Abstract: In order to discuss the Fourier-Sato transform of not necessarily conic sheaves, we compensate the lack of homogeneity by adding an extra variable. We can then extend a theorem by Kashiwara and Schapira on the Laplace transform for temperate holomorphic functions to obtain Paley-Wiener type results. As a key ingredient for this approach, we introduce the subanalytic sheaf of functions with exponential growth, which could be of independent interest.