  For the fractional Laplace equation, a surprising observation is the
non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat
sharp uniqueness condition for the fractional Laplace equation is established.
We derive the $L^p$-estimate for fractional Laplacian operators to better
understand this phenomena. Several weighted fractional Sobolev spaces appear
naturally. We then establish the embedding relations between these spaces.
These existence-uniqueness conditions and the spaces we introduce here are
intrinsically related to the fractional Laplacian. These are basic properties
to the fractional Laplace equations and can be useful in the study of related
problems.
