Duality in Segal-Bargmann spaces

摘要

For α>0, the Bargmann projection Pα is the orthogonal projection from L2(γα) onto the holomorphic subspace Lhol2(γα), where γα is the standard Gaussian probability measure on Cn with variance (2α)-n. The space Lhol2(γα) 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 Lholp(γαp/2) is an Lp- Segal-Bargmann space, but with the Gaussian measure scaled differently: (Lholp(γαp/2))*?Lholp'(γαp'/2) (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.