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\font\eightit=cmti8

\font\eightpoint=cmr8

\def\rmi{{\rm i}}

\def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3}

\def\sup{\mathop{\rm sup}}

\def\Dom{D}

\def\ve{\varepsilon}



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



\hbox to \hsize{\hfil\eightit 3/August/1996}

\vglue1truein

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

\centerline{\bf OPTIMAL HEAT KERNEL ESTIMATES FOR}

\centerline{\bf SCHR\"ODINGER OPERATORS WITH MAGNETIC FIELDS}

\centerline{\bf IN TWO DIMENSIONS}

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

\bigskip

\bigskip

\bigskip

{\baselineskip=2.5ex

\vfootnote{}{\eightpoint

\noindent To appear in: Commun. Math. Phys.\medskip}}



{\baselineskip=2.5ex

\vfootnote{}{\eightpoint

\noindent\copyright 1996 by the authors.

Reproduction of this article, in its entirety, by any means is permitted

for non-commercial purposes.}}



{\baselineskip = 12pt

\halign{\qquad\qquad#\hfil\qquad\qquad\qquad\qquad\quad\hfil&#\hfil\cr

{\bf Michael Loss}\footnote {$^*$}{Work supported by N.S.F. grant DMS-95-00840

and the

Erwin Schr\"odinger Institute} &

{\bf Bernd Thaller}\cr

&\cr

School of Mathematics & Institut f\"ur Mathematik\cr

Georgia Institute of Technology & Universit\"at Graz\cr

Atlanta, GA 30332, USA & A-8010 Graz, Austria\cr}}

\bigskip



\vskip .7 true in



\centerline{\bf Abstract}

\smallskip

{\rightskip=4pc \leftskip=4pc \tenrm\baselineskip=11pt\parindent=1pc

Sharp smoothing estimates are proven for

magnetic Schr\"odinger semigroups in two dimensions under the

assumption that the magnetic field is bounded below by some positive

constant $B_0$. As a consequence the $L^{\infty}$ norm of the associated

integral kernel is bounded by the $L^{\infty}$ norm of the Mehler kernel

of the Schr\"odinger semigroup with the constant magnetic

field $B_0$.\par}



\noindent



{\vglue 0.5cm}





\vfill\eject

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

\noindent{\bf 1. INTRODUCTION}

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

\medskip





\noindent 









A recurring problem in the theory of semigroups

is to prove smoothing estimates, i.e., estimates

of the semigroup acting as an operator from $L^p \to L^q$, $q \geq p$.

(see [S82] for a general review on semigroups)

In rare cases it is possible to render these

estimates in their sharp form. Among the examples

are Nelson's hypercontractive estimate [N66] and the

sharp smoothing properties of the heat kernel

(which follow from the sharp form of Young's inequality

[B75], [BL76]). A more recent example is furnished by the

Mehler kernel associated with the Schr\"odinger operator

of a charged particle in a constant magnetic field

which essentially is a result of Lieb [L90] although

not explicitly stated there (see Theorem~1.1).



An especially flexible approach to such problems, due to  

Gross [G76] (see also Federbush [F69]) and relevant for our paper, is the use 

of logarithmic Sobolev inequalities.

In fact there is complete equivalence between Gross' logarithmic Sobolev

inequality and Nelson's hypercontractive estimate. In a similar fashion,

Weissler [W78] proved the equivalence of the sharp smoothing estimates for the

heat semigroup and logarithmic Sobolev inequalities.



On a more abstract level it was realized in Simon and Davies [DS84] 

(see also [D89])

that the technique of logarithmic Sobolev inequalities can be used to

prove  ultracontractivity for Markov semigroups.



It is the aim of this note to implement Gross' method for magnetic

Schr\"odinger operators and to prove sharp smoothing estimates for the

associated semigroup.

The idea is similar to the one in [CL96] where it was realized that

Gross' logarithmic Sobolev inequality when viewed as a family of sharp

inequalities

on $\Bbb R^n$ can be used to obtain sharp smoothing

estimates of solutions of diffusion equations with a volume preserving drift.

