Duality in Segal-Bargmann Spaces

TL;DR

This paper investigates the boundedness and norm of the Bargmann projection on scaled $L^p$ spaces in Segal-Bargmann spaces, revealing duality relations and providing dimension-independent estimates.

Contribution

It extends the Bargmann projection to $L^p$ spaces with explicit norm calculations and clarifies the duality structure of $L^p$-Segal-Bargmann spaces.

Findings

Bargmann projection is bounded on $L^p( ext{scaled Gaussian})$ spaces.

Exact norm of the scaled $L^p$ Bargmann projection is computed.

Dual space of $L^p$-Segal-Bargmann space is isomorphic to a scaled $L^{p'}$ space.

Abstract

For $\alpha>0$, the Bargmann projection $P_\alpha$ is the orthogonal projection from $L^2(\gamma_\alpha)$ onto the holomorphic subspace $L^2_{hol}(\gamma_\alpha)$, where $\gamma_\alpha$ is the standard Gaussian probability measure on $\C^n$ with variance $(2\alpha)^{-n}$. The space $L^2_{hol}(\gamma_\alpha)$ 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 $L^p_{hol}(\gamma_{\alpha p/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian measure scaled differently: $(L^p_{hol}(\gamma_{\alpha p/2}))^* \cong L^{p'}_{hol}(\gamma_{\alpha p'/2})$ (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.