  This is a survey on the use of Fourier transformation methods in the
treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$
(0<a<1), and pseudodifferential generalizations P, over a bounded open set
$\Omega$ in $R^n$. The presentation starts at an elementary level. Two points
are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,d\Omega)$,
comes into the picture, related to the fact that the precise solution spaces
for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2)
The natural definition of a local nonhomogeneous Dirichlet condition
$\gamma_0(u/d^{a-1})=\varphi$. We also give brief accounts of some further
developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent
problems (for $Pu-\lambda u=f$), also with nonzero boundary conditions.
Integration by parts, Green's formula.
