Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains

On a Lipschitz domain D in R n , three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on D . The third theorem, on non-tangential limits of quotient of two positive harmonic functions in D , generalizes Doob’s relative Fatou theorem on a sphere. The main tools are maximum principle, Harnack inequality and differentiation of measures.