MULTIDIMENSIONAL TAUBERIAN THEOREMS FOR VECTOR-VALUED DISTRIBUTIONS

Stevan Pilipovic, Jasson Vindas

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

Keywords: Abelian and Tauberian theorems, vector-valued distributions, quasiasymptotics, slowly varying functions, Laplace transform, wavelet transform, regularizing transforms, asymptotic behavior of generalized functions

Classification (MSC2000): 40E05, 41A27; 26A12, 40E10, 41A60, 42C40, 46F10, 46F12

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