An example is the two

dimensional Navier--Stokes equations in the vorticity formulation.



Consider a single charged quantum particle in a magnetic field,

i.e., consider the Hamiltonian

%

$$

H(B)={1 \over 2}(\nabla+\rmi A)^2 \ ,

\eqno(1.1)

$$

%

where $A$ is a vector potential for the magnetic

field $B$, i.e., ${\rm curl}\, A= B$.





Note that the semigroup generated by (1.1) is not

Markovian, in fact not even positivity preserving.

Usually, this difficulty is overcome by the important

diamagnetic inequality ([K72], [S77,79], [HSU77],

see also [AHS77] and [CFKS87])

%

$$

\left|\bigl(e^{tH}u\big)(x)\right| \le \bigl

( e^{t\Delta/2}|u| \bigr)(x)\ ,

\eqno(1.2)

$$

%

which relates estimates on magnetic Schr\"odinger operator semigroup

to estimates on the heat semigroup.





The obvious disadvantage of using (1.2) is that all effects due to the magnetic

field are completely eliminated.

How, then, does the magnetic field affect the behavior of the semigroup?

The first paper that addressed this question is presumably [M86].

It is proved in [M86] that for a magnetic       Schr\"odinger operator in ${\Bbb

R}^3$

$$

\lim_{t \to \infty}{1 \over t}\,{\rm ln}\, \Vert e^{tH(B)} \Vert_{L^1 \to

L^{\infty}}

\leq - \Phi (C) B_0 \ ,

\eqno(1.3)

$$

for all curl-free magnetic fields $B$ that satisfy $0 < B_0 \leq |B| \leq C 

B_0$,

Here $|B|=\sqrt {B_1^2+B_2^2+B_3^2}$ with $B_1,B_2,B_3$ being the

components and the

function $\Phi$ is explicitly given in [M86].

It is noteworthy that the estimates were obtained by probabilistic techniques.



This work was subsequently improved in [E94]. The estimate (1.3) holds in any

dimension and without the condition that $B$ be curl-free.

In particular the following stronger estimate is true

$$

\lim_{t \to \infty}{1 \over t}\,{\rm ln}\, \Vert e^{tH(LB)} \Vert_{L^1 \to

L^{\infty}}

\leq - C_L\min_{x \in\Bbb R^n}|B(x)|L \ .

$$

Here the constant $C_L=1+O(L^{\alpha}\,{\rm ln}\, L)$ as $L \to \infty$ for some

explicitly given $\alpha<0$. Thus, this result is in a certain sense optimal

since by considering a constant magnetic field in two

dimensions one cannot improve the {\it exponent}\/ $\min_{x \in\Bbb

R^n}|B(x)|$. Again the

results in [E94] were obtained by probabilistic techniques.

Similar results can be found in [U94] where techniques from differential

geometry

were used.

\medskip

In this paper we consider the simpler problem for a magnetic field of constant

direction, or what amounts to the same, a magnetic field in two dimensions.

However, we try to retain as much information as possible about the

magnetic field in

the estimate on the semigroup.



A magnetic field in two dimensions is a scalar function $B$ which can be

expressed in terms of a (non-unique) vector potential $A$,

%

$$

B(x) = {\partial A_2(x)\over \partial x_1} - {\partial A_1(x)\over \partial

x_2}\ ,

\qquad A(x) = \bigl(A_1(x),A_2(x)\bigr)\ .

$$

%

For example, if $B(x)=B_0$ is a constant magnetic field then

%

$$

 A(x) = {B_0\over 2}\,\bigl(-x_2,x_1\bigr)\ .

\eqno(1.4)

$$

%

We want to estimate (for suitable $p$ and $q$ with $1\le p\le q \le \infty$)

%

$$

C(t;p,q)\,:=\,\sup_{u\in L^p}\,{\|u(t)\|_q \over \|u\|_p}

\qquad\hbox{where}\quad

u(t) = e^{tH}\,u\ ,

$$

%

which is just the norm of the Schr\"odinger semigroup $\exp(tH)$ as a mapping

from $L^p({\Bbb R}^2)$ to $L^q({\Bbb R}^2)$.

