Tauberian class estimates for vector-valued distributions

Abstract

© 2019 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. We study Tauberian properties of regularizing transforms of vector-valued tempered distributions. The transforms have the form Mf (x, y) = (f∗y)(x), where the kernel is a test function and .n.( E/y). We investigate conditions which ensure that a distribution that a priori takes values in a 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 Mf. (x, y). Our results generalize and improve earlier Tauberian theorems due to Drozhzhinov and Zavfyalov. Special attention is paid to finding the optimal class of kernels . for which these Tauberian results hold.