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\title{Sharp bounds on heat kernels of higher order uniformly elliptic
operators}
\author{G. Barbatis {\hskip 2cm}   E.B. Davies}
\date{}

\begin{document}
\maketitle

\section{Introduction}
In \cite{D2} Davies established Gaussian heat kernel bounds for a class of
higher order elliptic operators with measurable coefficients acting on
$L^2(\R^N)$. He considered uniformly elliptic
operators of order $2m$ of the general form
\begin{equation}
Hf(x)=\sum_{\sscr{\alpha}{\beta}{\leq}}(-1)^{\mid\alpha\mid}D^{\alpha}
\{a_{\ab}(x)D^{\beta}f(x)\}
\label{eq:h}
\end{equation}
and proved that under certain conditions the kernel $K(t,x,y)$ of the
corresponding parabolic equation satisfies an off-diagonal estimate of
the form
\begin{equation}
\mid K(t,x,y)\mid \leq c_1t^{-N/2m}\exp\left\{-c_2\frac{\mid x-y\mid
^{2m/(2m-1)}}{t^{1/(2m-1)}}+c_3t\right\}.
\label{eq:1}
\end{equation}
Our aim in this paper is to obtain precise quantitative bounds on the
constants $c_2$ and $c_3$ above in terms of the ellipticity ratio and
dimension when the coefficients are measurable. This problem has been well
studied in the case $m=1$. Davies \cite{D4} was the first to obtain the
optimal constant $c_2=1/4$ for uniformly elliptic second order operators with
real measurable coefficients, using a Riemannian
distance defined in terms of the
operator coefficients instead of the Euclidean distance.
This result has since been extended in various directions by
many different authors and the theory has reached a high level of
sophistication. See \cite{D5,R,V} for three accounts of that theory. 


In the case of higher order operators however, no such bounds on $c_2$
seem to exist, even if local
regularity assumptions on the coefficients are made \cite[p441]{R},
\cite{K}.
We show that the constant $c_3$ can be taken to be arbitrarily small
if the bottom of the $L^2$ spectrum of $H$ is zero, and that one can put
$c_3=0$ if $H$ is homogenous. Our main result, Theorem \ref{thm:candy},
provides an explicit lower bound on $c_2$ of the stated type. When we
apply our method to the simplest case, $H=(-\Delta)^m$, we obtain a value
for $c_2$ which is sharp in one dimension, and which we conjecture is also
sharp in higher dimensions. See Theorem \ref{thm:1}.


We also consider the difference between short time and long time heat
kernel bounds. While the
higher order terms dominate for short times, the lower order ones
determine the long time estimates; see Proposition \ref{prop:mj}.
The long time behaviour can change dramatically
according to whether the lower order part takes negative values or not.
Further information on the term $c_3t$ and
long time estimates of heat kernels can be found in \cite{D3}.

The method which we follow in this paper is 
superficially similar to that in [D4]. We still make the 
assumptions (\ref{eq:sol}) and (\ref{eq:comp}), which were the key 
estimates of \cite{D2}. However we identify
the constant $k_{\lambda}$ introduced in Lemma \ref{lem:lunch} as
the crucial quantity involved in the problem,
and re-express as many as possible of the estimates of 
\cite{D2} in terms of that constant. We do this at an 
abstract level for possible future applications. The 
value of the results depends both upon obtaining the 
most efficient possible estimates in terms of $k_{\lambda}$ and 
upon being able to find sharp estimates for $k_{\lambda}$ in 
particular circumstances. We progress towards this 
in two steps. In Section 4 we 
give a general form for $k_{\lambda}$ which is valid for all 
homogeneneous operators of order $2m$ acting on 
$L^2(\R^N)$. We then evaluate $k_{\lambda}$ 
precisely for the particular case $H=(-\Delta)^m$, 
and use this information to obtain the main theorems 
of the paper. The fact that the final estimate which 
we obtain for $H=(-\Delta)^m$ is sharp, clearly indicates the effectiveness
of our approach.

\section{Preliminary results}


If $\alpha$ is a multi-index and $x$ a vector, we use the standard notation
$D^{\alpha}$ and $x^{\alpha}$ for the partial differential operator
$\partial_1^{\alpha_1}\ldots\partial_N^{\alpha_N}$ and the number
$x_1^{\alpha_1}\ldots x_N^{\alpha_N}$ correspondingly.


Let $\Omega\subset\R^N$ be a Euclidean domain. We shall be considering
differential operators on $L^2(\Omega)$
of order $2m$ that are comparable to $(-\Delta)^m$
not in the quadratic form sense but in the stronger sense of comparable
coefficients. More precisely, let
$\{a_{\ab}(x)\}_{\mid\alpha\mid,\mid\beta\mid\leq m}$ be the
uniformly bounded (self-adjoint) complex measurable 
coefficient matrix of the operator $H$ given by (\ref{eq:h}).
We assume that there exists a positive real constant $\mu\geq 1$ such that
\begin{equation}
\sum_{\sscr{\alpha}{\beta}{=}}a_{0,\ab}\xi_{\alpha}\bar{\xi}_{\beta}
\leq\sum_{\sscr{\alpha}{\beta}{=}}a_{\ab}(x)\xi_{\alpha}\bar{\xi}_{\beta}
\leq\mu\sum_{\sscr{\alpha}{\beta}{=}}a_{0,\ab}\xi_{\alpha}\bar{\xi}_{\beta},
\label{eq:jim}
\end{equation}
for all $x\in\Omega$ and $\xi\in\oplus_{\mid\alpha\mid =m}\C$, where the
non-negative
constant coefficient matrix $A_0=\{a_{0,\ab}\}$ is such that
\[\inprod{(-\Delta)^mf}{g}=\int_{\Omega}\sum_{\sscr{\alpha}{\beta}{=}}
a_{0,\ab}
D^{\alpha}fD^{\beta}\bar{g}\; dx\]
for all functions $f,g\in C^{\infty}_c(\Omega)$.
In the following we denote by $c$ or $c_i$ various constants depending
upon $m,N$ and
\begin{equation}
\nu=\sup\{\|a_{\ab}\|_{\infty}\;\;\mid\;\mid\alpha\mid,\mid\beta\mid\leq
m\}
\label{eq:oxf}
\end{equation}
as well as the constant $b$ introduced below.

We point out that it may be the
case that zero is an eigenvalue of the matrix $A_0$; see the example below.
Under the above assumptions the operator $H$ is defined to be the
self-adjoint operator associated to the closed and symmetric
form $Q$ with domain $W^{m,2}_0(\Omega)$ given by
\[Q(f)=\int_{\Omega}
\sum_{\sscr{\alpha}{\beta}{\leq}}a_{\ab}(x)D^{\alpha}f(x)
D^{\beta}\bar{f}(x)\;dx.\]
We shall call such operators {\em superelliptic}.
Adding a sufficiently large constant to the operator we may assume that
$Q$ is positive, and the fact that it is closed is then a
consequence of the inequality
\begin{equation}
c^{-1}\|(-\Delta)^{m/2}f\|_2^2\leq Q(f)\leq c
\left(\|(-\Delta)^{m/2}f)\|_2^2+\|f\|_2^2\right)
\label{eq:sol}
\end{equation}
which is valid for some $c>0$ and all $f\in C^{\infty}_c(\Omega)$.

