Conditional Gauge and q-Green Function

Abstract

In Section 5.4, we proved the Gauge Theorem for a bounded domain D: if the gauge u(x) = E x{e q (τD)} ≢ ∞ in D, then it is bounded in \( \bar D \) . In this section, we shall prove the Conditional Gauge Theorem for a bounded Lipschitz domain D: if the conditional gauge u(x, z) = E x z {e q (τD)} ≢ ∞ in D × ∂D, then it is bounded in D × ∂D. For the gauge theorem, no assumption about the boundary is imposed, not even its regularity in the Dirichlet sense. By contrast, the conditional gauge theorem requires a certain smoothness of the boundary. Ad hoc assumptions on D and q may be and have been considered, but we shall settle the case in which D is a bounded Lipschitz domain in ℝd, d≥2 and q ∈ Jloc. For the case d = 1, see Theorem 9.9 and the Appendix to Section 9.2.KeywordsLipschitz DomainHarmonic MeasureBound Lipschitz DomainStrong Markov PropertyBoundary Harnack PrincipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.