Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions

TL;DR

This paper proves the equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions, establishes bounds for the supremum, and confirms the existence of extremal functions, with explicit calculations in special cases.

Contribution

It demonstrates the equivalence between critical and subcritical inequalities in fractional dimensions and constructs extremal functions, advancing the understanding of these inequalities.

Findings

Established critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions.

Proved the existence of extremal functions for both inequalities.

Explicitly calculated the critical supremum in certain cases.

Abstract

We establish critical and subcritical sharp Trudinger-Moser inequalities for fractional dimensions on the whole space. Moreover, we obtain asymptotic lower and upper bounds for the fractional subcritical Trudinger-Moser supremum from which we can prove the equivalence between critical and subcritical inequalities. Using this equivalence, we prove the existence of maximizers for both the subcritical and critical associated extremal problems. As a by-product of this development, we can explicitly calculate the value of the critical supremum in some special situations.