1$ and $Sf=0,$ then $f=0.$ (Reason: if $Sf(i\xi)=0$ for all $\xi\in\mathbb{R}^{d}$ then the Fourier transform of $f(x)\rho(x)$ is identically zero.) See Section 4 for a brief discussion of the motivation for the definition of this transform, and \cite{H4,F} for additional information. Observe from (\ref{sdef2}) that the Segal-Bargmann transform is very closely related to the Fourier transform. (In this form, $S$ is essentially the same as the Fourier-Wiener transform. See \cite{GM} or \cite{H3} for more information on the connection of the Fourier-Wiener transform to the Segal-Bargmann transform.) In considering $Sf\left( x+iy\right) ,$ one should interpret $x$ as a position variable and $y$ as a frequency variable. In the context of quantum mechanics, the frequency variable has the interpretation of momentum. The transform given here is essentially the finite-dimensional version of the transform described in \cite{S} (see also \cite{BSZ}), and differs from the transform of \cite{B1} by the ground state transformation. (See Section 4.) The use of the Gaussian measure instead of the Lebesgue measure of \cite{B1} is natural in light of the importance of the scale of $L^{p}$ spaces relative to a Gaussian measure, in connection with hypercontractivity and logarithmic Sobolev inequalities. (See Section 2.3 and Section 4.) Note that when working with the Gaussian measure $\rho\left( x\right) \,dx,$ the value of $p$ has a dramatic effect on the allowed growth at infinity of a function in $L^{p}$: a function in $L^{p}(\mathbb{R}^{d},\rho)$ can grow roughly like $e^{x^{2}/2p}.$ In the case $p=2,$ the image of $L^{p}(\mathbb{R}^{d},\rho)$ under $S$ is described by the following result \cite{B1,S}. \begin{theorem} \label{sb.thm}For any $f\in L^{2}(\mathbb{R}^{d},\rho)$, $Sf$ is a holomorphic function on $\mathbb{C}^{d}$ and \[ \int_{\mathbb{R}^{d}}\left| f\left( x\right) \right| ^{2}\rho\left( x\right) \,dx=\pi^{-d}\int_{\mathbb{C}^{d}}\left| Sf\left( z\right) \right| ^{2}e^{-\left| z\right| ^{2}}\,dz. \] Conversely, if $F$ is a holomorphic function on $\mathbb{C}^{d}$ for which \[ \pi^{-d}\int_{\mathbb{C}^{d}}\left| F\left( z\right) \right| ^{2}e^{-\left| z\right| ^{2}}\,dz<\infty, \] then there is a unique $f\in L^{2}(\mathbb{R}^{d},\rho)$ such that $F=Sf.$ \end{theorem} The main objective of this paper is to give as precise as possible a description of the image of $L^{p}(\mathbb{R}^{d},\rho)$ for $p\neq2.$ I know of no integrability condition that exactly characterizes the image for $p\neq2.$ Instead we will obtain necessary conditions on the image and slightly stronger sufficient conditions. If $F$ is in the image of $L^{p}(\mathbb{R}^{d},\rho)$ then H\"{o}lder's inequality will tell us that \begin{equation} \left| F(x+iy)\right| \leq C\,e^{y^{2}/2}e^{x^{2}/2(p-1)}.\label{first.bnd}% \end{equation} In the other direction we will show that if $F$ is holomorphic and if \begin{equation} \int_{\mathbb{C}^{d}}\left| F(x+iy)\right| e^{-y^{2}/2}e^{-x^{2}% /2(p-1)}\,dx\,dy<\infty\label{first.int}% \end{equation} then $F$ is the image of a unique $f$ in $L^{p}(\mathbb{R}^{d},\rho).$ We will give sharper results for certain ranges of $p.$ Note that the growth condition in the $y$-direction is the same for all $p,$ but the growth condition in the $x$-direction is very different for different values of $p.$ These results are described in Section 2.1. The sufficient condition (\ref{first.int}) is only polynomially stronger than the necessary condition in (\ref{first.bnd}). That is, if $F$ satisfies polynomially better bounds than (\ref{first.bnd}), say \[ \left| F(x+iy)\right| \leq C\,e^{y^{2}/2}e^{x^{2}/2(p-1)}\frac{1}{\left( 1+\left| x\right| \right) ^{d+\varepsilon}\left( 1+\left| y\right| \right) ^{d+\varepsilon}}% \] then (\ref{first.int}) will hold and $F$ will be in the image of $L^{p}(\mathbb{R}^{d},\rho).$ As a consequence of the polynomial closeness of the two conditions, we will obtain a single necessary-and-sufficient condition on the image of the ``$L^{p}$ Schwartz space,'' that is, the space of functions $f\in L^{p}(\mathbb{R}^{d},\rho)$ for which $x^{\alpha}\left( \partial/\partial x\right) ^{\beta}f\left( x\right) $ is in $L^{p}% (\mathbb{R}^{d},\rho)$ for all multi-indices $\alpha$ and $\beta.$ This result generalizes a result of Bargmann \cite{B2} for the case $p=2,$ and is described in Section 2.2. Finally, I address the matter of obtaining dimension-independent conditions. Nelson's hypercontractivity theorem gives us dimension-independent necessary conditions (for $1

1+q/2,$ and that $S$ does not map $L^{p}(\mathbb{R}^{d},\rho)$ into $L^{q}(\mathbb{C}^{d},\mu)$ if $q>2$ or $p<1+q/2.$ The present paper is an attempt to obtain a more precise characterization of the image of $L^{p}(\mathbb{R}^{d},\rho).$ \section{Statement of results} \subsection{Bounds on the transform of $L^{p}$ functions} \begin{theorem} \label{holder.thm}Fix $p$ with $1

