Speaker
Description
Previously (in 2011), the author together with A.A. Pekarski obtained an inequality connecting the quasinorms of rational functions with respect to the linear measure on $\mathbb{R}$ and the planar measure in the half-plane $\Pi=\{z\in\mathbb{C}:\Im z>0\}$. In this context, the rational functions belonged to the weighted Lebesgue space in $\Pi$, where the quasinorm is defined as follows
$$
\|f\|_{L_{p,\mu}(\Pi)}=\left(\int_{\Pi}(\Im z)^{p\mu-1}|f(z)|^p\,dm_2(z)\right)^{1/p},\quad p>0,\quad \mu>0.
$$
Here $m_2$ is the planar Lebesgue measure in $\mathbb{C}$.
The report will discuss some applications of the noted inequality. Furthermore, a generalization of this inequality for a domain whose boundary is a Lavrent'ev curve will be presented.
