# Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

@article{Palatucci2013ImprovedSE,
title={Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces},
journal={Calculus of Variations and Partial Differential Equations},
year={2013},
volume={50},
pages={799-829}
}
• Published 24 February 2013
• Mathematics
• Calculus of Variations and Partial Differential Equations
We obtain an improved Sobolev inequality in $$\dot{H}^s$$H˙s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in $$\dot{H}^s$$H˙s obtained in Gérard (ESAIM Control Optim Calc Var 3:213–233, 1998) using the abstract approach of dislocation spaces…
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