This norm is finite, because (1.2) implies that

%

$$

\| e^{tH} \|_{L^p\to L^q} \le \|e^{t\Delta/2}\|_{L^p\to L^q} < \infty \ .

$$

%





In case of a constant magnetic field $B(x)=B_0>0$ these calculations can be done

using the explicitly known integral kernel of the time evolution operator.

If the vector potential $A$ is chosen as in Eq.~(1.4) above,

the time evolution is given by

%

$$

u(t,x) = \int G(t,x,y)\,u(y)\,d^2\!y \ ,

$$

%

where $G(t,x,y)$ is the Mehler kernel [S79, p.~168]

%

$$

G(t,x,y) = {B_0\over 4}\,{1\over \pi \sinh\Bigl({B_0 t\over 2}\Bigr)}\,

\exp\left\{

-{B_0\over 4}\,\coth\Bigl({B_0 t\over 2}\Bigr)\,(x-y)^2

+\rmi\,{B_0\over 2}\,(x_1y_2 -x_2y_1)\right\} \ ,

$$

%

which is a complex, degenerate, centered Gaussian kernel

(in the terminology of [L90]).

In particular, we need the time evolution of the initial function

%

$$

u_0(x) = N_0\,\exp\left\{-{B_0\over 4a_0}\,x^2\right\},\qquad a_0>0 \ .

\eqno(1.5)

$$

%

The solution at time $t$ is again a Gaussian which can be written as

%

$$

u_0(t,x) = N(t)\,\exp\left\{-{B_0\over 4a(t)}\,x^2\right\} \ ,

$$

%

where

%

\def\SH{\sinh\left({B_0 t\over 2}\right)}

\def\CH{\cosh\left({B_0 t\over 2}\right)}

$$

N(t) = {N_0\,a_0\over a_0\,\CH + \SH} \ ,

\qquad

a(t) = {a_0\,\CH + \SH\over a_0\,\SH + \CH} \ .

$$

%

The function $a(t)$ is the unique solution of the initial value problem

%

$$

\dot a(t) = {B_0\over 2} \bigl(1-a(t)^2\bigr) \ ,\qquad a(0) = a_0 \ .

$$

%

Writing $a_0 = \tanh(\theta)$ (for $a_0<1$),

or $a_0=\coth(\theta)$ (for $a_0>1$), we obtain

%

$$

a(t) = \cases{\tanh\left(\theta + {B_0 t\over 2}\right)& for $a_0<1$,\cr

              1                                        & for $a_0=1$,\cr

              \coth\left(\theta + {B_0 t\over 2}\right)& for $a_0>1$.}

\eqno(1.6)

$$

%

For $a_0 = 1$, the function $u_0(x) = N_0\exp(-B_0x^2/4)$ is

the eigenfunction of $H$ corresponding to the eigenvalue $-B_0/2$.



The following theorem sets the stage for our investigation. It is

essentially a corollary of

a theorem in [L90]. We shall give the details in Sect.~2.

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

\medskip

\noindent

{\bf Theorem 1.1: }

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

{\it Let $B(x) = B_0>0$, all $x\in{\Bbb R}^2$. For $1<p\le q<\infty$,

there is a centered Gaussian function $u_0$ which maximizes the expression

$\|u(t)\|_q/\|u\|_p$ over all $u\in L^p({\Bbb R}^2)$.

The function $u_0$ is of the form (1.5), where $a_0$ depends on $p$, $q$,

$t$, and $B_0$.

