Embedding theorems for weighted classes of harmonic and analytic functions

The inequality \[ (\int_\Omega|u|^qd\mu)^{1/q} \leq C (\int_\Omega|u|^p\rho d\lambda)^{1/p}, \tag{1} \] is established for analytic (harmonic) functions. Here $\rho$ is a continuous weight functions, $\lambda$ the Lebesgue measure and $\mu$ – a Borel measure. Necessary and sufficient conditions on the measure $\mu$ are given for some concrete $\Omega$ and $\rho$.