1.$ I give one last sufficient condition that in some cases may be better than Theorem \ref{l1.thm} or Theorem \ref{interpol2.thm}. \begin{theorem} \label{l1lp.thm}Fix $p$ with $1

2.$ On the other hand, a holomorphic function $F$ satisfying \[ \int_{\mathbb{C}^{d}}\left| F\left( \sqrt{p-1}z\right) \right| ^{2}% \frac{e^{-\left| z\right| ^{2}}}{\pi^{d}}\,dz<\infty \] can grow roughly like $e^{y^{2}/2(p-1)}$ in the $y$-direction, whereas Theorem \ref{interpol1.thm} allows only growth like $e^{y^{2}/2}.$ Similarly for $p>2,$ the condition for Theorem \ref{hyper2.thm} requires slightly weaker growth conditions in the $x$-direction than Theorem \ref{interpol2.thm} (because $2>p^{\prime}$), but much stronger conditions in the $y$-direction ($e^{y^{2}/2(p-1)}$ versus $e^{y^{2}/2}$). It should be pointed out that in the infinite-dimensional case, bounds do not imply integrability. That is, even having bounds like $\left| F(x+iy)\right| \leq Ce^{\varepsilon(x^{2}+y^{2)}}$ for some small $\varepsilon$ is insufficient to guarantee that $F$ is in the Segal-Bargmann space. Thus in the infinite-dimensional case it is useful to have both pointwise bounds (from Theorem \ref{holder.thm}) and integrability conditions (from Theorem \ref{hyper1.thm}) on the image of $L^{p}.$ \section{Proofs} \subsection{Bounds on the transform of $L^{p}$ functions} \textit{Proof of Theorem \ref{holder.thm}}. Formula (\ref{sdef2}) expresses the Segal-Bargmann transform as \[ Sf\left( z\right) =e^{-z^{2}/2}\int_{\mathbb{R}^{d}}e^{z\cdot x}f\left( x\right) \rho\left( x\right) \,dx. \] Thus by H\"{o}lder's Inequality, \begin{equation} \left| Sf\left( z\right) \right| \leq\left| e^{-z^{2}/2}\right| \left\| f\right\| _{L^{p}(\mathbb{R}^{d},\rho)}\left\| e^{z\cdot x}\right\| _{L^{q}(\mathbb{R}^{d},\rho)}\label{holder}% \end{equation} where $1/p+1/q=1.$ Now, $\left| e^{z\cdot x}\right| =e^{\operatorname{Re}% z\cdot x}.$ Completing the square gives \begin{align*} \int_{\mathbb{R}^{d}}\left| e^{z\cdot x}\right| ^{q}\rho\left( x\right) \,dx & =\left( 2\pi\right) ^{-d/2}\int_{\mathbb{R}^{d}}% e^{q\operatorname{Re}z\cdot x}e^{-x^{2}/2}\,dx\\ & =e^{q^{2}(\operatorname{Re}z)^{2}/2}\left( 2\pi\right) ^{-d/2}% \int_{\mathbb{R}^{d}}e^{-(x-q\operatorname{Re}z)^{2}/2}\,dx\\ & =e^{q^{2}(\operatorname{Re}z)^{2}/2}. \end{align*} Thus $\left\| e^{z\cdot x}\right\| _{L^{q}(\mathbb{R}^{d},\rho )}=e^{q(\operatorname{Re}z)^{2}/2}.$ Then simplifying the bounds in (\ref{holder}) gives the estimate in Theorem \ref{holder.thm}. Now if $f\in L^{p}(\mathbb{R}^{d},\rho)$ is a polynomial, then using (\ref{sdef1}) we can see that $Sf$ is a polynomial, so the expression \begin{equation} \left| Sf\left( x+iy\right) \right| e^{-y^{2}/2}e^{-x^{2}/2(p-1)}% \label{lim.quant}% \end{equation} will certainly tend to zero at infinity. For any $f$ in $L^{p}(\mathbb{R}% ^{d},\rho)$ we can approximate $f$ in the $L^{p}$ norm by a sequence $f_{n}$ of polynomials. By the bounds just established, the expression (\ref{lim.quant}) for $f_{n}$ will converge uniformly to the same expression for $f.$ So in the limit (\ref{lim.quant}) still tends to zero at infinity, establishing (\ref{zero.lim}). $\square$ \textit{Proof of Theorem \ref{l1.thm}}. The uniqueness of $f$ follows from the injectivity of $S$ on $L^{p}(\mathbb{R}^{d},\rho),$ discussed in the introduction. For the existence we will make use of an inversion formula obtained from the following result. \begin{theorem} \label{st.thm}For all $p>1,$ the Segal-Bargmann transform $S$ is a unitary map of $L^{2}(\mathbb{R}^{d},\rho_{p/2})$ onto $\mathcal{H}L^{2}(\mathbb{C}% ^{d},\mu_{p/2}),$ where \[ \rho_{p/2}(x)=\left( \pi p\right) ^{-d/2}e^{-x^{2}/p}% \] and where \[ \mu_{p/2}\left( x+iy\right) =\left( \pi(p-1)\right) ^{-d/2}\pi ^{-d/2}e^{-x^{2}/(p-1)}e^{-y^{2}}. \] Here $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{p/2})$ denotes the space of holomorphic functions on $\mathbb{C}^{d}$ that are square-integrable with respect to $\mu_{p/2}\left( x+iy\right) \,dx\,dy.$ \end{theorem} This is Theorem 3.2 of \cite{DH} (with $s=p/2$ and $t=1$) and a special case of Theorem 1.2 of \cite{H3}. (See also \cite{Sen}.) Note that if $p=2,$ then $\rho_{p/2}$ is just the density $\rho,$ and similarly for $\mu_{p/2}.$ As discussed in the introduction, $L^{p}(\mathbb{R}^{d},\rho)$ is roughly the same as $L^{2}(\mathbb{R}^{d},\rho_{p/2}),$ so we expect the image of $L^{p}(\mathbb{R}^{d},\rho)$ to be roughly the same as the image of $L^{2}(\mathbb{R}^{d},\rho_{p/2}),$ namely, $\mathcal{H}L^{2}(\mathbb{C}% ^{d},\mu_{p/2}).$ For each $p$ there is an inversion formula for $S$ that maps $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{p/2})$ isometrically onto $L^{2}% (\mathbb{R}^{d},\rho_{p/2})$ and is zero on the orthogonal complement of $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{p/2})$ in $L^{2}(\mathbb{C}^{d}% ,\mu_{p/2}).$ This inversion formula gives good estimates on $S^{-1}F$ when $F$ has growth like $e^{y^{2}/2}e^{x^{2}/2(p-1)}.