A superelliptic operator is called {\em homogeneous} if it is of the
special form
\[Hf(x)=\sum_{\sscr{\alpha}{\beta}{=}}(-1)^{\mid\alpha\mid}D^{\alpha}
\{a_{\ab}(x)D^{\beta}f(x)\}.\]


It should be noted that for a given superelliptic operator $H$ the
representation (\ref{eq:h}) is not unique and that different coefficient
matrices can induce the same operator. For example, considering the operator
$\Delta^2$ on $\R^2$, one can write the expressions
\[\Delta^2=\partial_{11}^2\partial_{11}^2+\partial_{22}^2
\partial_{22}^2+\partial_{11}^2
\partial_{22}^2+\partial_{22}^2\partial_{11}^2\]
as well as
\[\Delta^2=\partial_{11}^2\partial_{11}^2+\partial_{22}^2\partial_{22}^2+
2\partial_{12}^2\partial_{12}^2.\]
which relate to the matrices
\[\left( \begin{array}{ccc}
 1&1&0\\
 1&1&0\\
 0&0&0
\end{array}\right) \quad\mbox{ and }\quad
\left(\begin{array}{ccc}
 1&0&0\\
 0&1&0\\
 0&0&2
\end{array}\right)\]
correspondingly. 

The results in this section apply to any operators satisfying the conditions
(\ref{eq:sol}),(\ref{eq:comp}) and (\ref{eq:cubi}). In addition to the
superelliptic operators defined above we refer to Section 10 of \cite{D2}
for another class of `elliptic' operators of order $2m$ satisfying these
conditions. We also note that
the only use of the second inequality in (\ref{eq:sol}) is to ensure
that the domain of $Q$ is $W^{m,2}_0(\Omega)$, and that all our results
can be re-expressed in a slightly more general context; see
Proposition \ref{prop:pot} below.



Following \cite{D2}, we define the class $\cE_m=\cE_m(\Omega)$
to consist of all bounded real-valued smooth functions $\phi$ satisfying
$\|\nabla\phi\|_{\infty}\leq 1$ and $\|D^{\alpha}\phi\|_{\infty}\leq b$
for all multi-indices $\alpha$ with
$2\leq\mid\alpha\mid\leq m$, where the positive real number $b$ is fixed
throughout the paper.


For $\phi\in\cE_m$ and $\lambda
\in\R$ the operator acting by multiplication by $e^{\lambda
\phi}$ is then a
bounded operator on $W^{m,2}_0(\Omega)$. We define the twisted,
complex-valued form $Q_{\lf}$ with domain $W^{m,2}_0(\Omega)$ by
\begin{equation}
Q_{\lf}(f,g)=Q(e^{\lf}f,e^{-\lf}g),
\label{eq:kel}
\end{equation}
so that
\begin{equation}
Q_{\lf}(f)=\int_{\Omega}
\sum_{\alpha,\beta}a_{\ab}(x)\{e^{-\lf}D^{\alpha}e^{\lf}f\}\{e^{\lf}D^{\beta}
e^{-\lf}\bar{f}\}dx.
\label{eq:1.3}
\end{equation}

We also denote the associated operator by $H_{\lf}$ so that
\begin{equation}
H_{\lf}=e^{-\lf}He^{\lf}.
\label{eq:5}
\end{equation}

The form $Q_{\lf}-Q$ is of order $2m-1$, and it is shown in \cite{D2} that
\begin{equation}
\mid Q_{\lf}(f)-Q(f)\mid \leq\epsilon Q(f) +\gamma_{\lambda}(\epsilon)
\|f\|_2^2.
\label{eq:comp}
\end{equation}
for all $0<\epsilon\leq 1$ and
$f\in W^{m,2}_0(\Omega)$, where $\gamma_{\lambda}(\epsilon)$ is a
polynomial of degree $2m$ as a function of $\lambda\in\R$.
This is proved by writing an explicit
expression for the difference $Q_{\lf}(f)-Q(f)$ for $f\in C^{\infty}_c(\R^N)$
and then using estimates of the form
\begin{equation}
\|\lambda^{k}D^{\gamma}f\|_2^2\leq\epsilon \|\nabla^{k+\mid\gamma\mid}
f\|_2^2
+c\epsilon^{-\mid\gamma\mid/k}\lambda^{2k+2\mid\gamma\mid}\|f\|_2^2
\label{eq:mj}
\end{equation}
which are proved by means of the Fourier transform.
See \cite{D2} for the proofs of the above statements.
We shall see that although $\epsilon$ can be taken to be
arbitrarily small, what is important is what happens when
$\epsilon$ is close to one; rather than (\ref{eq:comp}) we shall use
the weaker
\begin{equation}
\mid Q_{\lf}(f)-Q(f)\mid \leq\epsilon Q(f) +\gamma_{\lambda}\|f\|_2^2,
\label{eq:ter}
\end{equation}
valid for all $1/2\leq\epsilon\leq 1$ and $f\in W^{1,2}_0(\Omega)$.



%%%%%% main lemmas  %%%%

Before proceeding we note that different $\phi\in\cE_m$
may satisfy (\ref{eq:cubi}) below for different constants $k_{\lambda,\phi}$.
However, (\ref{eq:ter}) implies the crude bound
\[\sup_{\phi\in\cE_m} k_{\lambda,\phi}<+\infty.\]
In the rest of this section we shall write simply $k_{\lambda}$ instead of
$k_{\lambda,\phi}$ and re-introduce the notation $k_{\lambda,\phi}$ in
Proposition \ref{cor:1}. One has anyway $\sup_{\phi\in\cE_m}k_{\lambda,\phi}
<+\infty$ for all $\lambda\in\R$.


