Duality in Segal-Bargmann Spaces

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 ${\text{C}}^n$ 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 $L^p(\gamma_{\alpha p/2})$,andcalculatetheexactnormofthisscaled$L^p$ Bargmann projection. We use this to show that the dual space of the $L^p$-Segal-Bargmann space $L_{hol}^p({\gamma }_{\alpha p\mathrm{/}2})$ is an $L^{p^{\mathrm{\prime }}}$ 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.