$ Specifically, we write $Sf$ in the form \[ Sf\left( z\right) =\left( 2\pi\right) ^{-d/2}\int\frac{e^{-(z-x)^{2}/2}% }{\rho_{p/2}\left( x\right) }f\left( x\right) \rho_{p/2}\left( x\right) \,dx. \] We now take the adjoint of $S,$ viewed as an operator from the Hilbert space $L^{2}(\mathbb{R}^{d},\rho_{p/2})$ into the Hilbert space $L^{2}% (\mathbb{C}^{d},\mu_{p/2}),$ whose image is the holomorphic subspace of $L^{2}(\mathbb{C}^{d},\mu_{p/2}).$ Since $S$ is isometric, its adjoint is a one-sided inverse to $S.$ On the other hand, since we have expressed $S$ as an integral operator, its adjoint may be computed in the usual way by taking the complex conjugate of the integral kernel and reversing the roles of the variables. So we obtain on operator $S^{\ast,p}:L^{2}(\mathbb{C}^{d},\mu _{p/2})\rightarrow L^{2}(\mathbb{R}^{d},\rho_{p/2})$ given by \begin{align} S^{\ast,p}F(x) & =\left( 2\pi\right) ^{-d/2}\int_{\mathbb{C}^{d}}% \frac{e^{-(\bar{z}-x)^{2}/2}}{\rho_{p/2}\left( x\right) }F\left( z\right) \mu_{p/2}\left( z\right) \,dz\nonumber\\ & =ce^{x^{2}/p}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-\left( \tilde{x}-i\tilde{y}-x\right) ^{2}/2}F\left( \tilde{x}+i\tilde{y}\right) e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}}d\tilde{x}\,d\tilde{y}% .\label{inverse}% \end{align} Here $c$ is a constant which depends on $p$ and the dimension. More precisely, the formula (\ref{inverse}) for $S^{\ast,p}$ makes sense whenever all the integrals are convergent, for example if $F$ has compact support. For general $F$ in $L^{2}(\mathbb{C}^{d},\mu_{p/2})$ we compute $S^{\ast,p}$ by integrating over a compact set and then taking a limit in $L^{2}(\mathbb{R}^{d},\rho_{p/2}).$ If $F$ is in the holomorphic subspace of $L^{2}(\mathbb{C}^{d},\mu_{p/2})$ then since $S$ is isometric $SS^{\ast ,p}F=F,$ so that $S^{\ast,p}$ is computing the inverse of $S.$ We now re-write the integral (\ref{inverse}) so that it is expressed in terms of the Gaussian measure in Theorem \ref{l1.thm}. So \begin{align*} S^{\ast,p}F(x) & =c\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{x^{2}% /p}e^{-\left( \tilde{x}-i\tilde{y}-x\right) ^{2}/2}e^{-\tilde{x}^{2}% /2(p-1)}e^{-\tilde{y}^{2}/2}\cdot\\ & \cdot\left[ F\left( \tilde{x}+i\tilde{y}\right) e^{-\tilde{x}^{2}% /2(p-1)}e^{-\tilde{y}^{2}/2}\right] d\tilde{x}\,d\tilde{y}. \end{align*} A straightforward Gaussian integral shows that \[ \left\| e^{x^{2}/p}e^{-\left( \tilde{x}-i\tilde{y}-x\right) ^{2}% /2}e^{-\tilde{x}^{2}/2(p-1)}e^{-\tilde{y}^{2}/2}\right\| _{L^{p}% (\mathbb{R}^{d},\rho)}=const. \] independent of $\tilde{x}$ and $\tilde{y},$ where the norm is computed with respect to the $x$-variable. So if $F$ is in $L^{2}(\mathbb{C}^{d},\mu_{p/2})$ and such that (\ref{inverse}) converges absolutely, we obtain by putting the $L^{p}(\mathbb{R}^{d},\rho)$ norm inside the integral that \begin{equation} \left\| S^{\ast,p}F\right\| _{L^{p}(\mathbb{R}^{d},\rho)}\leq c\,\int _{\mathbb{C}^{d}}\left| F\left( \tilde{x}+i\tilde{y}\right) \right| e^{-\tilde{x}^{2}/2(p-1)}e^{-\tilde{y}^{2}/2}\,d\tilde{x}\,d\tilde {y}.\label{int.l1}% \end{equation} This estimate shows that $S^{\ast,p}$ extends to a continuous map of \begin{equation} L^{1}(\mathbb{C}^{d},e^{-\tilde{x}^{2}/2(p-1)}e^{-\tilde{y}^{2}/2}\,d\tilde {x}\,d\tilde{y})\label{l1.space}% \end{equation} into $L^{p}(\mathbb{R}^{d},\rho).$ I claim that if $F$ is holomorphic and in this $L^{1}$ space, then $SS^{\ast,p}F=F.$ To see this, approximate $F$ by $F_{n}(z)=F(\lambda_{n}z),$ with $\lambda_{n}$ tending to one from below. Then $F_{n}$ will converge to $F$ in the $L^{1}$ space (\ref{l1.space}), and using elementary pointwise bounds \cite[Sect. 2]{H4} we can establish that each $F_{n}$ is in $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu_{p/2}),$ so that $SS^{\ast,p}F_{n}=F_{n.}$ Then by (\ref{int.l1}) $S^{\ast,p}F_{n}$ will converge to $S^{\ast,p}F$ in $L^{p}(\mathbb{R}^{d},\rho),$ so that $SS^{\ast,p}F_{n}=F_{n}$ will converge uniformly on compact sets to $SS^{\ast,p}F.$ Since $F_{n}$ also converges in the $L^{1}$ space to $F,$ the pointwise limit and the $L^{1}$ limit must coincide, so $SS^{\ast,p}F=F. $ Thus for any holomorphic function $F$ in the $L^{1}$ space (\ref{l1.space}), there exists a function $f$ in $L^{p}(\mathbb{R}^{d},\rho)$ (namely, $f=S^{\ast,p}F$) for which $Sf=F.$ This establishes Theorem \ref{l1.thm}. $\square$ \textit{Proof of Theorem \ref{interpol1.thm}}. We have stated (Theorem \ref{st.thm}) that for each $p\in(1,\infty)$, $S$ is a unitary map of $L^{2}(\mathbb{R}^{d},\rho_{p/2})$ onto $\mathcal{H}L^{2}(\mathbb{C}^{d}% ,\mu_{p/2}).$ We can then construct an isometric Lebesgue measure map \[ S_{p}:L^{2}(\mathbb{R}^{d},dx)\rightarrow L^{2}(\mathbb{C}^{d},dx\,dy) \] by making a change of measure on both sides. That is, $S_{p}$ is defined to by \[ S_{p}f(z)=\mu_{p/2}(z)^{1/2}S(\rho_{p/2}(x)^{-1/2}f(x)). \] Computing explicitly we obtain, after some algebra, \begin{align} S_{p}f\left( \tilde{x}+i\tilde{y}\right) & =c\,e^{-\tilde{y}^{2}% /2}e^{-\tilde{x}^{2}/2(p-1)}\int_{\mathbb{R}^{d}}e^{-z^{2}/2}e^{z\cdot x}\left( e^{x^{2}/2p}f(x)\right) e^{-x^{2}/2}\,dx\nonumber\\ & =c\int_{\mathbb{R}^{d}}e^{-i\tilde{x}\cdot\tilde{y}}e^{ix\cdot\tilde{y}}% \exp\left\{ -\frac{p-1}{2p}\left[ x-\frac{p}{p-1}\tilde{x}\right] ^{2}\right\} f\left( x\right) \,dx,\label{sp.form}% \end{align} where $z=\tilde{x}+i\tilde{y}$ and $c$ is a constant depending on $p$ and the dimension. Clearly $S_{p}$ also makes sense on $L^{1}(\mathbb{R}^{d},dx)$ and defines a bounded linear map of $L^{1}(\mathbb{R}^{d},dx)$ into $L^{\infty }(\mathbb{C}^{d}).$ Therefore by interpolation (e.g. \cite[Thm. V.1.3]{SW}), $S_{p}$ is a bounded map of $L^{q}(\mathbb{R}^{d},dx)$ into $L^{q^{\prime}% }(\mathbb{C}^{d},dx\,dy)$ for all $q$ with $1\leq q\leq2.$ If $1

