\documentclass[twoside]{article}
\pagestyle{myheadings}
\markboth{On two reverse inequalities}
{ Fernando Galaz-Fontes \& Stephen Bruce Sontz }
\begin{document}
\setcounter{page}{103}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 103--111\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 On two reverse inequalities in the Segal-Bargmann space 
\thanks{ {\em Mathematics Subject Classifications:} 46N50, 47D03, 81S99.
\hfil\break\indent
{\em Key words:} Segal-Bargmann space, hypercontractivity, 
log-Sobelev inequality.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Published July 12, 2000. } }

\date{}
\author{Fernando Galaz-Fontes \& Stephen Bruce Sontz \\[3mm]
{\it Dedicated to Eyvind H. Wichmann on his 70th birthday}}
\maketitle
\begin{abstract} 
We review here two reverse inequalities in the
 Segal-Bargmann space: a reverse hypercontractivity estimate 
due to Carlen and a reverse log-Sobolev inequality due to the 
second author.
\end{abstract}

\section{Notation and Definitions}

We start with the scale of spaces
$$
 {\cal A}_p := \left\{ \phi : {\bf C}^n \rightarrow {\bf C} :
 \phi \mbox{ is holomorphic and } \|\phi\|_p < \infty \right\},
$$
where $0 < p < \infty$ and $\|\phi\|_p := \left( \int_{ {\bf C}^n }
d \mu_n(z) |\phi(z)|^p \right)^{1/p}$.  Here 
$$d \mu_n(z) = \pi^{-n} e^{ -|z|^2 } d^n x d^n y$$ is Gaussian
measure on ${\bf C}^n$, where $d^n x d^n y$ is the Lebesgue measure, 
and where $|z|$ is the Euclidean norm of $ z\in {\bf C}^n$. 
The spaces ${\cal A}_p$ for $ 1 \le p < \infty$ are Banach spaces with
the norm $\| \cdot \|_p$, and ${\cal A}_2$ is a Hilbert space 
known as the {\em Segal-Bargmann space}. (See \cite{B} and \cite{Se}.)
Moreover, ${\cal A}_2$ carries an irreducible representation of the
Weyl-Heisenberg group (exponentiated canonical commutation relations)
and as such is a Hilbert space ready for use in the quantum mechanics
of a system with $n$ degrees of freedom. The infinitesimal form of this
representation is given by the following definitions of the creation
and annihilation operators:
$$\displaylines{
     a_k^* \phi (z) := z_k \phi(z)  \cr
     a_k \phi (z) := { {\partial} \over {\partial z_k} } \phi
}$$
for $k=1, \dots , n$, where $z=(z_1, \dots , z_n) \in {\bf C}^n$
and $ { {\partial} \over {\partial z_k} } =
{ 1 \over 2 } \left (
{ {\partial} \over {\partial x_k} } - i 
{ {\partial} \over {\partial y_k} }
\right)$.
Moreover, the basic Hamiltonian operator in this formalism is defined
by 
$$
  N := \sum_{k=1}^n a_k^* a_k = \sum_{k=1}^n z_k 
    { {\partial} \over {\partial z_k} } 
$$
and is known as the {\em number operator}. It is unitarily
equivalent to the operator $H_{harm} - {n \over 2} I$, where
$H_{harm}$ is the (isotropic, normalized) quantum harmonic oscillator
Hamiltonian with $n$ degrees of freedom and $I$ is the identity
operator, both acting in $L^2 ( {\bf R}^n, d^n x)$. It turns out 
that $N$, when realized as an operator in ${\cal A}_2$, is an unbounded
self-adjoint operator satisfying $N \ge 0$. Consequently, for ${t \ge 0}$
by spectral theory we have 
