An example concerning parts and Newtonian capacity

Abstract:

We prove the following theorem: Let $\phi$ be an admissible function with $\phi ({0^ + }) = 0$ and p a nonnegative integer. Then there is a compact set X in the plane and $x \in X$ such that x is a nonpeak point of $R(X)$, \[ \sum {{2^{(p + 1)n}}\phi {{({2^{ - n}})}^{ - 1}}\mathcal {C}({A_n}(x)\backslash P(x)) < \infty ,} \] while $\Sigma {2^{(p + 1)n}}\phi {({2^{ - n}})^{ - 1}}\gamma ({A_n}(x)\backslash X) = \infty$, where ${A_n}(x) = \{ {2^{ - n - 1}} \leqslant |z - x| \leqslant {2^{ - n}}\} ,P(x)$ denotes the Gleason part of $x,\gamma$ the analytic capacity and $\mathcal {C}$ the Newtonian capacity.