(\overline{\Omega}, \omega_{\Omega})$.

Since $\phi$ is almost 1-1 on $\partial W$ with respect to $\omega_{W}$, it is apparent that $\tilde{\phi}$ has a well-defined inverse function on a set of full $\omega_{\Omega}$ measure. By Lemma \[l:key\] $A(\Omega)$ is boundedly pointwise dense in $H^{\infty}(\Omega)$. Choose a bounded sequence $\{f_{n}\}$ in $A(\Omega)$ such that $f_{n}\rightarrow \phi^{-1}$ on $\Omega$. Then one can show that $f_{n} \rightarrow \tilde{\phi}^{-1}$ in the weak-star topology of $L^{\infty}(\omega_{\Omega})$. Hence $\tilde{\phi}^{-1} \in A^{\infty}(\overline{W}, \omega_{\Omega})$.

The following is one of our key lemmas.

\[l:key2\] Suppose that each component of $K^{\circ}$ is multi-nicely connected and the harmonic measures of the components of $K^{\circ}$ are mutually singular. Let $U$ be a component of $\nabla A^{q}(K,\mu)$ and let $\Omega$ be the component of $K^{\circ}$ that contains $U$. Set $\tau = \mu|\overline{\Omega}$. Then $
A^{q}(\overline{\Omega},\tau)\subset A^{q}(K,\mu)$ and $ U \subset
\nabla A^{q}(\overline{\Omega},\tau)$.

We first assume that $A^{q}(K,\mu)$ is pure. Let $\{\Omega_{i}\}_{i=0}^{\infty}$ be the collection of all the components of $K^{\circ}$. Without loss of generality, let $\Omega_{0} = \Omega$. Suppose $h\in A^{q}(\overline{\Omega}, \tau)$ and choose a sequence $\{r_{n}\}$ in $A(\overline{\Omega})$ such that $r_{n}\rightarrow h $ in $L^{q}(\tau)$. Let $\omega = \sum_{i}
\frac{1}{2^{i}} \omega_{\Omega_{i}}$. Fix a function $r_{n}$. Extend both $h$ and $r_{n}$ to be functions on the 