a semigroup  $\left\{ e^{-tN} \right\}$ of 
contractions on ${\cal A}_2$. But more is true; this semigroup also
acts contractively on each ${\cal A}_p$.
To be perfectly clear here, we define 
$e^{-tN} \phi (z) := \phi (e^{-t} z )$ for any function
$\phi: {\bf C}^n \rightarrow {\bf C}$ and any $ t \in {\bf R}$,
where $ z \in {\bf C}^n$. This is valid since this formula is shown in \cite{B} 
to hold for $f \in {\cal A}_2$ and $t \ge 0$ where $e^{-tN}$ is 
already defined.  We actually have the following 
{\em hypercontractivity} result, which is so called since it implies 
contractivity in each ${\cal A}_p$. It says that
$$
  \| e^{-tN}\|_{ {\cal A}_p \rightarrow {\cal A}_q } 
  = \cases{
            1 & if $0 < q \le pe^{2t}$   \cr
            \noalign{\vskip3pt}
            \infty & if $q > pe^{2t}$     \cr
           }
$$
for $0 < p \le q$ and $ t \ge 0$. The credits for this result
begin with Janson \cite{J1} in 1983, continue with Carlen \cite{C} and
Zhou \cite{Z} in 1991, come back to Janson \cite{J2} in 1997 and 
conclude with Gross \cite{G2} in 1998 whose proof is the most direct in the
sense that the method is essentially that of Gross' original 
paper \cite{G1} on log-Sobolev inequalities. In fact, we do get a 
log-Sobolev inequality from the hypercontractivity result by 
taking $p=2$ and $q=pe^{2t}=2e^{2t}$ (the ``critical'' index)
and considering the inequality
$$
   \| e^{-tN} \phi \|_{2e^{2t}} \le \|\phi\|_2,
$$
which holds for $t\ge 0$. Differentiating this from the right
at $t=0$ (where it becomes an {\em equality}) gives the 
log-Sobolev inequality 
$$
   S(\phi) \le \left\langle \phi, N \phi \right\rangle,
$$
where 
$$
  S(\phi) = \int_{ {\bf C}^n }  d \mu_n(z)  |\phi(z)|^2 \log |\phi(z)|^2
          - \|\phi\|_2^2 \log \|\phi\|_2^2
$$
is the entropy of $\phi$ and
$$
  \left\langle \phi, N \phi \right\rangle = 
   \sum_{k=1}^n \int_{ {\bf C}^n }  d \mu_n(z) 
   \big| { {\partial \phi} \over { \partial z_k} } \big|^2
$$
is the (expected) energy of $\phi$. Here $\log$ means the natural
logarithm (base $e$), and we take $ 0 \log 0 = 0$.
Note that both $S(\phi)$ and $ \left\langle \phi, N \phi \right\rangle$ 
are defined for {\em all} $\phi \in {\cal A}_2$, and that 
$S(\phi) \ge 0$ (by Jensen's inequality) and 
$ \left\langle \phi, N \phi \right\rangle \ge 0$ hold. 
It is possible that $S(\phi) = \infty$ and 
$ \left\langle \phi, N \phi \right\rangle = \infty$ for some
$\phi \in {\cal A}_2$. We will have more to say about this later.

\section{Two Reverse Inequalities}

The reverse hypercontractivity inequality due to Carlen \cite{C} 
in the Segal-Bargmann space says
the following.
\vskip .2cm \noindent
{\em Theorem:} \cite{C}
We have for all $\phi \in {\cal A}_p$ 
\begin{equation}
\label{rh}
   \| e^{-tN} \phi\|_q \le ( A(t,p,q))^n \|\phi\|_p,
\end{equation}
where $A(t,p,q) = ( 1 - q e^{-2t}/p )^{-1/q}$, 
provided that $0 < q < p $ and ${ 1 \over 2 } \log (q/p) < t < 0$.
%\end{theorem}
It seems that this is the only reverse hypercontractivity inequality
that is explicitly given in the literature. 

