Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Abstract

Abstract. In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to O-regular variation at zero and at infinity with lower indices greater than −d − 2. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.Our results are expressed in terms of Pruitt's functions K(r) and L(r), measuring local activity and the amount of large jumps of the Lévy process, respectively.

Extension and trace for nonlocal operators

“…Theorem 2.1 is proved in Section 4 by using recent regularity results of Grzywny and Kwaśnicki [32] for L-harmonic functions.…”

“…Further, we claim that in this case the right-hand side of (4.10) is also infinite. Namely, we will check that R d (u(z) − u(y)) 2 ν(z, y) dz = ∞ for y ∈ B(x, δ 2 ), and then use the positivity of the Green function G U in U [32]. If z ∈ U c , then |z − y| ≤ C δ |z − x| with some C δ > 0.…”

Reduction of integration domain in Triebel–Lizorkin spaces

“…Assume A2 and A3. We will show that these assumptions enforce Oregular variation with positive lower index at 0 and, for unbounded Ω, at infinity by using [16,Proposition A.1]. Note that by A3 for r > 0, k ∈ Z, and z ∈ [2 k−1 r, 2 k r] we have φ(z) ≈ φ(2 k r).…”

Martin Kernels for Markov Processes with Jumps

“…This is based on estimates obtained recently in [9,10,18,20] and will be studied in detail in [19]. Other extensions can be obtained by allowing the Lévy kernel to depend on x or restricting it to a domain, as described in the following two examples.…”