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

We study the dualizability of sheaves on manifolds with isotropic singular supports Sh_Λ(M) and mcirosheaves with isotropic supports μ sh_Λ(Λ) 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_Λᵇ(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_Λᶜ(M)₀ when the wrap-once functor is an equivalence, for instance, when Λ is a full Legendrian stop or a swappable Legendrian stop.