Embedding theorems for weighted classes of harmonic and analytic functions

V. L. Oleinik

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 120-137

Résumé

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 $\rho$.