Tauberian class estimates for vector-valued distributions

Abstract

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M^{\mathbf{f}}_φ(x,y)=(\mathbf{f}\astφ_{y})(x)$, where the kernel $φ$ is a test function and $φ_{y}(\cdot)=y^{-n}φ(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform $M^{\mathbf{f}}_φ(x,y)$. Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov [Sb. Math. 194 (2003), 1599-1646]. Special attention is paid to find the optimal class of kernels $φ$ for which these Tauberian results hold.