Parabolic Potential Theory: Basic Facts

Abstract

The potential theory based on the Laplace operator, developed in the preceding chapters, will be called classical potential theory below. The potential theory based on the heat operator Δ̇ and its adjoint $\mathop{\Delta }\limits^{*}$, called parabolic potential theory, will be developed in Chapters XV to XIX. Concepts that are parabolic counterparts of classical concepts will be distinguished by dots or asterisks, depending on whether the concepts are related to Δ̇ or to $\mathop{\Delta }\limits^{*}$. Just as the domains of classical potential theory are subsets of ℝ^N, the domains of parabolic potential theory are subsets of “space time” ℝ^N+1, which we denote in this context by ℝ̇^N. Here N > 1, and the case N=1 is not exceptional. A point ξ̇=(ξ, s) of ℝ̇^N had space coordinate ξ in ℝ^N and time coordinate s-ord ξ (the ordinate of ξ̇), a point of ℝ. The point η̇: (η, t) will be said to be [strictly] below ξ̇: (ξ, s) if t<-s [t <s]. If ξ̇ is a point of an open subset Ḋ of ℝ̇^N, the set of points of Ḋ [strictly] below ξ ̇ relative to Ḋ is the set of points of Ḋ that are endpoints of continuous [strictly] downward-directed arcs from ξ̇. That is, η̇ is [strictly] below ξ̇ relative to Ḋ if and only if there is a continuous function f from [0, 1] into Ḋ for which f(0)=ξ̇, f(1)=η̇, and ord f is a [strictly] decreasing function. The upper [lower] half-space of ℝ̇^N is the set {ord ξ̇>0} [{ord ξ̇<0}] and the abscissa hyperplane is the set {ord ξ̇=0}