We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^f_{arphi}(x,y)=(f*arphi_y)(x)$, $(x,y)inmathbb{R}^nimesmathbb{R}_+$, with kernel $arphi_{y}(t)=y^{-n}arphi(t/y)$. 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 ${x_0}imesmathbb R^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.
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机译：我们证明了几个Tauberian定理，用于正规化向量值分布的变换。 $ f $的正则化变换由整数变换$ M ^ f _ { varphi}（x，y）=（f * varphi_y）（x）$，$（x，y） in mathbb {R } ^ n times mathbb {R} _ + $，内核为$ varphi_ {y}（t）= y ^ {-n} varphi（t / y）$。我们将结果应用到一类Cauchy问题的渐近稳定性分析，Laplace变换的Tauberian定理，分布空间中的拟症状的比较中，并给出了存在分布踪迹的充要条件。 $ {x_0 } times mathbb R ^ m $。此外，我们提出了Littlewood的Tauberian定理的新证明。
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