Abstract:
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 $\rho$.