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

The operator square root of the Laplacian \((-\lap)^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.