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In this work, we prove some trace theorems for function spaces with anonlocal 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 onthe boundary of a domain with sufficient regularity. The new generalized spacesare associated with norms that are characterized by nonlocal interactionkernels defined heterogeneously with a special localization feature on theboundary. Intuitively speaking, the class contains functions that are as roughas 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 nonlocalnorms that are weaker than the classical $W^{s, p}$ norm. These results areimprovement and refinement of the classical results since the boundary tracecan be attained without imposing regularity of the function in the interior ofthe domain. They also extend earlier results in the case of $p=2$. In themeantime, we prove Hardy-type inequalities for functions in the new generalizedspaces that vanish on the boundary, showing them having the same decay rate tothe boundary as functions in the smaller space $W^{s,p}(\Omega)$. APoincar\'e-type inequality is also derived. An application of the new theoryleads to the well-posedness of a nonlinear variational problem that allowspossible singular behavior in the interior with imposed smoother data on theboundary.

Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization