An interpolation problem for extended Gevrey regularity

We study an interpolation problem in extended Gevrey classes defined by the sequences $M_n^{τ,σ}=n^{τn^σ}$, $n\in\mathbb N$, $τ>0$, $σ>1$. We show that, under a suitable growth condition on a divergent sequence of derivative orders, estimates of extended Gevrey type imposed only along this sequence