Martinazzi, Luca.
(2015)
* Fractional Adams-Moser-Trudinger type inequalities.*

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Official URL: https://edoc.unibas.ch/69991/

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## Abstract

Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde{H}^{n/p,p}(\Omega)$ into Orlicz spaces for an arbitrary domain $\Omega$ with finite measure. In particular we prove

\[

\sup_{u\in \tilde{H}^{n/p,p}(\Omega), \|(-\Delta)^{n/(2p)} u\|_{L^p(\Omega)}\le 1} \int_\Omega e^{\alpha_{n,p} u^{p/(p-1)} \d x \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. $(-\Delta)^{n/(2p)} u \in L^{(p,q)}$). 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.

\[

\sup_{u\in \tilde{H}^{n/p,p}(\Omega), \|(-\Delta)^{n/(2p)} u\|_{L^p(\Omega)}\le 1} \int_\Omega e^{\alpha_{n,p} u^{p/(p-1)} \d x \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. $(-\Delta)^{n/(2p)} u \in L^{(p,q)}$). 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.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) 12 Special Collections > Preprints Fachbereich Mathematik |
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UniBasel Contributors: | Martinazzi, Luca |

Item Type: | Preprint |

Publisher: | Universität Basel |

Language: | English |

Last Modified: | 12 May 2019 21:07 |

Deposited On: | 28 Mar 2019 09:51 |

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