Subparabolic, Superparabolic, and Parabolic Functions on a Slab

If Ḋ is the slab ℝN×]0, δ[, with 0<δ<+∞. the restriction to Ḋ×Ḋ of Ġ satisfies the rather vague description of the Green function Ġ_Ḋ given in Section XV. 7 for smooth regions. It is therefore to be expected from XV (7.3) that the upper boundary of Ḋ if δ<+∞ is a parabolic measure null set and that parabolic measure on the lower boundary is given by $$ {\mathop{u}\limits^{.}_{{\mathop{D}\limits^{.} }}}(\mathop{\zeta }\limits^{.}, d\eta ) = b(s,\zeta - \eta ){l_N}(d\eta ) = \mathop{G}\limits^{.} (\mathop{\zeta }\limits^{.}, (\eta, 0)){l_N}(d\eta ) [\mathop{\zeta }\limits^{.} = (\zeta, s)] $$ so that if u̇ is parabolic on Ḋ with boundary function f in some suitable sense on the lower boundary and if u̇ is appropriately restricted, then 1.1$$ \dot u\left( {\dot \xi } \right) = \int\limits_{^{^{\mathbb{R}^N } } } \ell \left( {s,\xi - \eta } \right)f\left( \eta \right)l_N \left( {d\eta } \right){\text{ }}\left[ {\dot \xi = \left( {\xi ,s} \right)} \right] \cdot $$