1,$ the function $\ e^{-\left( z-x\right) ^{2}% /2}e^{x^{2}/2p}$ tends to zero rapidly as $x$ tends to infinity (for each fixed $z\in\mathbb{C}^{d}$). Thus Condition 2 and Lemma \ref{schwarz.lem} permit us to integrate by parts with no boundary terms, giving \begin{align*} S\frac{\partial f}{\partial x_{k}}\left( z\right) & =\left( 2\pi\right) ^{-d/2}\int_{\mathbb{R}^{d}}e^{-\left( z-x\right) ^{2}/2}\frac{\partial f}{\partial x_{k}}\,dx\\ & =-\left( 2\pi\right) ^{-d/2}\int_{\mathbb{R}^{d}}(z_{k}-x_{k})e^{-\left( z-x\right) ^{2}/2}f\left( x\right) \,dx. \end{align*} So \[ S\frac{\partial f}{\partial x_{k}}=-z_{k}Sf+S\left[ x_{k}f\left( x\right) \right] \] or \begin{equation} S\left( x_{k}-\frac{\partial}{\partial x_{k}}\right) f=z_{k}Sf=z_{k}% F.\label{create}% \end{equation} Now the assumptions on $f$ tell us that $(x_{1}-\partial/\partial x_{1})^{n_{1}}\cdots(x_{d}-\partial/\partial x_{d})^{n_{d}}f$ is in $L^{p}(\mathbb{R}^{d},\rho)$ for all positive integers $n_{1},\cdots,n_{d}.$ So applying (\ref{create}) repeatedly we see that $z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}F\left( z\right) $ is the transform of a function in $L^{p}(\mathbb{R}^{d},\rho).$ So both $F$ itself and $z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}F\left( z\right) $ satisfy the bounds given in Theorem \ref{holder.thm}, from which the bounds given in Condition 1 of Theorem \ref{schwarz.thm} easily follow. We now prove that Condition 1 implies Condition 2. Assume that $F$ is holomorphic and satisfies the bounds of Condition 1. Then by direct calculation $F$ is square-integrable with respect the measure $\mu _{p/2}\left( z\right) \,dz$ and therefore by Theorem \ref{st.thm} there exists a function $f$ with $Sf=F.$ We may calculate $f$ by the inversion formula (\ref{inverse}). Putting absolute values inside the integral and simplifying gives (as noted previously) \begin{align} & \left| S^{-1}F(x)\right| \nonumber\\ & \leq c\,e^{x^{2}/2p}\int_{\mathbb{R}^{d}}e^{-\frac{p}{2(p-1)}\left[ \tilde{x}-\frac{p-1}{p}x\right] ^{2}}\int_{\mathbb{R}^{d}}\left| F(\tilde {x}+i\tilde{y})e^{-\tilde{y}^{2}/2}e^{-\tilde{x}^{2}/2(p-1)}\right| \,d\tilde{y}\,d\tilde{x}.\label{direct.bnd}% \end{align} Substituting in the bounds then gives (for $n>d$) \begin{align*} \left| S^{-1}F(x)\right| & \leq C_{n}\,e^{x^{2}/2p}\int_{\mathbb{C}^{d}% }e^{-\frac{p}{2(p-1)}\left[ \tilde{x}-\frac{p-1}{p}x\right] ^{2}}\frac {1}{(1+\left| \tilde{x}\right| )^{n}(1+\left| \tilde{y}\right| )^{n}% }\,d\tilde{x}\,d\tilde{y}\\ & \leq D_{n}\,e^{x^{2}/2p}\int_{\mathbb{C}^{d}}e^{-\frac{p}{2(p-1)}\left[ \tilde{x}-\frac{p-1}{p}x\right] ^{2}}\frac{1}{(1+\left| \tilde{x}\right| )^{n}}\,d\tilde{x}% \end{align*} In particular the bounds guarantee that the integral is absolutely convergent. Now, it is not difficult to see that the integral \[ \int_{\mathbb{C}^{d}}e^{-\frac{p}{2(p-1)}\left[ \tilde{x}-\frac{p-1}% {p}x\right] ^{2}}\frac{1}{(1+\left| \tilde{x}\right| )^{n}}\,d\tilde {x}\,d\tilde{y}% \] decays like a constant times $\left| x\right| ^{-n}$ as $x$ tends to infinity. (The main contribution to the integral comes from the region near where $\tilde{x}=(p-1)x/p,$ and at that point the integrand is of order $\left| x\right| ^{-n}.$) Thus $S^{-1}F(x)$ is bounded by $c_{n}e^{x^{2}% /2p}(1+\left| x\right| )^{-n}$ for all $n.$ This shows that any polynomial times $f$ is in $L^{p}(\mathbb{R}^{d},\rho).$ We need then to consider the derivatives of $f.$ Using again the inversion formula (\ref{inverse}) we may write \[ f=\lim_{R\rightarrow\infty}f_{R}% \] where \[ f_{R}(x)=ce^{x^{2}/p}\int_{\left| \tilde{x}_{l}\right| \leq R}\int_{\left| \tilde{y}_{l}\right| \leq R}e^{-\left( \bar{z}-x\right) ^{2}/2}F\left( z\right) e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}}d\tilde{x}\,d\tilde{y}. \] Here $z=\tilde{x}+i\tilde{y}$ and $\left| \tilde{x}_{l}\right| \leq R$ means that we integrate over the set of points $\tilde{x}=\left( \tilde{x}% _{1},\cdots,\tilde{x}_{d}\right) $ in $\mathbb{R}^{d}$ where each coordinate is at most $R$ in absolute value. Differentiating this gives \begin{align*} \frac{\partial f_{R}}{\partial x_{k}} & =ce^{x^{2}/p}\int_{\left| \tilde {x}_{l}\right| \leq R}\int_{\left| \tilde{y}_{l}\right| \leq R}\left( \frac{\partial}{\partial x_{k}}e^{-\left( \bar{z}-x\right) ^{2}/2}\right) F\left( z\right) e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}}d\tilde {x}\,d\tilde{y}\\ & +\frac{2x_{k}}{p}f_{R}\left( x\right) . \end{align*} Note that $(\partial/\partial x_{k})\exp\left[ -(\bar{z}-x)^{2}/2\right] =-(\partial/\partial\bar{z}_{k})\exp\left[ -(\bar{z}-x)^{2}/2\right] .