More explicitly, with

$$

a_0 = d_{t,p,q} + \sqrt{d_{t,p,q}^2 + p-1} \ ,

\eqno(1.7)

$$

%

%

$$

d_{t,p,q} = {1\over 2 c_ts_t}\left({p\over q}-1 + s_t^2(p-2)\right) \ ,

$$

%

%

$$

c_t = \cosh\left\{ {B_0 t\over 2} \right\} \ , \qquad

s_t = \sinh\left\{ {B_0 t\over 2} \right\} \ .

$$

%

we have

%

$$

\eqalign{

C_0(t;p,q) &\equiv \sup_{u\in L^p}{\|e^{tH(B_0)}u\|_q\over \|u\|_p} =

{\|u_0(t)\|_q\over \|u_0\|_p} \cr

& = {N(t)\over N_0}\,\left({B_0\over 4a_0}{p\over\pi}\right)^{1/p}

\left({B_0\over 4a(t)}{q\over\pi}\right)^{-1/q} \ . \cr}

\eqno(1.8)

$$

%

}



\medskip

{\bf Remark 1:} The condition $p \leq q$ in Theorem 1.1 is necessary as the

following example

shows. Consider the function

%

$$

u(x)=N_0\sum_{a \in G}\exp(-B_0(x-a)^2/4) \ ,

$$

%

where $G$ is a finite square lattice in $\Bbb R^2$ whose lattice spacing is

large compared

to $1/B_0$. Thus the Gaussian functions in the sum have essentially

disjoint support

and hence

%

$$

\Vert u \Vert_p \sim N_0 \left({4 \pi \over B_0 p}\right)^{1/p}|G|^{1/p} \ ,

$$

%

where $|G|$ denotes the number of lattice points.

The function $u$ is an eigenfuction of the magnetic semigroup with

eigenvalue $e^{-B_0t/2}$.

Thus,

%

$$

\Vert u(t) \Vert_q \sim N_0 e^{-B_0t/2} \left({4 \pi \over B_0

q}\right)^{1/q}|G|^{1/q} \ ,

$$

%

and

%

$$

{\Vert u(t) \Vert_q \over \Vert u \Vert_p}\sim e^{-B_0t/2} \left({B_0 \over

4 \pi}\right)^

{1/p-1/q}{p^{1/p} \over q^{1/q}} |G|^{1/q-1/p} \ .

$$

%

The last expression tends to infinity with $|G|$ if $p>q$.





{\bf Remark 2:} Instead of using the results of [L90] one might approach

Theorem 1.1 a la Gross [G76],i.e., trying to reduce the  problem to an integral 

over infinitesimal time steps,

and estimating the semigroup over these time steps by employing

logarithmic Sobolev inequalities. It is clear that one would get some 

smoothing estimates but can one obtain them in the sharp form?

That there are some obstructions to reach this goal by this method 

can be seen as follows.







Eqs.~(1.5) and (1.7) determine the Gaussian function $u_0$

that yields the norm

of the magnetic heat kernel as an operator from $L^p$ to $L^q$. With this

Gaussian we may write

%

$$

C_0(t;p,q) = {\|u_0(t)\|_q\over \|u_0\|_p} =

{\|u_0(s)\|_r\over \|u_0\|_p}\, {\|u_0(t)\|_q\over \|u_0(s)\|_r}\ ,

$$

%

for some $r$ and $s$ with $p\le r\le q$, $0\le s\le t$. Hence it is obvious that

%

$$

C_0(t;p,q) \le C_0(s;p,r)\,C_0(t-s,r,q)\ .

$$

%

Clearly, the proposed method can only work, if there exists a number

$r$ with $p <r<q$ such that the above formula is an equality, i.e.,

%

$$

C_0(t;p,q) = \min_{p\le r\le q} \{C_0(s;p,r)\,C_0(t-s,r,q)\} \ .

\eqno(1.9)

$$

%

However, it is easy to see that Eq.~(1.9) cannot hold in general. For

example, it is wrong in the

specific case $q=p>2$, $0<s<t$.

{}From the explicit form of $C_0(t;p,q)$ given in Theorem~1.1 we find that

%

$$

C_0(t;p,q) = C_0(s;p,r(s)) \, C_0(t-s;r(s),q)

\eqno(1.10)

$$

%

holds for some $r(s)\in[p,q]$ if and only if the optimizer $u_0$ giving

$C_0(t;p,q)$ is also the

optimizer for $C_0(s;p,r(s))$ (i.e., $C_0(s;p,r(s))=\|u(s)\|_{r(s)}/\|u_0\|_p$).



