Analytic capacity, Hölder conditions and $\tau$-spikes

Abstract:

We consider the uniform algebra $R(X)$, for compact $X \subset {\text {C}}$, in relation to the condition ${I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) < + \infty$, where $0 \leq p \in {\mathbf {Z}},0 < \alpha < 1,\gamma$ is analytic capacity, and ${A_n}(x)$ is the annulus $\{ z \in {\text {C}}:{2^{ - n - 1}} < |z - x| < {2^{ - n}}\}$. We introduce the notion of $\tau$-spike for $\tau > 0$, and show that ${I_{p + \alpha }} = + \infty$ implies $x$ is a $p + \alpha$-spike. If $\mathring {X}$ satisfies a cone condition at $x$, and ${I_{p + \alpha }} < + \infty$, we show that the $p$th derivatives of the functions in $R(X)$ satisfy a uniform Hölder condition at $x$ for nontangential approach. The structure of the set of non-$\tau$-spikes is examined and the results are applied to rational approximation. A geometric question is settled.