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 $L^2_{hol}(γ_α)$, where $γ_α$ is the standard Gaussian probability measure on $\C^n$ with variance $(2α)^{-n}$. The space $L^2_{hol}(γ_α)$ is classically known as the Segal-Bargmann space. We show that $P_α$ extends to a bounded operator on $L^p(γ_{αp/2})$, 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 $L^p_{hol}(γ_{αp/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian measure scaled differently: $(L^p_{hol}(γ_{αp/2}))^* \cong L^{p'}_{hol}(γ_{αp'/2})$ (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.

24 pages