\begin{lemma}
Let $k_{\lambda}=k_{\lambda,\phi}$ be such that
\begin{equation}
\re Q_{\lf}(f)\geq -k_{\lambda}\|f\|_2^2
\label{eq:cubi}
\end{equation}
for some $\phi\in\cE_m$ and all $f\in W^{m,2}_0(\Omega)$. Then
\begin{eqnarray*}
&{\rm(i)}&\qquad\|e^{-H_{\lf}t}\|\leq e^{k_{\lambda}t}\\
\mbox{ and }&{\rm(ii)}&\qquad\|H_{\lf}e^{-H_{\lf}t}\|\leq 
\frac{c_{r,\epsilon}}{t}e^{(rk_{\lambda}+\epsilon)t}
\end{eqnarray*}
for all $r>1$ and $\epsilon>0$.
\label{lem:lunch}
\end{lemma}
{\em Proof : }It follows from the method of proof in \cite{D2} that there
exists $c>0$ such that $\gamma_{\lambda}\leq c(\lambda^{2m}+1)$ for all
$\lambda\in\R$. By looking first at $(-\Delta)^m$ and then at the general
case, one also sees that there exists $c'$ such that $k_{\lambda}\geq
c'(\lambda^{2m}-1)$ for all $\lambda\in\R$. We deduce that
there exist constants
$c_1$ and $c_2$ such that
\begin{equation}
\gamma_{\lambda}\leq c_1 k_{\lambda}+c_2
\label{eq:www}
\end{equation}
for all $\lambda\in\R$.
Now, let $f\in L^2$ and set $f_t=e^{-H_{\lf}t}f$. Then $f_t\in\dom
(H_{\lf})$ for all $t>0$ and
\begin{eqnarray*}
\frac{d}{dt}\|f_t\|_2^2&=& -\inprod{H_{\lf}f_t}{f_t}-
\inprod{f_t}{H_{\lf}f_t}\\
&\leq &2k_{\lambda}\|f_t\|_2^2,
\end{eqnarray*}
which implies (i). 

Now, it follows from (\ref{eq:ter}) that 
\begin{equation}
\re Q_{\lf}(f)\geq \frac{1}{2}Q(f)-\gamma_{\lambda}\|f\|_2^2
\label{eq:opera}
\end{equation}
so for $0\leq\eta\leq 1$ we have
\begin{eqnarray}
\re Q_{\lf}(f)&=&(1-\eta)\re Q_{\lf}(f)+\eta\re Q_{\lf}(f)\nonumber\\
&\geq&\frac{1-\eta}{2}Q(f)-(1-\eta)\gamma_{\lambda}\|f\|_2^2
  -\eta k_{\lambda}\|f\|_2^2
\label{eq:qq}
\end{eqnarray}
and hence
\[\re\{Q(f)-Q_{\lf}(f)\}\leq\frac{1+\eta}{2}Q(f)+[(1-\eta)
\gamma_{\lambda}+\eta k_{\lambda}]\|f\|_2^2.\]

Now, let $f\in L^2(\Omega)$ and $\theta\in (-\pi/2,\pi/2)$
be fixed and for $\rho>0$ set
\[f_{\rho}=\exp\{-H_{\lf}\rho e^{i\theta}\}f.\]
We then have
\begin{eqnarray*}
\frac{d}{d\rho}\|f_{\rho}\|_2^2 &=& -2\cos\theta\; Q(f_{\rho})+
2\re \left[e^{i\theta}(Q-
Q_{\lf})(f_{\rho})\right] \\
&=& -2\cos\theta\; Q(f_{\rho})+2\cos\theta\;
\re\left[Q(f_{\rho})-Q_{\lf}(f_{\rho})
\right]+2\sin\theta\;\im[Q_{\lf}(f_{\rho})]\\
&\leq &-2\cos\theta\; Q(f_{\rho})+2\cos\theta
 \left\{\frac{1+\eta}{2} Q(f_{\rho})+\right.\\
&&\left.+[(1-\eta)\gamma_{\lambda}+\eta
 k_{\lambda}]\|f_{\rho}\|_2^2\right\}+ 
2\sin\mid\theta\mid\left\{\frac{1}{2} Q(f_{\rho})
+\gamma_{\lambda}\|f_{\rho}\|_2^2\right\}\\
&=&\left\{(\eta-1)\cos\theta +\sin\mid\theta\mid\right\}Q(f_{\rho})+\\
&&
+\left\{2\cos\theta[(1-\eta)\gamma_{\lambda}
+\eta k_{\lambda}]+2\sin\mid\theta\mid
\gamma_{\lambda}\right\}\|f_{\rho}\|_2^2.
\end{eqnarray*}
Defining $\alpha\in (0,\pi/2)$ by
\[\tan\alpha=1-\eta\]
it follows that for $\mid\theta\mid\leq\alpha$ we have
\[(\eta-1)\cos\theta +\sin\mid\theta\mid\leq 0\]
and
\[2\cos\theta[(1-\eta)\gamma_{\lambda}+\eta k_{\lambda}]
 +2\sin\mid\theta\mid\gamma_{\lambda}\leq 4(1-\eta)\gamma_{\lambda}
+2k_{\lambda}.\]
Using (\ref{eq:www}) we conclude that
\[\frac{d}{d\rho}\|f_{\rho}\|_2^2\leq\{4(1-\eta)(c_1k_{\lambda}+c_2)+2k_{\la
mbda}
\}\|f_{\rho}\|_2^2.\]
Solving the differential inequality yields
\[\|f_{\rho}\|_2\leq\exp\left\{2(1-\eta)(c_1k_{\lambda}+c_2)+
k_{\lambda}\right\}\|f\|_2,\]
that is,
\[\|e^{-H_{\lf}z}\|\leq\exp\left\{[2(1-\eta)(c_1k_{\lambda}+c_2)+
k_{\lambda}]\mid z\mid\right\}\]
for all $\mid\theta\mid\leq\alpha$.

Now, let
\begin{equation}
\tau_{\lambda,\eta}=\frac{2(1-\eta)(c_1k_{\lambda}+c_2)+
k_{\lambda}}{\cos\alpha}
\label{eq:sunny}
\end{equation}
so that
\[\|e^{-(H_{\lf}+\tau_{\lambda,\eta})z}\|\leq 1\]
if $\mid\theta\mid\leq\alpha$.
It is a known result \cite[p64]{D1} that this implies
\[\|(H_{\lf}+\tau_{\lambda,\eta})e^{-(H_{\lf}+\tau_{\lambda,\eta})t}
\|\leq\frac{c}{\alpha t}.\]
for all $t>0$. Hence, for any $\delta>0$ we have
\begin{eqnarray*}
\|H_{\lf}e^{-H_{\lf}t}\|&\leq& \frac{c}{\alpha t}e^{\tau_{\lambda,\eta}t}
+\tau_{\lambda,\eta}e^{\tau_{\lambda,\eta}t}\\
&\leq& \frac{c}{\alpha t}e^{\tau_{\lambda,\eta}t}+\frac{c_{\delta}}{t}
e^{(1+\delta)\tau_{\lambda,\eta}t}\\
&\leq& \frac{c_{\eta,\delta}}{t}e^{(1+\delta)\tau_{\lambda,\eta}t}.
\end{eqnarray*}

But it follows from (\ref{eq:sunny})
that given $r>1$ and $\epsilon>0$ we can find $\eta$ close
enough to one and $\delta$ close enough to zero so that
\[(1+\delta)\tau_{\lambda,\eta}\leq rk_{\lambda}+\epsilon.\]
This proves (ii).$\qquad\qquad\Box$


{\bf Hypothesis : }
At this point a condition on the order $2m$ of the operator is necessary;
we assume from now on that $2m>N$.


