Multidimensional Tauberian theorems for vector-valued distributions

Abstract

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M_{\phi }^f(x,y)=(f\ast {\phi }_y)(x),$ $(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}$, 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 $\left\{x_0\right\}\times {\mathbb{R}}^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.

28 pages. arXiv admin note: substantial text overlap with arXiv:1012.5090

References in corpus (1)

• Distributional versions of Littlewood's Tauberian theorem

Cited by in corpus (5)

• Tauberian class estimates for vector-valued distributions
• Multiresolution expansions of distributions: Pointwise convergence and quasiasymptotic behavior
• A multidimensional Tauberian theorem for Laplace transforms of ultradistributions
• Structural theorems for quasiasymptotics of ultradistributions
• Local regularity in non-linear generalized functions