Eq.~(1.10)  can only hold for all $s\in(0,t)$ if the function $r(s)$ is

monotonic nondecreasing, with $r(0)=p$ and $r(t)=q$.

Given $u_0$ (and hence $a_0$) as the optimizer for $C_0(t;p,q)$, we can

easily determine $r(s)$ using

Theorem~1.1 by solving

%

$$

a_0 = d_{s,p,r(s)} + \sqrt{d_{s,p,r(s)}^2+p-1} \ .

$$

%

This gives the function

%

$$

r(s)={ p \over [a_0c_{s}+(1-p)s_{s}][c_{s}/a_0+s_{s}] } \ ,

\eqno(1.11)

$$

%

which is monotonic nondecreasing in the following cases:

%

$$

{q-p \over q(p-2)} \geq (s_t)^2 \qquad {\rm if} \ 2  \leq p \leq q \ ,

$$

%

%

$$

{q-p \over p(2-q)} \geq (s_t)^2 \qquad {\rm if} \ 2  \geq q \geq p \ .

$$

%

If $p \leq 2 \leq q$ the function is always monotonic increasing. Thus, the

formula

(1.9) holds whenever the above inequalities are satisfied and in these cases

Eq.~(1.10) holds for all $s \in [0,t]$ with $r(s)$ given as above.

The identity

%

$$

a(s) = d_{t-s,r(s),q} + \sqrt{d_{t-s,r(s),q}^2+r(s)-1}

$$

%

(with $a(s)$ as in (1.6))

shows that the optimizer $u_0$ for $C_0(s;p,r(s))$ is mapped by the time

evolution onto the optimizer $u_0(s)$

for $C_0(t-s;r(s),q)$, if $r(s)$ is chosen as in Eq.~(1.11).



\bigskip

The approach discussed at the beginning of Remark 2 has the great advantage of 

being quite

flexible and we shall use it to obtain smoothing estimates in the cases

where the magnetic field is not constant.

Certainly, one cannot expect to get such detailed information as in Theorem

1.1 in this case, however, we have the following result which is proved in

Sect.~4.



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

\medskip

\noindent

{\bf Theorem 1.2: }

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



{\it Assume that $B(x) \geq B_0 >0$ is continuous. Then the estimate

%

$$

C(t;p,q) = \|e^{tH(B)}\|_{L^p\to L^q} \le C_0(t;p,q)

$$

%

(with $C_0(t;p,q)$ given by Eq.~(1.8)) holds for



\smallskip\noindent

{\bf a}) all $t>0$, if $p\le 2$, $q\ge 2$,

\smallskip\noindent

{\bf b})        $t>0$ and $2<p<q\le\infty$, if

%

$$

{q-p\over q(p-2)} \ge \left(\sinh\left\{ {B_0 t\over 2} \right\}\right)^2  ,

\eqno(1.12)

$$

%

\smallskip\noindent

{\bf c})        $t>0$ and $1\le p<q<2$, if

%

$$

{q-p\over p(2-q)} \ge \left(\sinh\left\{ {B_0 t\over 2} \right\}\right)^2  .

\eqno(1.13)

$$

%

}

\medskip



{\bf Remark 1:} In the case where neither a), b) nor c) holds but $p \leq q$

we get a

nontrivial bound but it is not sharp.





{\bf Remark 2:} A simple consequence of Theorem 1.2 is the pointwise bound

on the

heat kernel

%

$$

\Vert e^{tH(B)} \Vert_{L^1 \to L^\infty} \leq {B_0 \over 4 \pi \sinh({B_0 t

\over 2})} \ ,

$$

%

which is best possible. In fact there is equality if $B(x) \equiv B_0$ as can be

seen from the Mehler kernel.



\medskip





Finally, also in Sect.~4, we extend our result on the ``diagonal" of the

magnetic

heat kernel to the ``off-diagonal" using a technique due to Davies [D89].

