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 (τ^D)} ≢ ∞ 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 (τ^D)} ≢ ∞ 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 ℝ^d, d≥2 and q ∈ J_loc. For the case d = 1, see Theorem 9.9 and the Appendix to Section 9.2.