   In \cite{C} Carlen notes but does not prove that the constant
$(A(t,p,q))^n$ is not optimal. However, it was recently shown in \cite{S2}
that this indeed is so, though the proof was by contradiction
(i.e., assume that the constant $(A(t,p,q))^n$ is optimal and derive
from that a false statement) and so one gets no further
improvement in the constant. But now we have an improvement in
the following result.


\paragraph{Proposition:} 
For $0<q<p$ and ${1 \over 2} \log (q/p) < t < 0$
we have
\begin{equation}
 \label{better}
 \| e^{-tN} \phi \|_q \le (B(t,p,q))^n \|\phi\|_p
\end{equation}
for all $\phi \in {\cal A}_p$ where
$$
B(t,p,q) = \left( 1 - {q \over p} \right)^{1/q}
 \left( { {p e^{2t} - q} \over {p-q} } \right)^{1/p} 
A(t,p,q).
$$
However, $\| e^{-tN}\|_{ {\cal A}_p \rightarrow {\cal A}_q } < 
(B(t,p,q))^n$, that is to say, the constant in (\ref{better}) is not optimal. 


\paragraph{Observation:} The hypotheses on $p$, $q$ and $t$ imply
that $B(t,p,q) < A(t,p,q)$, so that this is a better bound than Carlen's,
as claimed.

\paragraph{Proof:} One simply calculates
\begin{eqnarray}
\| e^{-tN} \phi \|_q^q &=& \int_{ {\bf C}^n } d \mu_n(z) 
|e^{-tN} \phi (z) |^q \nonumber \\
&=& \int_{ {\bf C}^n } d^n x ~ d^n y ~ \pi^{-n} e^{-|z|^2} 
| \phi ( e^{-t} z ) |^q \nonumber \\
&=& e^{2nt} \int_{ {\bf C}^n } d^n u ~ d^n v ~ \pi^{-n}  e^{-|w|^2}
 e^{(1-e^{2t})|w|^2} | \phi (w) |^q \nonumber \\
&\le& e^{2nt} \| ~|\phi|^q~ \|_r 
\left\{ \int_{ {\bf C}^n } d^n u ~ d^n v ~ \pi^{-n} ~ e^{-|w|^2} 
e^{ r^\prime (1-e^{2t}) |w|^2 } \right\}_,^{1/r^\prime} \nonumber
\end{eqnarray}
where we have used $w=u+iv=e^{-t}z$ and H\"older's inequality
for $ 1 < r < \infty$. 
Here the norm $\| \cdot \|_r$ is with respect to the measure $\mu_n$.
Taking $r=p/q$, we obtain
$$
\| e^{-tN} \phi \|_q^q \le e^{2nt} \|\phi\|_p^q 
\left( { {pe^{2t}-q} \over {p-q} } 
\right)^{-n\left( 1 - {q \over p} \right)}
$$
since the Gaussian integral converges for $p$, $q$ and $t$ satisfying
the hypotheses. Now the result follows by simple algebra manipulations.

