for classical Sobolev spaces.

Abstract

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.