Precise comparison with the motivic Mellin transform framework

Develop a precise comparison between the Grothendieck ring of varieties with characters and its integration-over-characters formalism introduced in this paper and the motivic Mellin transforms framework of Cluckers–Loeser–Nguyen–Vermeulen, rigorously identifying how the basic objects, operations (including Fourier/Poisson-type transforms), and integration procedures correspond or differ between the two settings.

Background

The paper introduces a new Grothendieck ring of varieties with characters to develop a motivic Fourier theory for split algebraic tori, including an integration operator over the character side, and a motivic Poisson formula.

The authors note similarities with the recently proposed motivic Mellin transforms framework, but a detailed, formal comparison between the two frameworks—identifying correspondences between constructions, functorial properties, and transform/integration mechanisms—has not yet been carried out.