Now let us show that the constant $(B(t,p,q))^n$ is not optimal.
As shown in \cite{S2}, the inequality
$$
  \|e^{-tN} \phi \|_q \le M(t,p,q,n) \|\phi\|_p,
$$
where $M(t,p,q,n)$ is the optimal constant for this inequality
(i.e., the operator
norm $\|e^{-tN}\|_{ {\cal A}_p \rightarrow {\cal A}_q }$), does have an
optimizer $\phi_0 \in {\cal A}_p$. This means that $\phi_0 \ne 0$ and
$$
  \|e^{-tN} \phi_0 \|_q = M(t,p,q,n) \|\phi_0\|_p.
$$
This is shown using a standard compactness argument. If
$(B(t,p,q))^n$ were the optimal constant, then we would have equality
at the step in the above argument where we applied the H\"older inequality.
This would imply that 
$$
   |\phi_0(w)|^q = c e^{ (r^\prime - 1) \left( 1 - e^{2t} \right) |w|^2 }
$$
for some constant $c > 0$ and all $ w \in {\bf C}^n$. But it is 
well known that a nonzero holomorphic function can not satisfy 
such an equality. To see this, simply note that we would have
for all $w \in {\bf C}^n$ that 
$$
   \phi_0 (w) = c_1 e^{ i \theta (w) } e^{ c_2 |w|^2} 
$$
for some phase $\theta(w) \in {\bf R}$, 
some $c_1 > 0$ and some $c_2 \ne 0$.
Then taking the real part of
(say, the principal branch of) the logarithm of $\phi_0 (w)$ 
gives a harmonic function (on some open
subset of ${\bf C}^n$). But on the other hand, this gives us 
$ \log c_1 + c_2 |w|^2$, which is harmonic if and only if the 
constant $c_2$ is zero.  QED \vskip .3cm 
The fact that $(B(t,p,q))^n$ is not optimal has to do with the
fact that the inequality (\ref{better}) actually holds for all
$\phi$ in $L^p ( {\bf C}^n, \mu_n)$, since the proof 
of the inequality (\ref{better}) given above never
uses the holomorphicity of $\phi$. Of course, the constant
$(B(t,p,q))^n$ is optimal for (\ref{better}) if $\phi$ is
allowed to run over all of $L^p({\bf C}^n, \mu_n)$. 

Another observation about
reverse hypercontractivity is that for the ``critical'' index we do not
have a bounded operator. This is unlike the hypercontractivity result
where we do have boundedness at the ``critical'' index, though not
compactness. Specifically, we have the next result.


\paragraph{Proposition:} For $0 < p < \infty$ and $t<0$, we have 
$$
  \| e^{-tN} \|_{ {\cal A}_p \rightarrow {\cal A}_q } = \infty
$$
for $ q = p e^{2t}$. 

\paragraph{Proof:}  Let $\phi_\alpha (z) = e^{ \alpha z^2 }$ for real
$\alpha$ and $ z \in {\bf C}^n$. 
Then 
$$
  \| \phi_\alpha\|_p = \left( 1 - \alpha^2 p^2 \right)^{-n/(2p)}
$$
provided that $ -1/p < \alpha < 1/p$. 
Since $e^{-tN} \phi_\alpha (z) = \phi_\alpha (e^{-t} z ) = 
e^{ \alpha e^{-2t} z^2 } = \phi_{ \alpha e^{-2t} } (z)$, we have
that $\|e^{-tN} \phi_\alpha \|_q = \| \phi_{ \alpha e^{-2t} } \|_q
= \left( 1 - \alpha^2 e^{-4t} q^2 \right)^{-n/(2q)}$ provided that
$ -1/q < \alpha e^{-2t} < 1/q$, which is equivalent to 
$ -1/p < \alpha < 1/p$ since $q = pe^{2t}$. 
Next, note that 
$$
{ {\|e^{-tN} \phi_\alpha\|_q} \over {\|\phi_\alpha\|_p} } =
\left( 1 - \alpha^2 p^2 \right)^{ n \left( e^{2t} -1 \right) 
/ (2pe^{2t})  
}. 
$$
But $t<0$ implies that $e^{2t} -1 <0$, and so taking the limit
as $|\alpha| \rightarrow 1/p$ gives us $+\infty$. 
QED

\vskip .3cm
   The other reverse inequality in the Segal-Bargmann space is a 
reverse log-Sobolev inequality.

\paragraph{Theorem:} For every $c>1$ there exists a constant $R(c)$
independent of the dimension $n$ such that 
\begin{equation}
\label{rlsi}
  \left\langle \phi, N\phi \right\rangle \le c S(\phi) +
  n R(c) \|\phi\|_2^2
\end{equation}
for all $\phi \in {\cal A}_2$. One can take
$R(c) = -1 + c \log \left( { {c} \over {c-1} } \right)$.
\vskip .3cm 

\noindent This was originally proved in \cite{S1}, but Gross has given a more
elegant proof, reproduced in \cite{S2}. His proof gives the constant
$R(c)$ quoted here, which is better than the constant found in \cite{S1}. 
However, the optimal value of the coefficient of the norm term 
remains unknown. This seems to be the only reverse log-Sobolev 
inequality in the literature. 

