A remark on continuous K-theory and Fourier-Sato transformation

In this note, we prove a generalization of Efimov’s computation for the universal localizing invariant of categories of sheaves with certain microsupport constraints. The proof is based on certain categorical equivalences given by the Fourier-Sato transform, which is different from the original proof. As an application, we compute the universal localizing invariant of the category of almost quasi-coherent sheaves on the Novikov toric scheme introduced by Vaintrob [ <italic>Coherent-constructible correspondences and log-perfectoid mirror symmetry for the torus</italic> , 2017. Preprint, available at https://math.berkeley.edu/ vaintrob/toric.pdf].