$μhom$ and multi-microlocal operators
Naofumi Honda, Luca Prelli
TL;DR
The paper extends microlocal analysis to simultaneous microlocalization along multiple submanifolds by introducing a universal multi-normal deformation framework for forest-type configurations. It defines the functor $\mu hom_{\widehat{\chi}}$ (and its sa-variant) to capture morphisms in the multi-microlocal setting and proves its compatibility with base change and composition, yielding canonical isomorphisms that link classical and multi-microlocalizations. It then constructs sheaves of multi-microlocal operators $\mathscr{E}^{\varpi,\mathbb{R}}_{\hat{\chi}}$ and $\mathscr{E}^{\varpi,f,\mathbb{R}}_{\hat{\chi}}$ (including tempered versions), showing they act on (tempered) multi-microfunctions and form rings, with a systematic reduction of redundant indices under transitive-type assumptions. The results provide a hierarchically organized, functorial framework for higher microlocal analysis and its operator theory in subanalytic settings, with potential applications to multi-growth sheaves and holomorphic function classes.
Abstract
In this paper, we construct the multi-microlocalization functor $μhom_{χ}$ of homomorphisms, which is a counterpart of the functor $μhom$ studied by M.Kashiwara and P.Schapira. Furthermore, using the new functor, we also introduce several sheaves of multi-microlocal operators which act on multi-microlocalized objects such as a multi-microfunction.