   Two applications of this reverse log-Sobolev inequality to the 
analysis of the Segal-Bargmann transform can be found in \cite{S1}.
Also, one surprising result of the reverse log-Sobolev inequality 
together with the (regular) log-Sobolev inequality is that the entropy
$S(\phi)$ is finite if {\em and only if} the energy 
$\left\langle \phi, N\phi \right\rangle$ is finite. 
However, the reverse log-Sobolev inequality 
may also be considered as a ``generalized'' Heisenberg 
uncertainty principle, which may be its more fundamental role. 
The idea here is that, by a theorem of Stone and von~Neumann, all 
irreducible representations of the Weyl-Heisenberg group are 
unitarily equivalent, and moreover by a unitary (and intertwining)
operator that is unique up to multiplication by a complex number of
modulus one. So all properties expressible in terms of the inner
product and the creation and annihilation operators 
alone are invariant under such Stone-von~Neumann operators.
However, certain quantities are representation dependent and may not
be preserved by the Stone-von~Neumann unitary operator. The usual
example given in an introductory physics course in quantum mechanics is
the {\em variance} ${\rm Var} (\phi)$ of a state 
$\phi$ in $L^2( {\bf R}^n , d^n x )$, the position space. 
By going to the momentum space representation (which is in fact another 
representation of the Weyl-Heisenberg group as is the position space), 
one finds that the
Stone-von~Neumann unitary operator in this case is the Fourier 
transform ${\cal F}$, and it is well known that 
the Fourier transform does {\em not} preserve variance. In fact,
we have the Heisenberg inequality 
${\rm Var}(\phi) {\rm Var} ( {\cal F}\phi) \ge c > 0$ for all states $\phi$
(i.e., $\|\phi\|_2 = 1$) where $c$ is a constant independent of
$\phi$ that depends only on how one normalizes Planck's constant. 
Other quantities (actually nonlinear functionals on the
Hilbert space) not necessarily preserved by the 
Stone-von~Neumann unitary operator are the $L^p$ norm of a state
and the entropy of a state, provided that the Hilbert space carries
enough extra structure to allow us to define these quantities, namely, 
that it is the $L^2$ space of a measure space. 
But the energy $\left\langle \phi, N \phi \right\rangle$ in the
Segal-Bargmann space (which also is a nonlinear functional in   
$\phi \in {\cal A}_2$) {\em is} preserved by the
Stone-von~Neumann unitary operators, since it has a definition in terms 
of creation and annihilation operators. So inequalities
(such as the regular or reverse log-Sobolev inequalities) express 
relations that hold in one particular representation, 
but not in all representations, and thereby 
serve to distinguish that representation from the others. In this sense
we can think of such inequalities as ``generalized'' Heisenberg
uncertainty principles. Returning to the position and momentum 
representations, we see for example 
that the Hausdorff-Young inequality (of the
Fourier transform ${\cal F}$ on Euclidean space) is a ``generalized''
Heisenberg inequality, which is related to the fact that ${\cal F}$ 
does not preserve $L^p$ norms for $p \ne 2$. 

