Tauberian class estimates for vector-valued distributions

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.Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1012.509

Multidimensional Tauberian theorems for vector-valued distributions

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x(0)} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem