Duality in Segal–Bargmann spaces

For α>0, the Bargmann projection Pα is the orthogonal projection from L2(γα) onto the holomorphic subspace , where γα is the standard Gaussian probability measure on Cn with variance (2α)−n. The space is classically known as the Segal–Bargmann space. We show that Pα extends to a bounded operator on Lp(γαp/2), and calculate the exact norm of this scaled Lp Bargmann projection. We use this to show that the dual space of the Lp-Segal–Bargmann space is an Lp′ Segal–Bargmann space, but with the Gaussian measure scaled differently: (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.