Fractional Adams-Moser-Trudinger type inequalities

TL;DR

This paper establishes a broad class of Adams-Moser-Trudinger inequalities for fractional Sobolev spaces on arbitrary domains, including sharp constants and extensions to Lorentz spaces, with applications to geometric problems.

Contribution

It generalizes Adams-Moser-Trudinger inequalities to fractional Bessel-potential spaces on arbitrary domains, including sharp constants and Lorentz space extensions.

Findings

Proved a sharp Adams-Moser-Trudinger inequality for fractional Sobolev spaces.

Extended the inequality to Lorentz spaces with fractional derivatives.

Applied results to problems involving $Q$-curvature.

Abstract

Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(\Omega)$ into Orlicz spaces for an arbitrary domain $\Omega\subset \mathbb{R}^n$ with finite measure. In particular we prove $$\sup_{u\in \tilde H^{\frac{n}{p},p}(\Omega), \;\|(-\Delta)^{\frac{n}{2p}}u\|_{L^{p}(\Omega)}\leq 1}\int_{\Omega}e^{\alpha_{n,p} |u|^\frac{p}{p-1}}dx \leq c_{n,p}|\Omega|, $$ for a positive constant $\alpha_{n,p}$ whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. $(-\Delta)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple, as they use Green functions for fractional Laplace operators and suitable cut-off procedures to reduce the fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai. We also discuss an application to the problem of prescribing the $Q$-curvature and some open problems.