ACTA issues

Hölder continuity of Green's functions

Lennart Carleson, Vilmos Totik

Acta Sci. Math. (Szeged) 70:3-4(2004), 557-608

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.)