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

\medskip

\noindent

{\bf Theorem 1.3: }

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

 {\it Assume that $B(x) \geq B_0 >0$ is continuous. Then the magnetic heat

kernel satisfies

the bound

%

$$

|e^{tH}(x,y)| \leq {B_0 \over 4 \pi \sinh({B_0 t \over 2})} e^{-{(x-y)^2

\over 2t}}  \ .

\eqno(1.14)

$$

%

}

\medskip

{\bf Remark:} The Gaussian decay on the right side of (1.14) is the one of

the heat

kernel, which is considerably weaker than the decay of the Mehler kernel.

Is it true that $|e^{tH}(x,y)|$ is bounded by the Mehler kernel $|G(t,x,y)|$?

The truth of this estimate would reveal a robust dependence of the magnetic

heat kernel on the magnetic field. This is an open problem.



\medskip

{\bf Acknowledgment:} We would like to thank Eric Carlen, Laszlo Erd\"os, and Vitali Vugalter

for many helpful discussions.





\bigskip\bigskip\bigskip

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

\noindent{\bf 2. PROOF OF THEOREM 1.1}

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

\medskip

\noindent

Let $G_t$ be the integral operator with the Mehler kernel $G(t,x,y)$.

We want to determine

%

$$

\sup_u {\| G_t u\|_q  \over \|u\|_p} = C_0(t;p,q),

$$

%

which is either finite or infinite.

By a result of E.~Lieb (see [L90], Theorem~4.1) it is sufficient to take

the supremum

over centered Gaussian functions of the form

%

$

\exp( - x\cdot J_\ve x )

$,

%

where $J_\ve$ is a complex-valued matrix, symmetric with

respect to the scalar product in ${\Bbb R}^2$, and with a strictly positive

real part.

Even for this case the calculation is still somewhat tedious.

Instead it is convenient to rework the proof of Theorem~4.1 in [L90].



Let $t>0$ be arbitrary.

We approximate $G_t$ by the operator $G_t^\ve$ with kernel

%

$$

G_t^\ve (x,y) = {\rm e}^{-\ve x^2}\,G(t,x,y),

$$

%

which for $\ve>0$ is a non-degenerate, centered Gaussian kernel.

According to [L90], Theorem~3.4, there is a unique (up to a multiplicative

constant)

centered Gaussian function $u_\ve$ which is the maximum of

$\| G_t^\ve u\|_q/\|u\|_p$ over all $u\in L^p({\Bbb R}^2)$.

The function $u_\ve$ is of the form

%

$

u_\ve(x) = \exp( - x\cdot J_\ve x )

$,

%

where $J_\ve$ is a (possibly complex-valued) matrix which is symmetric with

respect to the scalar product in ${\Bbb R}^2$ and has a strictly positive

real part.

But since the integral operator $G_t^\ve$ commutes with

rotations,

the unique maximum $u_\epsilon$ must also be rotationally invariant.

Hence $J_\ve= \alpha_\ve {\bf 1} + {\rm i}\beta_\ve {\bf 1}$,

where $\alpha_\ve>0$, and $\beta_\ve$ is real.

Since the integrals of Gaussian functions can be evaluated explicitly,

we can evaluate the quotient $\|G_t^\ve u\|_q/\|u\|_p$ for

$u(x)=\exp[-(\alpha + \rmi \beta)\,x^2]$

and maximize this expression over all $\alpha$ and $\beta$.

The maximum is obtained for $(\alpha,\beta)=(\alpha_\ve,\beta_\ve)$,

with $\alpha_\ve>0$ and $\beta_\ve = 0$.



Since $\exp(-\ve x^2)\le 1$ we find for any Gaussian function $u$

%

$$

\|G_t^\ve u\|_q \le \|G_tu\|_q

$$

%

and hence

%

$$

C^\ve={\|G_t^\ve u_\ve\|_q\over \|u_\ve\|_p} \le C_0(t;p,q),

$$

%

where $u_\ve = \exp(-\alpha_\ve x^2)$ is the unique Gaussian maximizer for

$G_t^\ve$.



