  We examine three equivalent constructions of a censored symmetric purely
discontinuous L\'evy process on an open set $D$; via the corresponding
Dirichlet form, through the Feynman-Kac transform of the L\'evy process killed
outside of $D$ and from the same killed process by the Ikeda-Nagasawa-Watanabe
piecing together procedure. By applying the trace theorem on $n$-sets for
Besov-type spaces of generalized smoothness associated with complete Bernstein
functions satisfying certain scaling conditions, we analyze the boundary
behaviour of the corresponding censored L\'evy process and determine conditions
under which the process approaches the boundary $\partial D$ in finite time.
Furthermore, we prove a stronger version of the 3G inequality and its
generalized version for Green functions of purely discontinuous L\'evy
processes on $\kappa$-fat open sets. Using this result, we obtain the scale
invariant Harnack inequality for the corresponding censored process.
