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8 we can conclude that + + + u(A+ %̄,Λ (z0 , t0 )) ≤ cu(A%,Λ (0, 0)), (3. 13) with boundary data ϕ ∈ C(∂Ω) and let u = uϕ be the corresponding Perron-Wiener-Brelot solution. 6) + γ cu(A+ %,Λ ) ≤ c(%/%̃) u(A%̃,Λ ) − 2 cc̄(%/%̃)γ u(A− %̃,Λ ) ≤ c c̄u(A%,Λ ), where c, 1 ≤ c < ∞, depends only on N, M . POLIDORO where −A12 (τ /δ)x̃ + A11 (τ /δ)(x̃ − x) + δ −1 A12 (τ /δ)(ỹ − y), δx (τ ) = δy (τ ) = τ (A11 (τ /δ)(x̃ − x) + A21 (τ /δ)(x̃ − x)) +Ã12 (τ /δ)(ỹ − y − δ x̃) + A22 (τ /δ)(ỹ − y − δ x̃), (3. By elementary estimates and the Harnack inequality we see that c̄−1 ≤ %q m+ ≤ c̄, (6. 34) (ii) inf BK (A− δ%,Λ (z0 ,t0 ),%/c) inf BK (A+ %,Λ (z0 ,t0 ),%/c) u ≥ c−1 u, sup u, BK (A− %,Λ (z0 ,t0 ),%/c) and (3. The coefficients ai,j and ai are bounded continuous functions and B = (bi,j )i,j=1,. 3, represent the the main (novel) technical components of the paper. By essentially the same argument we have that − A− 1,Λ (0, 0) ∈ A(z,t) (C2,η,Λ (0, 0)), (3. A weak comparison principle and its consequences The main purpose of this section is to prove Lemma 5. Now, using that Ωf,2r0 is an admissible local LipK domain, with LipK -constant M , we first note that d ≈ |x1 − f (x0 , y 0 , t)|, with constants of comparison depending only on N and M , and then that there exists c = c(N, M ), 1 ≤ c < ∞, such that c−1 ≤ τ0 ≤ c. 28) v(A+ %̃1 ,Λ ) ≤ sup BK (A+ %̃1 ,Λ ,ε%̃1 ) u, v(A− %̃1 ,Λ ) ≥ inf BK (A+ %̃ 1 ,Λ u. t−2 −3 Im 2 Im t −1 (C(t))−1 = 12 Using this notation we have that (2. 42) (i) τj + ε̄2 β < δ or (ii) τj + ε̄2 β ≥ δ, where β is the constant appearing in Lemma 2. However, arguing as in the proof of statement (ii) in Lemma 4. , in Lipschitz type domains defined by a function f which is independent of (z̄2 , . 22) c−1 v(A− %,Λ ) u(A+ %,Λ ) ≤ v(A+ v(z, t) %,Λ ) ≤c , u(z, t) u(A− %,Λ ) whenever (z, t) ∈ Ωf,%/c (z0 , t0 ). However, in the context of the equation Ku = 0 it is also possible to give a more intuitive construction of Harnack chains, a construction that gives a non sharp, but equivalent, exponent. Consider the path (γ(τ ), t−τ ) : [0, t − t̃] → RN +1 . Then there exists η = η(N, M ), 0 < η 1, such that if we introduce ± ± C%,2η,Λ (z0 , t0 ), C̃%,2η,Λ (z0 , t0 ), as in (3