Duality in Segal–Bargmann spaces

Abstract

For α > 0 , the Bargmann projection P α is the orthogonal projection from L 2 ( γ α ) onto the holomorphic subspace L hol 2 ( γ α ) , where γ α is the standard Gaussian probability measure on C n with variance ( 2 α ) − n . The space L hol 2 ( γ α ) is classically known as the Segal–Bargmann space. We show that P α extends to a bounded operator on L p ( γ α p / 2 ) , and calculate the exact norm of this scaled L p Bargmann projection. We use this to show that the dual space of the L p -Segal–Bargmann space L hol p ( γ α p / 2 ) is an L p ′ Segal–Bargmann space, but with the Gaussian measure scaled differently: ( L hol p ( γ α p / 2 ) ) ⁎ ≅ L hol p ′ ( γ α p ′ / 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.