Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab

Abstract:

This paper investigates the boundary behaviour of positive solutions of the equation $Lu = 0$, where $L$ is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on $X = {{\mathbf {R}}^n} \times (0,T)$, where $0 < T < \infty$. In particular, the notion of semithinness for the potential theory on $X$ associated with $L$ is introduced, and the relationships between fine, semifine and parabolic convergence at points of ${{\mathbf {R}}^n} \times \{ 0 \}$ are studied. The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of $Lu = 0$ on $X$ has parabolic limits Lebesgue-almost-everywhere on ${{\mathbf {R}}^n} \times \{ 0 \}$. Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.