We shall need the following lemma from \cite{D2}:
\begin{lemma}
If $2m>N$ then $f\in W^{m,2}_0(\Omega)$ implies $f\in L^{\infty}(\Omega)$ and
\begin{equation}
\|f\|_{\infty}\leq c\|(-\Delta)^{m/2}f\|_2^{{N/2m}}\|f\|_2^{1-{N/2m}}.
\label{eq:skata}
\end{equation}
Moreover, $e^{-Ht}$ is ultracontractive and
\begin{equation}
\|e^{-Ht}f\|_{\infty}\leq ct^{-N/4m}\|f\|_2
\label{eq:skata1}
\end{equation}
for all $t>0$ and $f\in L^2$.
\end{lemma}
{\em Proof : }This is proved in \cite{D2} for the case $\Omega=\R^N$; the
first estimate then follows for general $\Omega$ by using the inclusion
$W^{m,2}_0(\Omega)\subset W^{m,2}(\R^N)$. The second then follows as in
\cite{D2}.



\begin{lemma}
The semigroup $\exp(-H_{\lf}t)$ is ultracontractive and
\[\|e^{-H_{\lf}t}\|_{\infty,2}\leq c_{r,\epsilon}
t^{-N/4m}e^{(rk_{\lambda}+\epsilon)t}\]
for all $r>1$ and $\epsilon>0$.
\label{lem:ear}
\end{lemma}
{\em Proof : }Let $f\in L^2$ and set $f_t=e^{-H_{\lf}t}f$.
Using the estimates of the last lemma we have

\begin{eqnarray*}
\|f_t\|_{\infty}&\leq& cQ(f_t)^{N/4m}\|f_t\|_2^{1-N/2m}\\
&\leq& c\left\{\re Q_{\lf}(f_t)+\gamma_{\lambda}\|f_t\|_2^2\right\}
^{{N/4m}}\|f_t\|_2^{1-{N/2m}}\\
&\leq& c\left\{\|H_{\lf}f_t\|_2\|f_t\|_2 +(c_1k_{\lambda}+c_2)
\|f_t\|_2^2\right\}^{{N/4m}}
\|f_t\|_2^{1-{N/2m}}\\
&\leq&c\left\{\frac{c_{r,\epsilon}}{t}e^{(rk_{\lambda}+\epsilon)t}
  e^{k_{\lambda}t}+(c_1k_{\lambda}+c_2)e^{2k_{\lambda}t}\right\}^{{N/4m}}
  e^{k_{\lambda}t(1-{N/2m})}\|f\|_2\\
&=&ct^{-N/4m}\{c_{r,\epsilon}e^{(r-1)k_{\lambda}t+\epsilon t}
  +(c_1k_{\lambda} +c_2)t\}^{{N/4m}}e^{k_{\lambda}t}\|f\|_2.\\
\end{eqnarray*}
Given any $r'>1$ and $\epsilon'>0$ one can find $r$ close enough one and
$\epsilon$ close enough to zero so that the last term is smaller than
\[c_{r',\epsilon'}t^{-N/4m}e^{(r'k_{\lambda}+\epsilon')t}\|f\|_2,\]
as required. $\qquad\qquad\qquad\Box$


The starting point for our main theorems will be the following
\begin{proposition}
For any $r>1$ and $\epsilon>0$ there exists a constant
$c_{r,\epsilon}$ such that
\begin{equation}
\mid K(t,x,y)\mid\leq c_{r,\epsilon} 
t^{-N/2m}\exp\left\{\lambda (\phi(x)-\phi(y))
+(rk_{\lambda,\phi}+\epsilon)t\right\},
\label{eq:pict}
\end{equation}
for all $\lambda\in\R$ and all $\phi\in\cE_m$.
\label{cor:1}
\end{proposition}
{\em Proof : }Lemma \ref{lem:ear} implies that the kernel $K_{\lf}(t,x,y)$
of $e^{-H_{\lf}t}$ satisfies
\[\mid K_{\lf}(t,x,y)\mid\leq c_{r,\epsilon} 
t^{-N/2m}\exp\{(rk_{\lambda,\phi}+\epsilon)t\}.\]
But it follows from (\ref{eq:5}) that
\[K_{\lf}(t,x,y)=e^{-\lambda\phi(x)}K(t,x,y)e^{\lambda\phi(y)},\]
hence
\begin{equation}
\mid K(t,x,y)\mid\leq c_{r,\epsilon} 
t^{-N/2m}\exp\left\{\lambda(\phi(x)-\phi(y))+(rk_{\lambda,\phi}
+\epsilon)t\right\}
\end{equation}
as required. $\qquad\qquad\Box$



%%%%%%%%%%%  linear phi's  %%%%%%%%%%%%%

\section{Linear $\phi$'s}
Up to this point we have taken the function $\phi$ to lie in $\cE_m$.
This choice guarantees that the map $f\mapsto e^{\lf}f$ is a bounded
automorphism of $W^{m,2}_0(\Omega)$.
Since it is also an automorphism of $L^2(\Omega)$, it
induces a canonical functional calculus for the operator $H_{\lf}$ by
the equation
\begin{equation}
f(H_{\lf})=e^{-\lf}f(H)e^{\lf}.
\label{eq:hypnos}
\end{equation}
In order to obtain sharp constants below in the case $\Omega=\R^N$,
it is necessary to
consider functions $\phi$ that are linear. Since such functions are not
bounded some extra arguments are needed; these arguments are unnecessary
if the domain $\Omega$ is bounded. 


