Abstract. Wiener type characterizations are given for Hölder continuity of Green's functions at boundary points of the underlying domains. For Hölder continuity with some positive exponent it is shown that if $0$ is a boundary point of a domain $G\subset{\overline{\bf C}}$ and $G$ satisfies the cone condition at $0$, then Green's function $g_G(\cdot,a)$ is Hölder continuous at $0$ if and only if the sequence ${\cal N}_{\partial G}(\varepsilon )$ of those $n\in{\bf N}$ for which $\mathop{\rm cap} (\partial G\cap D_{2^{-n}}(0))\ge\varepsilon 2^{-n}$, is of positive lower density in ${\bf N}$. For $G=\overline{\bf C}\setminus E$ with $E\subseteq[0,1]$ the optimal Hölder 1/2 smoothness holds at 0 if and only if $\sum_k 2^k(\mathop{\rm cap} (I_k)-\mathop{\rm cap} (E_k))< \infty $ where $I_k=[0,2^{-k}]$, and $E_k$ is the union of $I_k\cap E$ with $[0,\varepsilon2^{-k}]\cup[(1-\varepsilon )2^{-k},2^{-k}]$ for some $\varepsilon < 1/3$. The corresponding uniform results are also true, and similar statements hold in higher dimensions.

AMS Subject Classification (1991): 30C85, 31A15

Keyword(s): Green's functions, Hölder continuity, logarithmic capacity, harmonic measure, cone condition, Cantor sets, Wiener type characterization

Received September 16, 2003, and in final form September 17, 2004. (Registered under 5833/2009.)