Integral transforms with exponential kernels and Laplace transform

Let X<--(f)Z-->Y-g be a correspondence of complex manifolds. We study integral transforms associated to kernels exp(phi), with phi meromorphic on Z, acting on formal or moderate cohomologies. Our main application is the Laplace transform. In this case, X is the projective compactification of the vector space V similar or equal to C-n, Y is its dual space, Z = X x Y and phi(z, w) = (z, w). We obtain the isomorphisms: Fx(W)O(V) similar or equal to F<^>[n]x(W)O(V*), THom(F, O-V)similar or equal to THom(F<^>[n], O-V*) where F is a conic and R-constructible sheaf on V and F<^> is its Fourier-Sato transform, Some applications are discussed.