Integral transforms with exponential kernels and Laplace transform

Let $X \underset{f}{\longleftarrow} Z \underset{g}{\longrightarrow} Y$ be a correspondence of complex manifolds. We study integral transforms associated to kernels $\exp(\varphi)$, with $\varphi$ 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 $V \simeq {\Bbb C}^n$, $Y$ is its dual space, $Z=X\times Y$ and $\varphi(z,w) =\langle z,w \rangle$. We obtain the isomorphisms:

where $F$ is a conic and ${\Bbb R}$-constructible sheaf on $V$ and $F^\wedge$ is its Fourier-Sato transform. Some applications are discussed.