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

Abstract

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 Holder-continuous coefficients on X= R" X (0, 7"), where 0 < T < oo. 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 R" X {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 R" X (0). Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions. 0. Introduction. Let £ be a second-order linear parabolic differential operator having divergence structure on X = R" X (0, T) where 0 < T < oo. The coefficients of L are assumed to be such that the classical fundamental solution exists, the results in [2] for classical solutions hold, and the solutions of Lu = 0 form a strong harmonic space in the sense of Bauer [3].In the particular case of the heat operator Ax -9/3i, Doob [11] proved the almost everywhere convergence through parabolic regions of quotients of positive solutions on X. Hattemer [16] showed that if £ c R", « is a solution of the heat equation on X, and for each b e £, u is bounded on a parabolic region with vertex b, then u has finite parabolic limits almost everywhere on £ (cf. For certain parabolic operators with divergence structure, Johnson [19] proved the Lebesgue-almost-everywhere convergence through parabolic regions of positive solutions on X.In this paper, Johnson's result for £ is deduced from the abstract Fatou-NaimDoob theorem. Also, a Carleson-type result is established for solutions of Lu = 0 defined on a union of parabolic regions. The methods employed in this paper were inspired mainly by those in [6,21 and 22].