Let $\Omega\subset\R^N$ be unbounded and let
\[\phi(x)=a\cdot x\]
where $a$ is a vector of unit length. We cannot use (\ref{eq:kel})
to define $Q_{\lf}$ since multiplication by $e^{\lf}$ does not leave
$W^{m,2}_0(\Omega)$ invariant. We can however compute the RHS of
(\ref{eq:1.3}) formally, and a simple calculation yields
\begin{equation}
Q_{\lf}(f)=\int_{\Omega}\sum_{\alpha,\beta}a_{\ab}(x)\sum_
{{\scriptstyle
\gamma_1+\delta_1=\alpha \atop\scriptstyle
\gamma_2+\delta_2=\beta}}c'_{\gamma_1,\delta_1}c'_{\gamma_2,\delta_2}
(\lambda a)^{\gamma_1}(-\lambda a)^{\gamma_2}
D^{\delta_1}fD^{\delta_2}\bar{f}\; dx
\label{eq:inter}
\end{equation}
where
\[c'_{\gamma,\delta}=\frac{(\gamma+\delta)!}{\gamma !\;\delta !}.\]
The RHS of (\ref{eq:inter}) is well defined for $f\in W^{m,2}_0(\Omega)$
and we use this formula to define the (closed) form $Q_{\lf}$. 
Note that the functional calculus (\ref{eq:hypnos}) is no longer valid.
However, we still have the following
\begin{proposition}
Assume $2m>N$ and let $k_{\lambda}$ be such that
\[\re Q_{\lf}(f) \geq -k_{\lambda}\|f\|_2^2\]
for some linear function $\phi$ and all $f\in W^{m,2}_0(\Omega)$. Then
\[\mid K(t,x,y)\mid\leq c_{r,\epsilon} 
t^{-N/2m}\exp\left\{\lambda(\phi(x)-\phi(y))
+(rk_{\lambda}+\epsilon)t\right\}\]
for all $r>1$ and $\epsilon>0$.
\label{prop:cup}
\end{proposition}
{\em Proof  : }Let $(\Omega_n)$ be an increasing sequence of bounded
domains such that $\cup\Omega_n=\Omega$. For each $n$ we denote by $H_n$
the operator on $L^2(\Omega_n)$ induced by $H$ and satisfying Dirichlet
boundary conditions. So $H_n$ is the operator associated to the form
$Q_n$ obtained by restricting $Q$ to $W^{m,2}_0(\Omega_n)$. For our given
linear $\phi$ the twisted form $Q_{n,\lf}$ has already been defined, and we
set
\[-k_{\lambda,n}=\inf \re Q_{\lf}(f)\]
where the infimum is taken over all functions $f\in C^{\infty}_c(\Omega_n)$
with  $\|f\|_2=1$. It is immediate that the sequence $(k_n)$ is increasing
and that 
\[\lim k_{\lambda,n}\leq k_{\lambda}\]
(with an actual equality holding if $k_{\lambda}$ is chosen optimally).
Since $\phi$ is bounded on each $\Omega_n$ we can apply our earlier results
to the operators $H_n$ and conclude that
\[\mid K_n(t,x,y)\mid\leq c_{r,\epsilon} t^{-N/2m}\exp\left\{\lambda
(\phi(x)-\phi(y))
+(rk_{\lambda,n}+\epsilon)t\right\}\]
for all $r>1,\epsilon>0,t>0$ and $x,y\in\Omega_n$.

Now the sequence $Q_n$ is a decreasing
sequence and $Q_n(f)\rightarrow Q(f)$ for all $f\in C^{\infty}_c(\Omega)$.
This implies \cite[p8]{D5} that
\[(H_n+1)^{-1}\longrightarrow (H+1)^{-1}\]
strongly (where $(H_n+1)^{-1}$ is now a pseudo-resolvent) and from this
follows that
\[e^{-H_nt}\longrightarrow e^{-Ht}\]
strongly, where, again, $e^{-H_nt}$ is interpreted as being zero on
$L^2(\Omega\setminus\Omega_n)$ so that its kernel is 
\[K'_n(t,x,y)=\chi_{\Omega_n}(x)K_n(t,x,y)
\chi_{\Omega_n}(y).\]
For $f,g\in C^{\infty}_c(\Omega)$ we then have
\begin{eqnarray*}
&&\mid\int_{\Omega\times\Omega}K(t,x,y)f(x)g(y)dxdy\mid \\
&=&\mid\inprod{e^{-Ht}f}{g}\mid\\
&=&\lim_n\mid\inprod{e^{-H_nt}f}{g}\mid\\
&=&\lim_n\mid\int_{\Omega_n\times\Omega_n}K'_n(t,x,y)f(x)g(y)dxdy\mid \\
&\leq&\limsup_n
    c_{r,\epsilon} t^{-N/2m}\!\int_{\Omega_n\times\Omega_n}
    \!\!\exp\left\{\lambda (\phi(x)-\phi(y))
   +(rk_{\lambda,n}+\epsilon)t\right\}\mid f(x)g(y)\mid dxdy \\
&\leq&c_{r,\epsilon} t^{-N/2m}\int_{\Omega\times\Omega}
     \exp\left\{\lambda(\phi(x)-\phi(y)) +(rk_{\lambda}+\epsilon)t
    \right\}\mid f(x)g(y)\mid dxdy.
\end{eqnarray*}
Since $f$ and $g$ are arbitrary, this implies that
\[\mid K(t,x,y)\mid\leq c_{r,\epsilon}
t^{-N/2m}\exp\left\{\lambda (\phi(x)-\phi(y))
+(rk_{\lambda}+\epsilon)t\right\}\]
as required.$\qquad\qquad\Box$


>From now on we shall restrict our attention to operators acting on the
whole of $\R^N$ and to functions $\phi$ belonging to the set
\[\cE_{lin}=:\{x\mapsto a\cdot x\mid\; a\in\R^N ,\;\mid a\mid \leq 1\}\].


%%%%%%%%%%%%%%%  Homogeneous operators   %%%%%%%%%%%%%

\section{Homogeneous operators}
In this section we shall consider the case of homogeneous operators
and we shall only consider functions $\phi\in\cE_{lin}$. In this case every
term in (\ref{eq:inter}) has $\mid\gamma_1+\delta_1\mid =$
$\mid\gamma_2+\delta_2\mid =m$ and thus it follows from (\ref{eq:mj}) that
the function $\gamma_{\lambda}(\epsilon)$
in (\ref{eq:comp}) can be taken to be of the special form
\begin{equation}
\gamma_{\lambda}(\epsilon)=\gamma(\epsilon)\lambda^{2m}.
\label{eq:nicesun}
\end{equation}
Similarly, the estimate (\ref{eq:cubi}) can be taken to be of the form
\begin{equation}
\re Q_{\lf}(f)\geq - k\lambda^{2m}\|f\|_2^2.
\label{eq:flamengo}
\end{equation}

The determination of the smallest constant $k$ for which (\ref{eq:flamengo})
holds is non-trivial and is studied for various particular situations
below. We have the following
\begin{lemma}
Let $2m>N$ and let $H$ be a homogeneous operator
satisfying (\ref{eq:flamengo}) for all functions $\phi\in\cE_{lin}$.
We then have
\[\mid K(t,x,y)\mid \leq c_rt^{-N/2m}\exp\left\{-\frac{2m-1}{2mr}(2km)^{
-1/(2m-1)}\frac{\mid x-y\mid ^{2m/(2m-1)}}{t^{1/(2m-1)}}
\right\}\]
for all $r>1$.
\label{lem:11}
\end{lemma}

