Thomson was unable to offer any result for $R^{q}(K,\mu)$ that is beyond $P^{q}(\mu)$ (that is, the disk case in our setting) (see [@jm p. 505]).

To tackle the problem, we first need to restrict our effort on those $K$ such that the components of $K^{\circ}$ are finitely connected. In fact, the author shows in [@comm Theorem 2] even when $K$ is a simplest kind of infinitely connected domains, such as a “road-runner”, our main theorem (Theorem \[t:main\]) for $A^{q}(K,\mu)$ could fail. In this paper, we seek a necessary and sufficient condition on $K$ so that a Thomson type of structure theorem holds for $A^{q}(K,\mu)$.

A domain is called a circular domain if its boundary consists of finitely many disjoint circles. We call a domain $U$ multi-nicely connected if there is a circular domain $W$ and a conformal map $\alpha$ from $W$ onto $U$ such that $\alpha$ is almost 1-1 on $\partial W$ with respect to the arclength measure.

Our main theorem, Theorem \[t:main\], extends Thomson’s theorem to $A^{q}(K,\mu)$ in the case when the components of $K^{\circ}$ are multi-nicely connected and the harmonic measures of the components of $K^{\circ}$ are mutually singular. We also show that the condition of Theorem \[t:main\] 