Tauberian class estimates for vector-valued distributions

TL;DR

This paper investigates conditions under which vector-valued tempered distributions can be characterized as taking values in Banach spaces using Tauberian estimates of their regularizing transforms, generalizing previous theorems.

Contribution

It provides new Tauberian theorems for vector-valued distributions, characterizing Banach space-valued distributions via class estimates and identifying optimal kernels for these results.

Findings

Generalizes earlier Tauberian theorems of Drozhzhinov and Zav'yalov

Establishes conditions for distributions to take values in Banach spaces

Identifies optimal kernels for Tauberian estimates

Abstract

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M^{\mathbf{f}}_{\varphi}(x,y)=(\mathbf{f}\ast\varphi_{y})(x)$, where the kernel $\varphi$ is a test function and $\varphi_{y}(\cdot)=y^{-n}\varphi(\cdot/y)$. 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 $M^{\mathbf{f}}_{\varphi}(x,y)$. 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 $\varphi$ for which these Tauberian results hold.