{\em Proof : }From Proposition \ref{prop:cup} we have
\[\mid K(t,x,y)\mid \leq c_{r,\epsilon}
t^{-N/2m}\exp\left\{\lambda(\phi(x)-\phi(y))
+(rk\lambda^{2m}+\epsilon)t\right\}\]
for all $\phi\in\cE_{lin}$. Optimizing over all such $\phi$ yields
\[\mid K(t,x,y)\mid \leq c_{r,\epsilon}
t^{-N/2m}\exp\left\{-\lambda\mid x-y\mid
+(rk\lambda^{2m}+\epsilon)t\right\}\]
and optimizing over $\lambda$ by putting
\[\lambda=\left(\frac{\mid x-y\mid}{2mrkt}\right)^{\frac{1}{2m-1}}\]
yields
\begin{equation}
\mid K(t,x,y)\mid \leq c_{r,\epsilon}
t^{-N/2m}\exp\left\{-\frac{2m-1}{2mr}(2km)^{
-1/(2m-1)}\frac{\mid x-y\mid ^{2m/(2m-1)}}{t^{1/(2m-1)}}+\epsilon t
\right\}.
\label{eq:414}
\end{equation}
Using a scaling argument we eliminate the term $\epsilon t$: Let $\delta>0$
be fixed, let $U$ be the unitary operator given by
\[Uf(x)=\delta^{N/2}f(\delta x)\]
and set
\[H'=\delta^{-2m}U^{-1}HU.\]
Then $\{a_{\ab}(\delta x)\}$ is a coefficient matrix for $H'$ and the heat
kernel $K'(t,x,y)$ is related to $K(t,x,y)$ by
\[K(t,x,y)=\delta^N K'(\delta^{2m}t,\delta x,\delta y).\]
Applying (\ref{eq:414}) to $K'(t,x,y)$ we get
\begin{eqnarray*}
&&\hspace{-1.5cm}\mid K(t,x,y)\mid\\
&\leq&c_{r,\epsilon}\delta^N (\delta^{2m}t)^{-N/2m}\\
&&\exp\left\{-\frac{2m-1}{2mr}(2km)^{
-1/(2m-1)}\frac{(\delta\mid x-y\mid) ^{2m/(2m-1)}}
{(\delta^{2m}t)^{1/(2m-1)}}+\epsilon\delta^{2m}t
\right\}\\
&=&c_{r,\epsilon}t^{-N/2m}\exp\left\{-\frac{2m-1}{2mr}(2km)^{
-1/(2m-1)}\frac{\mid x-y\mid ^{2m/(2m-1)}}{t^{1/(2m-1)}}+\epsilon\delta^{2m}t
\right\}.
\end{eqnarray*}
The result now follows by letting $\delta\rightarrow 0.\qquad\Box$

{\bf Powers of the Laplacian}

A special case of homogeneous operators that can be treated in more detail
is the case where $H=(-\Delta)^m$.

\begin{lemma}
Let
\begin{equation}
k_m=\left(\sin\frac{\pi}{4m-2}\right)^{-(2m-1)}.
\label{eq:313}
\end{equation}
We have 
\[\re Q_{\lf}(f)\geq -k_m\lambda^{2m}\|f\|_{2}^2.\]
for all $\phi\in\cE_{lin}$ and all $f\in W^{m,2}(\R^N)$.
\label{lem:20}
\end{lemma}
{\em Proof : }Since $\dom(H_{\lf})$ is a form core for $Q_{\lf}$, it is enough
to prove that
\[\re \inprod{H_{\lf}f}{f}\geq -k_m\lambda^{2m}\|f\|_{2}^2\]
for all $f\in\dom(H_{\lf})$.
For any multi-indices $\alpha$, $\beta$ and $\gamma$ we set
\[c_{\alpha}=\frac{(\alpha_1+\cdots +\alpha_N)!}{\alpha_1!\ldots\alpha_N!},
\qquad\quad c_{\beta\gamma}'=\frac{(\beta+\gamma)!}{\beta!\gamma!}\]
so that
\begin{eqnarray*}
\Delta^m(e^{\lf}f)&=&\sum_{\mid\alpha\mid=m}c_{\alpha}D^{2\alpha}(e^{\lf}f)\\
&=&\sum_{\mid\alpha\mid=m}
c_{\alpha}\sum_{\beta+\gamma =2\alpha}
c_{\beta\gamma}'(D^{\gamma}e^{\lf})D^{\beta}f
\end{eqnarray*}
and hence
\begin{eqnarray*}
e^{-\lf}(-\Delta)^me^{\lf}f&=& (-1)^m
{\sum}'(e^{-\lf}D^{\gamma}e^{\lf})D^{\beta}f \\
&=& (-1)^m{\sum}'(\lambda a)^{\gamma}D^{\beta}f
\end{eqnarray*}
where $\sum'=\sum_{\mid\alpha\mid=m}c_{\alpha}\sum_{\beta+\gamma=2\alpha}
c_{\beta\gamma}'$.
In the Fourier space this acts by multiplication by the complex-valued
polynomial
\begin{eqnarray*}
P(\xi)&=&(-1)^m{\sum}'(\lambda a)^{\gamma}(i\xi)^{\beta}\\
&=&(-1)^m\sum_{\mid\alpha\mid=m}c_{\alpha}(\lambda a+i\xi)^{2\alpha}\\
&=&(-1)^m\{(\lambda a_1+i\xi)^2+\cdots+(\lambda a_N +i\xi_N)^2\}^m\\
&=&(\xi^2-\lambda^2+2i\lambda a\cdot\xi)^m\\
&=:&\lambda^{2m}\hat{P}(\xi/\lambda)
\end{eqnarray*}
where $\hat{P}(\xi)=(\xi^2+2ia\cdot\xi -1)^m$. The minimum of
$\re\hat{P}(\xi)$
is attained when $\xi=\mu a$ for an appropriate $\mu\in\R$, and for such
a $\xi$ we get
\begin{equation}
P(\xi)=\lambda^{2m}(\mu+i)^{2m}.
\label{eq:1.2}
\end{equation}

Writing $\mu+i=re^{i\theta}$ where $r>0$ and $0<\theta <\pi$
we have $r^2=\sin^{-2}
\theta$
so that
\[\re(\mu+i)^{2m}=(\sin\theta)^{-2m}\cos 2m\theta.\]
The result follows by minimizing the above with respect to $\theta$.
$\qquad\quad\Box$


The restriction on $N$ in the following theorem is almost surely not
necessary.
\begin{theorem}
Let $2m>N$ and
\[\sigma_m=(2m-1)(2m)^{-2m/(2m-1)}\sin(\frac{\pi}{4m-2}).\]
The heat kernel $K(t,x,y)$ of $(-\Delta)^m$ satisfies the bound
\begin{equation}
\mid K(t,x,y)\mid\leq c_{r}t^{-N/2m}
\exp\left\{-\sigma_m\frac{\mid x-y\mid^{2m/(2m-1)}}
{rt^{1/(2m-1)}}\right\}
\label{eq:22}
\end{equation}
for all $r>1$.
\label{thm:1}
\end{theorem}
{\em Proof : }Follows from Corollary \ref{cor:1} and Lemma \ref{lem:lunch}.
$\qquad\Box$

{\bf Remark }The constant $\sigma_m$ is optimal for $N=1$ and all $m$
and we conjecture that it is optimal for all values of $N$ and $m$.
First, we have $\sigma_1=1/4$, which is known to be optimal.
Moreover, in one dimension one can use the tools of
asymptotic analysis to find the large $x$ asymptotics of
\[S_m(x)=:K(1,x,0)=\int_{-\infty}^{\infty}e^{ix\xi -\xi^{2m}}d\xi \]
and one sees that $\sigma_m$ is optimal. For example, for $m=2$ the method
of steepest descent \cite{F,M} yields
\begin{equation}
S_2(x)\sim 2^{11/6}3^{-1/2}\pi^{1/2}x^{-2/3}\cos\left(\frac{2^{1/3}3^{3/2}}
{16}x^{4/3}-\frac{\pi}{3}\right)e^{-2^{-11/3}3x^{4/3}}
\label{eq:101}
\end{equation}
as $x\rightarrow +\infty$. Since $K(t,x,y)=t^{-1/4}S_2(t^{-1/4}(x-y))$,
(\ref{eq:101}) and a simple argument
also show that one cannot put $r=1$ in (\ref{eq:22}).
Formulas similar to (\ref{eq:101}) can be established for
$m>2$, but if the dimension is higher, then obtaining an asymptotic
estimate becomes much harder. Of course, the method fails completely
if one considers operators with variable coefficients.





