Free interpolation in the spaces of analytic functions with derivative of order $s$ in a Hardy space

We consider the problem of free interpolation for the spaces of analytic functions with derivative of order $s$ in the Hardy space $H^p$. For the sets that satisfy the Stolz condition, we obtain a condition necessary for interpolation: if $1\leq p<\infty$, then the set must be a union of $s$ sparse sets. For $p=\infty$, we obtain a necessary and sufficient condition for interpolation: the set must be a union of $s+1$ sparse sets. In this case, we construct an extension operator.