\section{Differentiating Hypercontractivity}

   Much as the log-Sobolev inequality arises from the hypercontractivity 
inequality by differentiation, we would like to be able to derive the
reverse log-Sobolev inequality by differentiating the reverse
hypercontractivity inequality. Unfortunately, this does not
work out as anticipated. Let us write
$$
  M(t,p,q) := \| e^{-tN}\|_{ {\cal A}_p \rightarrow {\cal A}_q }
$$
even though this notation omits the (possible) dependence on the
dimension $n$. 
(See \cite{S2} for a discussion of this dependence on $n$.) So we have
\begin{equation}
\label{best}
  \| e^{-tN} \phi \|_q \le M(t,p,q) \|\phi\|_p
\end{equation}
for all $\phi \in {\cal A}_p$. Moreover, for 
${1 \over 2} \log (q/p) < t < 0$ and $ 0 < q < p$, we have that
$M(t,p,q)$ is finite. However, it remains an open problem to find an
explicit formula for $M(t,p,q)$. Proceeding with (\ref{best}) above, we can not
take $q=p e^{2t}$ (the ``critical'' index) since we have already
shown that $M(t,p,q) = \infty$ in that case. Instead, we choose
some function $s: (-\epsilon, 0] \rightarrow (1,\infty)$ for
some $\epsilon >0$ such that: (a) $s(t) < p e^{2t}$ for
$-\epsilon < t < 0$; (b) $s(0) = p$; and (c) the derivative of
$s$ from the left at $t=0$, denoted $s^\prime(0^-)$, exists. 
(Here we are only considering the case $p>1$.) 
Using condition (a), we have 
\begin{equation}
\label{best2}
  \| e^{-tN} \phi \|_{s(t)} \le M(t,p,s(t)) \|\phi\|_p
\end{equation}
for $-\epsilon < t < 0$. Moreover, at $t=0$ both the sides of this
inequality become $\|\phi\|_p$, using condition (b) and 
$M(0,p,s(0))=1$. So we can take the left-sided
derivative at $t=0$ on both sides of (\ref{best2}) 
to get a new inequality.
(Warning: The sense of the inequality in (\ref{best2}) reverses on 
taking this derivative since $t=0$ is the {\em right} end point.)  
Using the chain rule together with condition (c), we get formally
\begin{equation}
\label{candidate}
   Re \left\langle \phi_p, N \phi \right\rangle \le
{ {1} \over {p^2} } s^\prime(0^-) S_p(\phi) - \kappa_p \|\phi\|_p^p,
\end{equation}
where $$\kappa_p := { {d} \over {dt} } \biggm|_{t=0^-} M(t,p,s(t)).$$
Moreover, $\phi_p = {\rm sgn} (\phi) |\phi|^{p-1}$, a usual notation in
$L^p$ analysis, and $S_p(\phi) := S ( |\phi|^{p/2} )$ is called the
{\em index-p entropy} of $\phi$. 

   One would now like to take $p=2$ so that 
$ Re \left\langle \phi_p, N \phi \right\rangle = 
\left\langle \phi, N \phi \right\rangle $ and $S_2(\phi) = S(\phi)$, 
and then (\ref{candidate}) reduces to
what seems to be a perfectly well-behaved reverse log-Sobolev 
inequality. But actually what is happening here is that 
$\kappa_p = -\infty$ for $p \ge 2$, that is, the derivative used to
define $\kappa_p$ does not exist as a finite number. 
(The proof is given in \cite{S1}.) But there may be an escape hatch here.
First, it may be that the argument above 
for (\ref{candidate}) is nontrivial for
$1<p<2$, namely, $\kappa_p$ is finite. 
But to verify this it would behoove us 
to find an explicit formula for $M(t,p,q)$. Then one would hope that 
one could take the limit of (\ref{candidate}) as $p$ increases to $2$, 
and that the resulting limit would be the reverse log-Sobolev 
inequality. The upshot of this paragraph is a conjectural approach 
to relating reverse hypercontractivity to reverse log-Sobolev,
but with some sturdy open problems.
The following proposition, which we have already referred to, 
may be useful in addressing these problems.

\paragraph{Proposition:} If $ 1 \le q < p$ and 
${ 1 \over 2} \log (q/p) < t < 0$, then 
$e^{-tN} : {\cal A}_p \rightarrow {\cal A}_q$ is a
compact operator and has a maximizer $\phi_0 \ne 0$ in
${\cal A}_p$. \vskip .3cm 

\noindent The proof is given in \cite{S2}. 
If one could identify one of these maximizers $\phi_0$, 
then one could read off
the value of $M(t,p,q)$ as
$ \| e^{-tN} \phi_0\|_q / \|\phi_0\|_p$. But this is also an
open problem which does not appear to be trivial.
\vskip .3cm 

We conclude with a list of open problems.  \begin{enumerate}
\item  Is the derivation of (\ref{candidate}) valid for $1 < p < 2$? 