{\bf Variable coefficients}

Let $H$ be superelliptic and homogeneous of order $2m$ where $2m>N$.
So $H$ has a representation
\[Hf(x)=\sum_{\sscr{\alpha}{\beta}{=}}
 (-1)^{\mid\alpha\mid}D^{\alpha}\{a_{\alpha\beta}(x) D^{\beta}
f(x)\}\]
where the self-adjoint matrix $A(x)=\{a_{\ab}(x)\}$ is such that
\begin{equation}
A_0\leq A(x)\leq\mu A_0
\label{eq:nice}
\end{equation}
for some constant $\mu\geq 1$ and all $x\in\R^N$, where $A_0$ is a constant
matrix representing the operator $(-\Delta)^m$. This of course implies
\begin{equation}
H_0\leq H\leq\mu H_0
\label{eq:sleepy}
\end{equation}
in the quadratic form sense.

\begin{theorem}
Assume $2m>N$. The kernel $K(t,x,y)$ satisfies the estimate
\[\mid K(t,x,y)\mid\leq c_{r}t^{-N/2m}\exp\left\{-\rho(\mu,m)
\frac{\mid x-y\mid ^{2m/(2m-1)}}{rt^{1/(2m-1)}
}\right\}\]
for all $r>1$, where
\[\rho(\mu,m)=(2m-1)(2m)^{-2m/(2m-1)}\mu^{1/(2m-1)}
\left\{k_m+ c(\mu-1)\mu^m\right\}^{-1/(2m-1)}\]
and $k_m$ is given by (\ref{eq:313}). In particular
\[\rho(\mu,m)=\sigma_m -O(\mu-1)\]
as $\mu\rightarrow 1$.
\label{thm:candy}
\end{theorem}
{\em Proof : }Let $\phi\in\cE_{lin}$ and $f\in W^{m,2}$ 
be fixed. We define the square-integrable vector valued function
\[\{v_{\lambda,\alpha}\}_{\mid\alpha\mid=m}=\{\tlf{\alpha}f(x)\}
_{\mid\alpha\mid=m},\]
so that by (\ref{eq:1.3})
\[Q_{\lf}(f)=\int A(x)v_{\lambda}(x)\cdot v_{-\lambda}(x)\;dx\]
where the dot denotes the standard inner product in
$\oplus_{\mid\alpha\mid =m}\C$. Writing $\phi(x)=a\cdot x$ we have
\begin{eqnarray*}
\tlf{\alpha}f&=& \sum_{\gamma+\delta=\alpha}c_{\gamma,\delta}'(\tlf{\gamma})
D^{\delta}f\\
&=&\sum_{\gamma+\delta=\alpha}c_{\gamma,\delta}'(\lambda a)^{\gamma}
(D^{\delta}f),
\end{eqnarray*}
and hence we can write
\[v_{\lambda}=v_{\lambda}^++v_{\lambda}^-,\]
\[v_{-\lambda}=v_{\lambda}^+-v_{\lambda}^-\]
where
\[v_{\lambda,\alpha}^+=\sum_{\scriptstyle  {\gamma+\delta =\alpha}
\atop\scriptstyle {\mid\gamma\mid\;\; \mbox{\small even}}}
c_{\gamma,\delta}'(\lambda a)^{\gamma}
D^{\delta}f,\]
\[v_{\lambda,\alpha}^-=\sum_{\scriptstyle  {\gamma+\delta =\alpha}
\atop\scriptstyle {\mid\gamma\mid\;\; \mbox{\small  odd}}}
c_{\gamma,\delta}'(\lambda a)^{\gamma}
D^{\delta}f.\]
Thus
\begin{eqnarray*}
Q_{\lf}(f)&=&\int Av_{\lambda}\cdot v_{-\lambda}\;dx \\
&=&\int A(v^+_{\lambda}+v^-_{\lambda})\cdot(v^+_{\lambda}-v^-_{\lambda})\;dx.
\end{eqnarray*}

Hence from (\ref{eq:nice}) we have
\begin{eqnarray*}
\re Q_{\lf}(f)&=&\int (Av_{\lambda}^{+}\cdot v_{\lambda}^{+}-
Av_{\lambda}^{-}\cdot v_{\lambda}^{-})dx \\
&\geq& \int A_0 v_{\lambda}^{+}\cdot v_{\lambda}^{+}\; dx
-\mu \int A_0 v_{\lambda}^{-}\cdot v_{\lambda}^{-}dx\\
&=&\re Q_{0,\lf}(f)-(\mu-1)\int A_0v^-_{\lambda}\cdot v^-_{\lambda} \;dx \\
&\geq&\re Q_{0,\lf}(f)-c_0(\mu-1)\|v^-_{\lambda}\|_2^2
\end{eqnarray*}
where the constant $c_0$ is independent of $\phi$ and $\mu$.

The vector $v_{\lambda}^-$ only contains derivatives of order $< m$, and it
follows from (\ref{eq:mj}) that
\[\|v_{\lambda}^{-}\|_2^2\leq \epsilon Q_0(f)+
c\epsilon^{1-m}\lambda^{2m}\|f\|_2^2\]
for all $\epsilon>0$ and all $f\in W^{m,2}(\R^N)$.


