We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^{ f}_{ \varphi}(x,y)=(f\ast\varphi_{ y})(x),$ $(x,y)\in\mathbb{ R}^{ n}\times\mathbb{ R}_{ +}$, with kernel $\varphi_{ y}(t)=y^{ -n}\varphi(t/y)$. 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 $\left\{ x_0\right\}\times \mathbb R^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.
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