Duality, K\"unneth formulae, and integral transforms in microlocal geometry

We study the dualizability of sheaves on manifolds with isotropic singular supports Sh_ (M) and mcirosheaves with isotropic supports sh_ () and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports Sh_ᵇ (M) ₀, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects Sh_ᶜ (M) ₀ when the wrap-once functor is an equivalence, for instance, when is a full Legendrian stop or a swappable Legendrian stop.