Tauberian class estimates for vector-valued distributions

Abstract

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M_{\phi }^{\mathbf{f}}(x,y)=(\mathbf{f}\ast {\phi }_y)(x)$, where the kernel $\phi$ is a test function and ${\phi }_y(\cdot )=y^{-n}\phi (\cdot /y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform $M_{\phi }^{\mathbf{f}}(x,y)$. Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov [Sb. Math. 194 (2003), 1599-1646]. Special attention is paid to find the optimal class of kernels $\phi$ for which these Tauberian results hold.

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

References in corpus (1)

• Optimal Tauberian constant in Ingham's theorem for Laplace transforms

Cited by in corpus (3)

• A multidimensional Tauberian theorem for Laplace transforms of ultradistributions
• Multiresolution expansions and wavelets in Gelfand-Shilov spaces
• Besov regularity in non-linear generalized functions