$ Making this substitution and writing $\partial/\partial\bar{z}_{k}% =(1/2)(\partial/\partial\tilde{x}_{k}+i\partial/\partial\tilde{y}_{k})$ we may integrate by parts to get \begin{align} \frac{\partial f_{R}}{\partial x} & =ce^{x^{2}/p}\int_{\left| \tilde{x}% _{l}\right| \leq R}\int_{\left| \tilde{y}_{l}\right| \leq R}e^{-\left( \bar{z}-x\right) ^{2}/2}\cdot\nonumber\\ & \cdot F\left( z\right) \frac{\partial\left( e^{-\tilde{x}^{2}% /(p-1)}e^{-\tilde{y}^{2}}\right) /\partial\bar{z}_{k}}{e^{-\tilde{x}% ^{2}/(p-1)}e^{-\tilde{y}^{2}}}e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}% }d\tilde{x}\,d\tilde{y}\label{1diff}\\ & +\frac{2x_{k}}{p}f_{R}\left( x\right) +\text{ boundary terms.}\nonumber \end{align} Note that we do not get a term involving the derivative of $F,$ since $\partial F/\partial\bar{z}_{k}=0.$ Note also that the factor just after $F\left( z\right) $ is linear in $\tilde{x}$ and $\tilde{y}.$ There will be four boundary terms, in each of which the integrand will be $e^{-\left( \bar{z}-x\right) ^{2}/2}F\left( z\right) e^{-\tilde{x}^{2}/(p-1)}% e^{-\tilde{y}^{2}}.$ Two terms will have integration over all of the variables except $x_{k},$ with $x_{k}$ evaluated at $\pm R,$ and the other two will be similar with the roles of $x_{k}$ and $y_{k}$ reversed. Our estimates on $F$ are sufficient to show that these boundary terms go to zero as $R$ tends to infinity. So $\lim_{R\rightarrow\infty}\partial f_{R}/\partial x_{k}$ exists, and the limit is easily seen to be locally uniform in $x.$ Since also $\lim_{R\rightarrow\infty}f_{R}=f$, we conclude that $f$ is differentiable and that \begin{align} \frac{\partial f}{\partial x_{k}} & =ce^{x^{2}/p}\int_{\mathbb{R}^{d}}% \int_{\mathbb{R}^{d}}e^{-\left( \bar{z}-x\right) ^{2}/2}\cdot\nonumber\\ & \cdot F\left( z\right) \frac{\partial\left( e^{-\tilde{x}^{2}% /(p-1)}e^{-\tilde{y}^{2}}\right) /\partial\bar{z}_{k}}{e^{-\tilde{x}% ^{2}/(p-1)}e^{-\tilde{y}^{2}}}e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}% }d\tilde{x}\,d\tilde{y}+\frac{2x_{k}}{p}f\left( x\right) .\label{2diff}% \end{align} Now we have already noted that $x^{\alpha}f\left( x\right) $ is in $L^{p}(\mathbb{R}^{d},\rho).$ Since the factor after $F\left( z\right) $ grows only linearly, our estimates on $F$ will show that the first term is also in $L^{p}(\mathbb{R}^{d},\rho).$ Thus $\partial f/\partial x_{k}$ is in $L^{p}(\mathbb{R}^{d},\rho)$ as well. But now we may start our analysis anew with the function $f$ replaced by $\partial f/\partial x_{k}$ and with the inversion formula (\ref{inverse}) replaced by (\ref{2diff}). A similar argument will show that $x^{\alpha}\left( \partial f/\partial x_{k}\right) \in L^{p}(\mathbb{R}^{d},\rho)$ for all $\alpha,$ and then that $x^{\alpha }\left( \partial^{2}f/\partial x_{l}\partial x_{k}\right) \in L^{p}% (\mathbb{R}^{d},\rho).$ We may keep on differentiating repeatedly, at each stage getting a sum of terms of the form \[ p_{1}\left( x\right) e^{x^{2}/p}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}% }e^{-\left( \bar{z}-x\right) ^{2}/2}F\left( z\right) p_{2}(\tilde {x},\tilde{y})e^{-\tilde{x}^{2}/(p-1)}e^{-\tilde{y}^{2}}d\tilde{x}\,d\tilde{y}% \] where $p_{1}$ and $p_{2}$ are polynomials. Our estimates on $F$ are then sufficient to give the required bounds on $(\partial/\partial x)^{\alpha}f. $ $\square$ Since the Segal-Bargmann transform $S$ is given in (\ref{sdef1}) as a convolution, if $f$ satisfies Condition 2 then we will have $S\left[ (\partial/\partial x)^{\beta}f\right] =(\partial/\partial z)^{\beta}F$ for all $\beta.$ But $(\partial/\partial x)^{\beta}f$ again satisfies Condition 2, and therefore $(\partial/\partial z)^{\beta}F$ satisfies Condition 1. Thus we see that if $F$ satisfies Condition 1 then so do all the derivatives of $F.$ \subsection{Dimension-independent bounds} \textit{Proof of Theorem \ref{hyper1.thm}}. We consider the number operator $N$ defined on (a suitable dense subspace of) $L^{p}(\mathbb{R}^{d},\rho)$ by \[ N=\sum_{j=1}^{d}\left[ -\frac{\partial^{2}}{\partial x_{j}^{2}}+x_{j}% \frac{\partial}{\partial x_{j}}\right] . \] We consider also the associated semigroup $\exp(-tN),$ which is a contraction semigroup on $L^{p}(\mathbb{R}^{d},\rho)$ for all $1

