The Harnack inequality explained in the lecture implies that for any ball BR(x4) of R", if u E C² (N) N C(N) is any harmonic function on N where N is a domain with Br(x) C R", then for all x E BR(x.), 1- 1+ u(x.) < u(x) < (1– -u(x.) -1 (1+ O True

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