Duality in Segal-Bargmann Spaces

Abstract

For $\alpha >0$, the Bargmann projection $P_{\alpha }$ is the orthogonal projection from $L^2({\gamma }_{\alpha })$ onto the holomorphic subspace $L_{hol}^2({\gamma }_{\alpha })$, where ${\gamma }_{\alpha }$ is the standard Gaussian probability measure on $\C^n$ with variance $(2\alpha {)}^{-n}$. The space $L_{hol}^2({\gamma }_{\alpha })$ is classically known as the Segal-Bargmann space. We show that $P_{\alpha }$ extends to a bounded operator on $L^p({\gamma }_{\alpha p/2})$, 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 $L_{hol}^p({\gamma }_{\alpha p/2})$ is an $L^{p^{\prime }}$ Segal-Bargmann space, but with the Gaussian measure scaled differently: $(L_{hol}^p({\gamma }_{\alpha p/2}){)}^{\ast }\cong L_{hol}^{p^{\prime }}({\gamma }_{\alpha p^{\prime }/2})$ (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