An interpolation problem for extended Gevrey regularity

Abstract

We study an interpolation problem in extended Gevrey classes defined by the sequences $M_n^{\tau,\sigma}=n^{\tau n^\sigma}$, $n\in\mathbb N$, $\tau>0$, $\sigma>1$. 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 imply corresponding estimates for all derivative orders. In particular, we explicitly quantify the change of the parameters in the resulting estimates. To this end, we first establish an equivalence between two definitions of the extended Gevrey classes, one involving the supergeometric factor $h^{n^\sigma}$ and another in which this factor is omitted. We then establish a corresponding interpolation principle for extended Gelfand--Shilov spaces, including a symmetric characterization and a formulation in terms of the associated function of the defining sequence.