On the nuclearity of weighted spaces of smooth functions

TL;DR

This paper investigates conditions under which weighted spaces of smooth functions are nuclear, facilitating the application of the Schwartz kernel theorem and tensor product techniques in analysis.

Contribution

It provides new sufficient conditions on weights that ensure the nuclearity of weighted smooth function spaces, expanding their functional analytic properties.

Findings

Identifies specific weight conditions for nuclearity

Enhances understanding of tensor product applications in analysis

Supports the use of Schwartz kernel theorem in weighted spaces

Abstract

Nuclearity plays an important role for the Schwartz kernel theorem to hold and in transferring the surjectivity of a linear partial differential operator from scalar-valued to vector-valued functions via tensor product theory. In this paper we study weighted spaces $\mathcal{EV}(\Omega)$ of smooth functions on an open subset $\Omega\subset\mathbb{R}^{d}$ whose topology is given by a family of weights $\mathcal{V}$. We derive sufficient conditions on the weights which make $\mathcal{EV}(\Omega)$ a nuclear space.