Widths of the unit ball in $H^\infty$ in weighted spaces $L_q(\mu)$

The study of the widths of the unit ball in the Hardy space $H^\infty$ in weighted spaces $L_q(\mu)$ is carried out. Sharp lower estimates of these widths in terms of the capacity of the support of the measure $\mu$ are obtained. The precise values of the widths are calculated for Blaschke lemniscates. For the measures $d\mu =p\,dS$, where $dS$ is plane Lebesgue measure and $p$ is a positive continuous weight, an asymptotic formula is found.