An Inequality for Fourier–Laplace Transforms of Entire Functions, and the Existence of Exponential Frames in Fock Space

We study mapping properties of the Fourier–Laplace transform between certain spaces of entire functions. We introduce a variant of the classical Fock space by integrating against the Monge–Ampère measure of the weight function and show that the norm of the Fourier–Laplace transform, in a dual Fock space, dominates the norm of the function. Equality holds when the weight function is an Hermitean form. As an application we get a criterium for the existence of frames of exponential functions in Fock space.