\item Find an explicit formula for $M(t,p,q)$.

\item Does an inequality of the form (\ref{candidate}) hold for $p\ne 2$?

\item  For each $c > 1$, 
what is the optimal constant for the coefficient of the norm
term in (\ref{rlsi})? Does this optimal constant depend on $n$? 
Is there
some value of $c > 1$ such that this optimal constant is zero? 

\item  Find the optimal constant in the reverse hypercontractivity
inequality (\ref{rh}). 

\item What are the maximizers in the last proposition?  In particular,
it would be helpful to know if it is true that every
$n$-dimensional maximizer can be expressed as
a product of one dimensional maximizers. 
Specifically, this means that for any 
$z=(z_1, \dots , z_n) \in {\bf C}^n$ we can write 
any maximizer $\phi_0$ as
in the above proposition in the form
$\phi_0(z) = \psi_1(z_1) \cdots  \psi_n(z_n)$, where each $\psi_j$ is
a maximizer for dimension $n=1$. This is what Zhou 
proves in his article \cite{Z}
for the case of (regular) hypercontractivity in the   
Segal-Bargmann space. 
\end{enumerate}


\paragraph{Acknowledgments.}  The second author would like to thank the 
organizers, Peter Hislop and Henry Warchall, of the International 
Symposium ``Mathematical Physics and Quantum Field Theory,'' 
held in honor of the seventieth birthday of 
Eyvind H. Wichmann on 11-13 June, 1999 at the University of
California, Berkeley, for the opportunity
to present a talk there based on the material of this article.
He also thanks Eric Carlen and Leonard Gross for many useful 
comments.

\begin{thebibliography}{10}

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\bibitem{C} E.A.~Carlen, ``Some integral identities and inequalities for entire 
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\bibitem{G1} L.~Gross, ``Logarithmic Sobolev inequalities,'' 
Am. J. Math. {\bf 97}, 1061-1083 (1975).  

\bibitem{G2} L.~Gross, ``Hypercontractivity over complex manifolds,'' 
Acta Math. {\bf 182} (1999), 159-206. 

\bibitem{J1} S.~Janson, 
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\bibitem{J2} S.~Janson,``On complex hypercontractivity,'' 
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\bibitem{Se} I.E.~Segal, 
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in {\em Proceedings of the Summer Seminar, Boulder, Colorado, 1960} 
 (M.~Kac, Ed.), Vol. 2, Lectures in Applied Mathematics, 
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\bibitem{S1} S.B.~Sontz,
``A reverse log-Sobolev inequality in the Segal-Bargmann space,''
J. Math. Phys. {\bf 40}, 1677-1695 (1999). 

\bibitem{S2} S.B.~Sontz,
``On some reverse inequalities in the Segal-Bargmann space, preprint,
submitted to the Proceedings of the 1999 ICDEMP, Birmingham, Alabama,
March, 1999. 

\bibitem{Z} Z. Zhou, 
``The contractivity of the free Hamiltonian semigroup in the $L_p$ space of
entire functions,'' J. Funct. Anal. {\bf 96}, 407-425 (1991).

\end{thebibliography}

\noindent{\sc Fernando Galaz-Fontes} \\ 
Centro de Investigaci\'on en Matem\'aticas \\
Guanajuato, Gto., Mexico \\
email: galaz@fractal.cimat.mx \smallskip

\noindent{\sc Stephen Bruce Sontz}\\
Universidad Aut\'onoma Metropolitana-Iztapalapa \\
M\'exico, D.F. 09340, Mexico\\
email: sontz@xanum.uam.mx

\end{document}