1.$) Using the $L^{2}$ isometry property of $S$ and hypercontractivity we conclude that \[ \left\| Sf(e^{-t}z)\right\| _{L^{2}(\mathbb{C}^{d},\mu)}=\left\| e^{-tN}f\right\| _{L^{2}(\mathbb{R}^{d},\rho)}\leq\left\| f\right\| _{L^{p}(\mathbb{R}^{d},\rho)}% \] for all $t\geq-\frac{1}{2}\log(p-1).$ If we take $t=-\frac{1}{2}\log(p-1), $ then $e^{-t}=\sqrt{p-1}$ and we conclude that \[ \left\| Sf(\sqrt{p-1}z)\right\| _{L^{2}(\mathbb{C}^{d},\mu)}\leq\left\| f\right\| _{L^{p}(\mathbb{R}^{d},\rho)}. \] This is the content of Theorem \ref{hyper1.thm}.$~\square$ \textit{Proof of Theorem \ref{hyper2.thm}}. Suppose $t$ is some positive number. Suppose $F$ is a holomorphic function such that the function $F_{t}$ defined by \[ F_{t}(z)=F(e^{t}z) \] is in $L^{2}(\mathbb{C}^{d},\mu).$ Define $f_{t}=S^{-1}F_{t}$ and $f=S^{-1}F.$ Again by \cite[Sect. 3e]{B1} we have \[ f=e^{-tN}f_{t}. \] Hypercontractivity then implies that for all $p$ such that $t\geq\frac{1}% {2}\log(p-1)$ we have $f\in L^{p}(\mathbb{R}^{d},\rho)$ and \[ \left\| f\right\| _{L^{p}(\mathbb{R}^{d},\rho)}\leq\left\| f_{t}\right\| _{L^{2}(\mathbb{R}^{d},\rho)}. \] Thus \[ \left\| f\right\| _{L^{p}(\mathbb{R}^{d},\rho)}\leq\left\| f_{t}\right\| _{L^{2}(\mathbb{R}^{d},\rho)}=\left\| F_{t}\right\| _{L^{2}(\mathbb{C}% ^{d},\mu).}% \] Now fix some $p>2$ and take $t=\frac{1}{2}\log(p-1)$, so that $e^{t}% =\sqrt{p-1}.$ We conclude that \[ \left\| S^{-1}F\right\| _{L^{p}(\mathbb{R}^{d},\rho)}\leq\left\| S^{-1}F_{t}\right\| _{L^{2}(\mathbb{R}^{d},\rho)}=\left\| F_{t}\right\| \] where (with this value of $t$) \[ \left\| F_{t}\right\| ^{2}=\int_{\mathbb{C}^{d}}\left| F\left( \sqrt {p-1}z\right) \right| ^{2}\frac{e^{-\left| z\right| ^{2}}}{\pi^{d}}\,dz. \] This is the content of Theorem \ref{hyper2.thm}.$~\square$ \section{A primer on the Segal-Bargmann transform} In this section I explain some of the whys and wherefores of the Segal-Bargmann transform, specifically: the connection with the windowed Fourier transform, the connection with coherent states, and the motivation for using Gaussian measure on the domain space. The Segal-Bargmann transform, the windowed Fourier transform, and coherent states are all parts of ``harmonic analysis in phase space.'' (I have taken this phrase from the title of the excellent book \cite{F} by Gerald Folland.) On the one hand, these objects in the simplest case are equivalent to one another, so that the same results have often been proved several times in different settings. On the other hand, the types of generalizations considered by the different groups are quite different. Broadly speaking, harmonic analysis in phase space is the attempt to combine information about a function and its Fourier transform into a single object. In signal processing one considers a function of time and the associated phase space is the plane, thought of as time-frequency space. In quantum mechanics one considers a ``wave function'' on (say) $\mathbb{R}^{d}$ and the associated phase space is $\mathbb{R}^{2d},$ thought of as position-momentum space. In either case instead of considering the Fourier transform of a function $f$ in its entirety one can consider ``the Fourier transform of $f$ near a point $x$,'' which will then be a function of both $x $ and the frequency variable. Both the Segal-Bargmann transform and the windowed Fourier transform give a way of making sense out of this. In contrast to the wavelet expansion, both the Segal-Bargmann and windowed Fourier transforms give information in a \textit{fixed} distance scale around the point $x.$ The distinctive feature of the Segal-Bargmann transform is that it is brings in holomorphic methods, viewing $\mathbb{R}^{2d}$ as the complex space $\mathbb{C}^{d}.$ \subsection{Windowed Fourier transforms and holomorphic convolution transforms} In the simplest (Gaussian) case the Segal-Bargmann transform and the windowed Fourier transform are essentially the same thing. However there is a difference in perspective that can be illustrated by considering two families of transforms that have the Gaussian transform as their unique point of intersection. First consider the general windowed Fourier transform. Given a Schwartz function $\phi$ we consider a transform $\mathcal{F}_{\phi}$ given by \[ \mathcal{F}_{\phi}\left( f\right) \left( a,b\right) =\int_{\mathbb{R}^{n}% }f\left( x\right) \phi\left( x-a\right) e^{ib\cdot x}\,dx. \] If for example $\phi$ is compactly supported near the origin then $\mathcal{F}_{\phi}$ is the Fourier transform of $f$ in a ``window'' near $x=a.$ It is not hard to show, using the Plancherel theorem, that $\mathcal{F}_{\phi}$ is (up to a constant) an isometry of $L^{2}% (\mathbb{R}^{d})$ into $L^{2}\mathbb{(}\mathbb{R}^{2d}).$ The image may be characterized by a certain reproducing kernel condition. See \cite[Sect. 3.1]{F} or \cite[p. 8]{FS}. Meanwhile, consider ``holomorphic convolution transforms'' as follows. Let $\nu$ be a strictly positive continuous function on $\mathbb{R}^{d}$ with faster-than-exponential decay at infinity. Let \[ \sigma\left( k\right) =\int_{\mathbb{R}^{d}}e^{2k\cdot b}\nu\left( b\right) \,db \] and let \[ \phi\left( z\right) =\frac{1}{2\pi}\int_{\mathbb{R}^{d}}\frac{e^{-ik\cdot z}}{\sqrt{\sigma\left( k\right) }}\,dk\quad z\in\mathbb{C}^{d}. \] Then $\phi$ is an entire holomorphic function on $\mathbb{C}^{d}$ \cite[Sect. 10]{H1}. Given a function $\phi$ constructed in this way, define a transform $C_{\phi} $ by \[ C_{\phi}\left( f\right) \left( z\right) =\int_{\mathbb{R}^{d}}\phi\left( z-x\right) f\left( x\right) \,dx,\quad z\in\mathbb{C}^{d}. \] Then, again using just the Plancherel theorem, one can show \cite[Thm. 8]{H1} that $C_{\phi}$ is a unitary map of $L^{2}(\mathbb{R}^{d})$ onto the space of holomorphic functions on $\mathbb{C}^{d}$ that are square-integrable with respect to the measure $\nu\left( b\right) \,da\,db,$ where $z=a+ib.