Representation Theorems and Fatou Theorems for Second‐order Linear Parabolic Partial Differential Equations

It was pointed out to me by Professor J. Chabrowski that there are errors in Lemmas 4 and 5 on pp. 331–33. In both cases the auxiliary functions introduced need not satisfy the equation. However, all of the results remain valid with a slight strengthening of the hypotheses. Assume throughout the paper that a ij ɛ C 2,α , b i ɛ C 1,α , c ɛ C α , 0 < α < 1. Lemma 4 becomes a consequence of Theorem 1 and Lemma 5 must be restated as Lemma 5. Let u be a regular solution of Lu = 0 in R n × (0, T ) which satisfies the global constraint ∥ u (·, t )∥ < M , 0 < t ⩽ T , Where 1 ⩽ p < + ∞. If 0 < t 0 < t 0 < t ⩽ T , u ( x,t ) = ∫ Γ ( x , t ; ξ , t 0 ) u ( ξ , t 0 ) d ξ .Proof . Both right‐and left‐hand sides are solutions of the equation Lv = 0, V ( x,t 0 ) = u ( x,t 0 ), and both satisfy∫ t 0 T∫ R n| v ( x , t ) exp ( ‐ β ~ | x | 2 ) | d x d t < +∞, forβ ~large enough. Thus, by Theorem 16, p. 29, of Friedman's Parabolic partial differential equations , the equality follows. Then the proof of Theorem 1 proceeds as before, but now it must be noted that Γ( x , t ;·, t 0 ) → Γ( x , t ;·,0) strongly inL p ′ ‐ ‐ βas t 0 → 0. One must make a similar adjustment to Lemmas 4′, 5′, and Theorem 3, using Chabrowski's Theorem 1, p. 228 ( Ann. Polon. Math. 24 (1971)).