Weighted function spaces: convolutors, multipliers, and mollifiers

TL;DR

This paper characterizes smooth Gelfand-Shilov type function spaces and their convolutor and multiplier spaces using mollification, with a focus on the roles of translation-invariant Banach spaces and weight functions.

Contribution

It introduces a new moment-wise decomposition property and provides complete characterizations of these function spaces via mollification techniques.

Findings

Characterization of Gelfand-Shilov type spaces through mollification.

Introduction of the moment-wise decomposition property.

Complete descriptions of convolutor and multiplier spaces.

Abstract

We study smooth function spaces of Gelfand-Shilov type, with global behavior governed through a translation-invariant Banach function space and localized via a weight function system. We clarify the roles of the translation-invariant Banach function space, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces-as well as the corresponding convolutor and multiplier spaces-through mollification. For this purpose, we introduce the moment-wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.