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We prove the boundary Harnack principle and the Carleson type estimate for ratios of solutions u/v of non–divergence second order elliptic equations Lu = aijDiju + biDiu = 0 in a bounded domain Ω ⊂ R n. We assume that bi ∈ Ln(Ω) and Ω is a Hölder domain of order α ∈ (0, 1) satisfying a strong regularity condition.

THE BOUNDARY HARNACK PRINCIPLE IN HÖLDER DOMAINS WITH A STRONG REGULARITY