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

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

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.