Censored symmetric Lévy-type processes

We examine three equivalent constructions of a censored symmetric purely discontinuous Lévy process on an open set D; via the corresponding Dirichlet form, through the Feynman–Kac transform of the Lévy 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 behavior of the corresponding censored Lévy 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 Lévy processes on κ-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.

A trace theorem for Martinet-type vector fields

In [Formula: see text] we consider the vector fields [Formula: see text] where [Formula: see text]. Let [Formula: see text] be the (closed) upper half-space and let [Formula: see text] be a function such that [Formula: see text] for some [Formula: see text]. In this paper, we prove that the restriction of [Formula: see text] to the plane [Formula: see text] belongs to a suitable Besov space that is defined using the Carnot–Carathéodory metric associated with [Formula: see text] and [Formula: see text] and the related perimeter measure.

The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs

In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension.DOI 10.14258/izvasu(2018)4-19