Mukai's theorem states that if $X$ is an abelian variety, and $\check{X}$ is the dual abelian variety, then the Fourier-Mukai transform corresponding to the Poincare line bundle on $X \times \check{X}$ is a derived equivalence. This is analogous to Pontryagin duality for locally compact abelian topological groups.

Orlov's theorem states (I think) that if $X$ and $Y$ are smooth projective varieties, then every derived equivalence $D^b(X) \to D^b(Y)$ is induced by a Fourier-Mukai transform.

Is there an analysis analogue of Orlov's theorem? If abelian varieties are analogous to locally compact abelian groups, then are varieties analogous to locally compact topological spaces? What is the analogue of projective variety? What is the analogue of smooth? Is it true that any isomorphism of algebras of functions (suitably defined) on [[*whatever the analogue of a smooth projective variety is*]] is induced by some kind of integral transform?

(I came up with these questions while writing up my answer to this question about the Fourier-Mukai transform.)