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We investigate mapping properties for the Bargmann transform on modulationspaces whose weights and their reciprocals are allowed to grow faster thanexponentials. We prove that this transform is isometric and bijective frommodulation spaces to convenient Lebesgue spaces of analytic functions. We usethis to prove that such modulation spaces fulfill most of the continuityproperties which are well-known when the weights are moderated. Finally we usethe results to establish continuity properties of Toeplitz andpseudo-differential operators in the context of these modulation spaces.

The Bargmann transform on a broad family of Banach spaces, with applications to Toeplitz and pseudo-differential operators