  In this work, we prove some trace theorems for function spaces with a
nonlocal character that contain the classical $W^{s,p}$ space as a subspace.
The result we obtain generalizes well known trace theorems for
$W^{s,p}(\Omega)$ functions which has a well-defined $W^{1-s/p, p}$ trace on
the boundary of a domain with sufficient regularity. The new generalized spaces
are associated with norms that are characterized by nonlocal interaction
kernels defined heterogeneously with a special localization feature on the
boundary. Intuitively speaking, the class contains functions that are as rough
as an $L^p$ function inside the domain of definition but as smooth as a
$W^{s,p}$ function near the boundary. Our result is that the $W^{1-s/p, p}$
norm of the trace on the boundary such functions is controlled by the nonlocal
norms that are weaker than the classical $W^{s, p}$ norm. These results are
improvement and refinement of the classical results since the boundary trace
can be attained without imposing regularity of the function in the interior of
the domain. They also extend earlier results in the case of $p=2$. In the
meantime, we prove Hardy-type inequalities for functions in the new generalized
spaces that vanish on the boundary, showing them having the same decay rate to
the boundary as functions in the smaller space $W^{s,p}(\Omega)$. A
Poincar\'e-type inequality is also derived. An application of the new theory
leads to the well-posedness of a nonlinear variational problem that allows
possible singular behavior in the interior with imposed smoother data on the
boundary.
