Fractional Adams-Moser-Trudinger type inequalities

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 $$\underset{u\in {\tilde{H}}^{\frac{n}{p},p}(\Omega ),\Vert (-\Delta {)}^{\frac{n}{2p}}u{\Vert }_{L^p(\Omega )}\le 1}{\sup }{\int }_{\Omega }{e}^{{\alpha }_{n,p}|u{|}^{\frac{p}{p-1}}}dx\le c_{n,p}|\Omega |,$$ 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.

Added references

References in corpus (4)

• Nonlocal equations with measure data
• Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension
• Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature
• A fractional Moser-Trudinger type inequalitiy in one dimension and its critical points

Cited by in corpus (15)

• Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity
• Improved Adams-type inequalities and their extremals in dimension 2m
• Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension
• Adams' inequality with exact growth in the hyperbolic space ${\mathbb{H}}^4$ and Lions lemma
• Asymptotic morphisms and superselection theory in the scaling limit II: analysis of some models
• Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation
• Constant $Q$-curvature metrics with a singularity
• A fractional Moser-Trudinger type inequalitiy in one dimension and its critical points
• The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms
• Remarks on the fractional Moser-Trudinger inequality
• Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases
• Moser functions and fractional Moser-Trudinger type inequalities
• Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition
• The Nehari manifold method for Fractional Kirchhoff problem involving singular and exponential nonlinearity
• Multiplicity results for fractional magnetic problems involving exponential growth