  We describe a procedure to introduce Sobolev spaces and the semigroup
generated by the fractional Dirichlet Laplacian on an arbitrary domain of
$\R^d$. In particular, the well-definedness of the spaces of both
non-homogeneous and homogeneous type together with their duality properties,
embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also
show the continuity and the smoothing property of the semigroup.
