We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators.
The extension is given by a suitable Poisson integral and solves the corresponding nonlocal …

We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional
Laplacian (-Δ) s in bounded open Lipschitz sets in the small order limit s→ 0+. While it is …

We investigate densities of vaguely continuous convolution semigroups of probability
measures on ℝ d. First, we provide results that give upper estimates in a situation when the …

In this work we study nonlocal operators and corresponding spaces with a focus on
operators of order near zero. We investigate the interior regularity of eigenfunctions and of …

We study the asymptotic formulas and estimates for the transition densities of isotropic
unimodal convolution semigroups of probability measures on under the assumption that its …

We revisit a Harnack inequality for antisymmetric functions that has been recently
established for the fractional Laplacian and we extend it to more general nonlocal elliptic …

Abstract We give Hardy–Stein and Douglas identities for nonlinear nonlocal Sobolev–
Bregman integral forms with unimodal Lévy measures. We prove that the corresponding …

We study translation-invariant integrodifferential operators that generate Lévy processes.
First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and …

The goal of this paper is to establish Green function estimates for a class of purely
discontinuous symmetric Markov processes with jump kernels degenerate at the boundary …

The theory of one-dimensional stochastic differential equations driven by Brownian motion is
classical and has been largely understood for several decades. For stochastic differential …