Using the lower bound that we obtained in Lemma \ref{lem:20} for
$\re Q_{0,\lf}$ together with (\ref{eq:qq}) and (\ref{eq:nicesun})
we have for $0\leq\eta\leq 1$
\begin{eqnarray*}
\re Q_{\lf}(f)&\geq& \{\frac{1-\eta}{2}-c_0(\mu-1)\epsilon\}Q_0(f) -\\
&&-\{c(1-\eta)+\eta k_m
+c(\mu-1)\epsilon^{1-m}\}\lambda^{2m}\|f\|_2^2
\end{eqnarray*}
and taking
\[\epsilon=\frac{1-\eta}{2c_0(\mu-1)}\] 
we get
\[\re Q_{\lf}(f)\geq -\left\{c(1-\eta)+\eta k_m 
  +c(\mu-1)^m(1-\eta)^{1-m}
  \right\}\lambda^{2m}\|f\|_2^2.\]
Choosing $\eta\in (0,1)$ so that
\[1-\eta =(\mu-1)/\mu\]
we conclude that
\begin{equation}
\re Q_{\lf}(f)\geq -\mu^{-1}\left\{k_m+ c(\mu-1)\mu^m
\right\}\lambda^{2m}\|f\|_2^2
\end{equation}
The result now follows by applying Lemma \ref{lem:11}.
$\qquad\quad\Box$

%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-homogeneous operators}
Up to this point we have only considered homogeneous (in the form sense)
operators. One of the common properties of the bounds this far obtained is
that they all involve a single term in the
exponent, of the general form $c\mid x-y\mid ^{2m/(2m-1)}/t^{1/(2m-1)}$. This
property is destroyed if one considers non-homogeneous operators.

We shall look at a simple example. Let $m_1>m_2$ and let
\[H=H_1+H_2=:(-\Delta)^{m_1}+(-\Delta)^{m_2}.\]
Note that the heat kernel of $H$ is now of the special form
$K(t,x,y)=K_t(x-y)$ and one has
\begin{equation}
K_t=K_{1,t}*K_{2,t}
\label{eq:quilt}
\end{equation}
where $K_{i,t}(x-y)$, $i=1,2$ is the heat kernel of $H_i$. This however
depends on the fact that the operators $H_i$ have constant coefficients.
Rather than using (\ref{eq:quilt}), we prefer to use the method discussed
earlier, which can also be applied in the variable coefficient case.
See also the note below.



The off-diagonal behaviour of the heat kernel depends on the ratio
$\mid x-y\mid /t$: if the ratio is large (small times) then $H_1$ is the
dominant component; if it very small, then $H_2$ is.
More precisely, let
\[L_r(u)=\inf_{\lambda\in\R}\left\{
-\lambda u+r(k_1\lambda^{2m_1}+k_2\lambda^{2m_2})\right\}\]
be the Legendre transform of the function
\[\lambda\mapsto r(k_1\lambda^{2m_1}+k_2\lambda^{2m_2}),\]
which can be computed numerically. We have the following

\begin{proposition}
Let $2m_1>N$. For any $r>1$ we have
\[\mid K(t,x,y)\mid \leq c_rt^{-N/2m_1}\exp\{tL_r(\mid x-y\mid /t)\}.\]
In particular, setting $\rho=(2m_1-2m_2)/(2m_2-1)$ we have
{\rm(i)} for large $\mid x-y\mid/t$
\[tL(\mid x-y\mid /t)
=-\sigma_1\mid x-y\mid^{2m_1/(2m_1-1)}t^{-1/(2m_1-1)}
\left(1-c(\mid x-y\mid /t)^{-\rho}\right),\]
and {\rm(ii)} for small $\mid x-y\mid/t$,
\[tL(\mid x-y\mid /t)=-\sigma_2\mid x-y\mid^{2m_2/(2m_2-1)}
t^{-1/(2m_2-1)}\left(1-c(\mid x-y\mid /t)^{\rho}\right).\]
\label{prop:mj}
\end{proposition}
{\em Proof : }We have from Lemma \ref{lem:20}
\[\re Q_{i,\lf}(f)\geq -k_i\lambda^{2m_i}\|f\|_2^2 \;,\qquad i=1,2\]
and hence
\[\re Q_{\lf}(f)\geq -\left(k_1\lambda^{2m_1}+k_2\lambda^{2m_2}
\right)\|f\|_2^2.\]
Hence, by Lemma \ref{cor:1} we obtain a pointwise bound
on the kernel of the corresponding semigroup
\begin{equation}
\mid K(t,x,y)\mid \leq c_rt^{-N/2m_1}\exp (rA_{r,\lambda}),\quad\all r>1,
\label{eq:and}
\end{equation}
where
\[A_{r,\lambda}=-\lambda\mid x-y\mid
+r(k_1\lambda^{2m_1}+k_2\lambda^{2m_2})t,\]
and the first assertion follows by taking the infimum over $\lambda$.

As for the two asymptotic estimates, the first one follows from
(\ref{eq:and}) by choosing
\[\lambda=\left[(2m_1k_1)^{-1}\mid x-y\mid t^{-1}\right]^{1/(2m_1)}\]
while the dual choice proves the second.$\qquad
\Box$

The same arguments can be used to obtained
similar bounds when one considers
the sum of general homogeneous operators with variable coefficients. The
expression in the
exponential will then involve some extra terms, namely the lower bounds
on $\re Q_{\lf}$ that were obtained in the proof of Theorem
\ref{thm:candy}. We do not pursue the details.


As one would expect, adding a non-negative potential to an operator does not
pose any problems for heat kernel estimates.
Let $H_0$ be a general superelliptic operator (not necessarily homogeneous).
Given a non-negative potential $V\in L^1_{{\rm loc}}(\R^N)$,
one can define
$H=H_0+V$ to be the operator associated to the (closed) form $Q$ defined by
\[\dom(Q)=\{f\in\dom(Q_0)\;\mid \int V\mid f\mid^2 dx <+\infty\}\]
and
\[Q(f)=Q_0(f)+\int V\mid f\mid^2 dx,\qquad f\in\dom(Q).\]
Although $\dom(Q)$ does not necessarily coincide with
$W^{m,2}$, the theory still applies. Estimate
(\ref{eq:comp}) is valid since $Q_{\lf}-Q=Q_{0,\lf}-Q_0$, and although the
second inequality in (\ref{eq:sol}) no longer holds, one easily checks
that this is not a problem and that the proofs of Lemmas 1-3 are
still valid. Hence we have
\begin{proposition}
Any heat kernel bound obtained for $H_0$ by means of Proposition~\ref{cor:1}
is also valid for $H$.
\label{prop:pot}
\end{proposition}
{\em Proof : }By hypothesis, we have an estimate on $Q_{0,\lf}$ of the form
\[\re Q_{0,\lf}(f)\geq -k_{\lambda}\|f\|_2^2.\]
Hence
\begin{eqnarray*}
\re Q_{\lf}(f) &=&\re Q_{0,\lf}(f)+\int V\mid f\mid ^2\; dx\\
&\geq&-k\|f\|_2^2
\end{eqnarray*}
and the result follows from Lemma \ref{cor:1}.$\qquad\Box$




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\eject
{\bf Acknowledgments }We would like to acknowledge support under EPSRC
grant number GR/K00967 for this work.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}

Department of Mathematics \newline
King's College \newline
Strand \newline
London WC2R 2LS \newline
England 

e-mail: \newline
G. Barbatis: udah027@bay.cc.kcl.ac.uk \newline
E.B. Davies: udah210@bay.cc.kcl.ac.uk

\end{document}
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