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 f(x, y) = (f � 'y)(x), (x, y) 2 R n × R+, with ker- nel 'y(t) = y−n'(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 dis- tribution on {x0} × R m . In addition, we present a new proof of Littlewood's Tauberian theorem.