$ One can modify the definition to \[ \tilde{C}_{\phi}\left( f\right) \left( a+ib\right) =\sqrt{\nu\left( b\right) }C_{\phi}\left( f\right) \left( a+ib\right) . \] Then $\tilde{C}_{\phi}$ maps $L^{2}(\mathbb{R}^{d})$ isometrically into $L^{2}(\mathbb{R}^{2d}),$ and image of $\tilde{C}_{\phi}$ is precisely the set of functions of the form $F\left( z\right) \sqrt{\nu\left( b\right) },$ where $F $ is holomorphic. If we take $\nu\left( b\right) =\pi^{-d/2}\exp(-b^{2})$ then $\phi\left( z\right) =\left( 2\pi\right) ^{-d/2}\exp\left( -z^{2}/2\right) $ and \[ \tilde{C}_{\phi}\left( f\right) \left( a+ib\right) =e^{-ia\cdot b}% \int_{\mathbb{R}^{d}}f\left( x\right) e^{-\left( x-a\right) ^{2}% /2}e^{ib\cdot x}\,dx. \] In this case, except for the ``postmultiplier'' $e^{-ia\cdot b},$ we have just a windowed Fourier transform with a Gaussian window. The map $C_{\phi}, $ with $\phi$ Gaussian, is a form of the Segal-Bargmann transform, normalized in a somewhat different way from the transform $S$ discussed in the rest of the paper. (The two forms differ simply by a change of measure on the domain side and the range side. On the domain side the change of measure is the ground state transformation discussed below. On the range side the change of measure is the map $F\left( z\right) \rightarrow c\,e^{z^{2}/4}F(z/\sqrt{2})$ which brings about a change from the fully Gaussian measure $\mu$ used with the $S$ version of the transform to the partly Gaussian measure $\pi^{-d/2}\exp (-b^{2})\,da\,db$ used with the $C_{\phi}$ version of the transform. See \cite[Sects. 6.2 and 6.3]{H4} for more details.) Note that in both the windowed Fourier transform and the holomorphic convolution transform we are integrating a function $f$ against a family of functions obtained from a single function $\phi.$ In the case of the windowed Fourier transform these functions are obtained from $\phi$ by translations in position and in frequency. In the case of the holomorphic convolution transform the functions are obtained from $\phi$ by translations in position and analytic continuation. It is not hard to show that one gets equivalent results precisely if $\phi$ is a Gaussian. That is, suppose $\phi$ is a Schwartz function on $\mathbb{R}^{d}$ that admits an entire analytic continuation to $\mathbb{C}^{d},$ with moderate growth in the imaginary directions. Suppose also that there exist constants $c_{a,b}$ such that \[ \phi\left( x-(a+ib)\right) =c_{a,b}\phi\left( x-a\right) e^{ib\cdot x}% \] for all $a,b\in\mathbb{R}^{d}.$ Then necessarily $\phi$ is of the form \[ \phi\left( x\right) =\alpha e^{-x^{2}/2}e^{i\beta\cdot x}% \] for some $\alpha,\beta.$ Thus the windowed Fourier transform can be re-cast in holomorphic terms only in the Gaussian case. \subsection{Coherent states} The transforms in the previous subsection consist of integrating a function $f$ against a family of functions parameterized by points $\left( a,b\right) $ in the phase space $\mathbb{R}^{2d}.$ These functions are typically localized in position near $a$ and in frequency near $b.$ Functions of this sort are often called \textit{coherent states} in the physics literature, with the name suggesting the phase space localization. The Gaussian wave packets $e^{-\left( x-a\right) ^{2}/2}e^{ib\cdot x}$ are the \textit{canonical coherent states} studied by Schr\"{o}dinger and von Neumann in the early days of quantum mechanics (though without using that name). These states behave in a very classical way under the time-evolution of a quantum harmonic oscillator. In general coherent states in physics are thought of as quantum states that approximate as well as possible a classical particle with a fixed position $a$ and momentum $b.$ Many generalizations of the canonical coherent states have been considered; see for example the book \cite{KS}. The generalizations often involve geometry or Lie group representation theory. For example, Perelomov \cite{P} replaces the Heisenberg group action (i.e. translations in position and in frequency) by an irreducible unitary representation $\pi$ of a Lie group $G$ acting on a Hilbert space $H.$ The coherent states are then the vectors of the form $\pi\left( g\right) \phi_{0},$ where $\phi_{0}$ is a fixed non-zero vector in $H.$ In a different direction, J. Rawnsley, M. Cahen, and S. Gutt, building on ideas of F. Berezin, have considered ``quantization'' of K\"{a}hler manifolds $M$ (e.g. \cite{RCG}). In this case one builds a Segal-Bargmann type Hilbert space of holomorphic functions over $M,$ and the coherent states are the elements $\chi_{m}$ of the Segal-Bargmann space such that $\left\langle \chi_{m},F\right\rangle =F\left( m\right) ,$ for $m\in M$ and $F$ in the Segal-Bargmann space over $M.