The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

Abstract

We investigate mapping properties for the Bargmann transform on an extended family of modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are valid for modulation spaces with moderate weights. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators on these modulation spaces, and on Gelfand–Shilov spaces.