Geodesic
Tauberian class estimates for vector-valued distributions
Sbornik. Mathematics, Tome 210 (2019) no. 2, pp. 272-296
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We study Tauberian properties of regularizing transforms of vector-valued tempered distributions. The transforms have the form $M^\mathbf f_\varphi(x,y)=(\mathbf f\ast\varphi_y)(x)$, where the kernel $\varphi$ is a test function and $\varphi_y(\cdot)=y^{-n}\varphi(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in a 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 $M^\mathbf f_\varphi(x,y)$. Our results generalize and improve earlier Tauberian theorems due to Drozhzhinov and Zav'yalov. Special attention is paid to finding the optimal class of kernels $\varphi$ for which these Tauberian results hold. Bibliography: 24 titles.
Keywords: regularizing transforms, class estimates, Tauberian theorems, vector-valued distributions, wavelet transform. 
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S. Pilipović; J. Vindas. Tauberian class estimates for vector-valued distributions. Sbornik. Mathematics, Tome 210 (2019) no. 2, pp. 272-296. http://geodesic.mathdoc.fr/item/SM_2019_210_2_a4/

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