$ The Perelomov and K\"{a}hler constructions intersect in the case of homogeneous K\"{a}hler manifolds, yielding for example the Bott-Borel-Weil construction of the irreducible representations of compact semisimple Lie groups. Still another type of generalized coherent states has been considered by the author, namely, coherent states on compact Lie groups defined in terms of heat kernels. See \cite{H1,H5}. Typically in the physics literature it is assumed that one has a ``resolution of the identity'' for the coherent states \cite{KS}. Suppose that the coherent states are a family of vectors $\psi_{\alpha}$ in a Hilbert space $H,$ parameterized by points $\alpha$ in some parameter space $P.$ Then a resolution of the identity means a decomposition of the identity operator on $H$ as \begin{equation} I=\int_{P}\left| \psi_{\alpha}\right\rangle \left\langle \psi_{\alpha }\right| \,d\mu\left( \alpha\right) \label{res.id}% \end{equation} for some measure $\mu$ on $P.$ Here $\left| \psi_{\alpha}\right\rangle \left\langle \psi_{\alpha}\right| $ means the orthogonal projection onto the vector $\psi_{\alpha}.$ Given such a resolution of the identity we have \[ v=\int_{P}\psi_{\alpha}\left\langle \psi_{\alpha},v\right\rangle \,d\mu\left( \alpha\right) \] for all $v\in H.$ That is, each $v\in H$ has a preferred (but not unique!) expansion in terms of the coherent states, the coefficients of which are computed simply by taking the inner product of $v$ with each $\psi_{\alpha}.$ Furthermore we have \begin{align*} \left\langle v,v\right\rangle & =\int_{P}\left\langle v,\psi_{\alpha }\right\rangle \left\langle \psi_{\alpha},v\right\rangle \,d\mu\left( \alpha\right) \\ & =\int_{P}\left| \left\langle \psi_{\alpha},v\right\rangle \right| ^{2}\,d\mu\left( \alpha\right) . \end{align*} This means that we can define a generalized Segal-Bargmann transform $S:H\rightarrow L^{2}(P,\mu)$ by $S\left( v\right) \left( \alpha\right) =\left\langle \psi_{\alpha},v\right\rangle ,$ and $S$ will be an isometry of $H$ into (but not onto) $L^{2}(P,\mu).$ Conversely, if one has an isometric transform of Segal-Bargmann type, one can reformulate the isometricity as a resolution of the identity. So at some level Segal-Bargmann transforms are equivalent to coherent states. Nevertheless there is a different emphasis in the two settings. \subsection{The ground state transformation} One special feature of the Segal-Bargmann transform is that it is commonly normalized so that its domain is an $L^{2}$ space over $\mathbb{R}^{d}$ with respect to a Gaussian measure. The transformation from Lebesgue measure to Gaussian measure is a special case of the \textit{ground state transformation}% , which has a long and distinguished history in mathematical physics. (See for example \cite[Sect. 4]{G}, \cite{DS}, or \cite[Sects. 3.2-3.4]{GJ}.) The Gaussian wave packets (or coherent states) $\exp[-\left( z-x\right) ^{2}/2]$ are evidently obtained from the function $\phi_{0}\left( x\right) :=\exp[-x^{2}/2]$ by complex translations. The function $\phi_{0}$ is the ground state of a quantum harmonic oscillator. This means that \[ -\Delta\phi_{0}\left( x\right) +x^{2}\phi_{0}\left( x\right) =d\phi _{0}\left( x\right) , \] where $d$ is the minimum of the spectrum of the operator $-\Delta+x^{2}$ acting in $L^{2}(\mathbb{R}^{d},dx).$ In studying this operator it is convenient to divide all functions by the ground state, thus considering the map $f\rightarrow f/\phi_{0},$ which is a unitary map of $L^{2}(\mathbb{R}% ^{d},dx)$ onto $L^{2}(\mathbb{R}^{d},\phi_{0}^{2}\left( x\right) \,dx).$ More generally the operation of dividing by the ground state arises naturally in the path-integral formulation of quantum mechanics (not just for the harmonic oscillator), and is necessary if one wants to let the dimension $d$ tend to infinity. This infinite-dimensional limit is what arises in quantum field theory, and indeed Segal \cite{S} worked in the $d=\infty$ setting from the beginning. Another reason for the change from Lebesgue to Gaussian measure is that it is in the setting of Gaussian measures that the story of hypercontractivity and logarithmic Sobolev inequalities gets told. (See \cite{DGS} for a summary of this story.) In particular, the scale of $L^{p}$ spaces with respect to Gaussian as opposed to Lebesgue measure is extremely important. See, for example, the last part of Section 2 for an application of hypercontractivity. This brings us to the Gaussian measure formulation of the Segal-Bargmann transform $S,$ which is a unitary map of $L^{2}(\mathbb{R}^{d},\rho)$ onto $\mathcal{H}L^{2}(\mathbb{C}^{d},\mu).$ In this form the measure on $\mathbb{R}^{d}$ is Gaussian and the measure on $\mathbb{C}^{d}$ is Gaussian in both the real and imaginary directions. This form differs from that considered by Bargmann in \cite{B1} simply by the ground state transformation. 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