Sharp Estimates on the Green Functions of Perturbations of Subordinate Brownian Motions in Bounded κFat Open Sets

Panki Kim, Hyunchul Park, Renming Song

Research output: Contribution to journalArticle

Abstract

In this paper we study perturbations of a large class of subordinate Brownian motions in bounded κ-fat open sets, which include bounded John domains. Suppose that X is such a subordinate Brownian motion and that J is the Lévy density of X. The main result of this paper implies, in particular, that if Y is a symmetric Lévy process with Lévy density JY satisfying {pipe}JY(x) - J(x){pipe} ≤ c max {{pipe}x{pipe}-d + ρ, 1} for some c > 0,ρ ∈ (0, d), then for any bounded John domain D the Green function GDY of Y in D is comparable to the Green function GD of X in D. One of the main tools of this paper is the drift transform introduced in Chen and Song (J Funct Anal 201:262-281, 2003). To apply the drift transform, we first establish a generalized 3G theorem for X.

Original languageEnglish (US)
Pages (from-to)319-344
Number of pages26
JournalPotential Analysis
Volume38
Issue number1
DOIs
StatePublished - Jan 1 2013

Keywords

  • 3G theorem
  • Boundary Harnack principle
  • Generalized 3G theorem
  • Green functions
  • Harmonic functions
  • Harnack inequality
  • Subordinate Brownian motion

ASJC Scopus subject areas

  • Analysis

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