A note on kernel functions of Dirichlet spaces

For a planar domain Ω, we consider the Dirichlet spaces with respect to a base point $\zeta \in \Omega$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on $\Omega \times \Omega$. This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.