Let \(b\) be a holomorphic map from the open disc \(\mathbb{D}\) into itself. Then for each \(\alpha \in \mathbb{T}\) there exists a unique positive measure \(\tau_{\alpha}\) such that \[ \frac{1 - |b (z) |^2}{|\alpha - b (z) |^2} = \int_{\mathbb{T}} P_z (\xi) d \tau_{\alpha} (\xi) \] holds where \(P\) denotes the Poisson kernel. The author generalizes some results on the operator \(A_b\) on \(C (\mathbb{T})\) given by \(A_b f (\alpha) = \int_{\mathbb{T}} f (\xi) d \tau_{\alpha} (\xi) \) to the same operator but defined now on the subspace \(\Lambda_{\omega} \subset C (\mathbb{T})\) assoziated to the nonnegative continuous function \(\omega : [ 0, \infty [ \rightarrow [ 0, \infty [\) satisfying some additional conditions. \(\Lambda_{\omega}\) is the space of all continuous functions \(f\) whose module of continuity \(\omega (f, \cdot)\) is dominated by \(\omega\). The norm on \(\Lambda_{\omega}\) is defined by \[ \|f \|_{\omega} = \|f \|_{\infty} + \sup_{\delta} \frac{\omega (f, \delta)}{\omega (\delta)}. \] The author proves that \(A_b\) maps \(\Lambda_{\omega}\) into itself and is bounded. Moreover \(A_b \mid \Lambda_{\omega}\) is compact iff all \(\tau_{\alpha}\) are diffuse.