By an explicit calculation one sees that

%

$$

\lim_{\ve \to 0} C^\ve \equiv C^0 = {\| G_t u_0 \|_q \over \|u_0\|_p},

$$

%

with $u_0(x) = \exp(-\alpha x^2)$, $\alpha = \lim_{\ve\to 0} \alpha_\ve$.

Of course, we have $C^0\le C_0(t;p,q)$.



Finally, for any Gaussian function $u$, the limit

%

$$

\lim_{\ve \to 0 } {\|G_t^\ve u\|_q\over \|u\|_p} = {\| G_tu\|_q\over \|u\|_p}

$$

%

exists by Lebesgue's dominated convergence, and from

%

$$

{\|G_t^\ve u\|_q\over \|u\|_p} \le C^\ve \quad \hbox{for all $\ve > 0$}

$$

%

we find immediately that

%

$$

{\| G_t u\|_q\over \|u\|_p} \le \lim_{\ve \to 0} C^\ve = C^0

\quad\hbox{for all Gaussian functions $u$}.

$$

%

Hence, also $\sup \|G_t u\|_q/\|u\|_p = C_0(t;p,q) \le C^0$.

This proves $C^0 = C_0(t;p,q)$.

A little calculation easily gives the explicit value of this constant.





\vfill\eject

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

\noindent{\bf 3. A DIFFERENTIAL INEQUALITY}

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

\medskip



\noindent In the following we assume that the magnetic field $B$ is given by a

differentiable

vector field $A$ as in Eq.~(1.1) and satisfies

$B(x) \ge B_0 > 0$, for all $x \in {\Bbb R}^2$. In particular we choose the

vector

potential to be in $L^4_{\rm loc}(\Bbb R^2)$ and then by the

Leinfelder--Simader Theorem

[LS81] the formal expression  (1.1) defines a selfadjoint operator on some

domain

$\Dom(H)$ with

$C^{\infty}_0(\Bbb R^2)$ as a core.



If we set $u(s)=e^{Hs}u_0$

then $u(s)$ is a solution of

%

$$

{d\over ds}\,u = H\,u={1 \over 2}(\nabla + \rmi A)^2\,u \ .

\eqno(3.1)

$$

%

If $u_0 \in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$ then we

have that

$u(s)\in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$.

This follows from the diamagnetic inequality (1.2)

for each $s\in (0,t]$ and the explicit form of the heat kernel.



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

\medskip

\noindent

{\bf Theorem 3.1: }

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

\noindent

{\it Let $r:(0,t]\to {\Bbb R}$ be a twice differentiable function with

$r(s) \geq 2$ and $\dot r(s) \ge 0$ for all $s\in (0,t]$.

Then

%

$$

{d\over ds}\,\ln \|u(s)\|_{r(s)} \le - L(r(s),\dot r(s)) \ ,

\eqno(3.2)

$$

%

for all $s\in (0,t]$, where

%

$$

L(r, \dot r) =  {\dot r\over r^2}\,\left\{

2+\ln{4\pi a(r,\dot r)\over B_0 r}

\right\}  + {a(r,\dot r) B_0\over r} \ .

\eqno(3.3)

$$

%

and

%

$$

a(r,\dot r) = {1\over B_0 r}\bigl( \sqrt{\dot r^2 + B_0^2 r^2 (r-1)}

- \dot r \bigr).

$$

%

Equality holds in Eq~(3.2) for $B(x) = B_0$ and

%

$$

u(s,x) = \exp\left( - {B_0\over 4 a\bigl(r(s),\dot r(s)\bigr)} \,x^2 \right) \ .

$$

%

}

\medskip\noindent

{\it Proof\/}: Pick $u_0 \in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$.

Since ${d\over d\tau}\,\ln k\|u(\tau)\|_{r(\tau)}$ does not depend on $k>0$, we

assume during the following calculation, without loss of generality, that

at the time

$\tau=s>0$ the solution is normalized such that

$\|u(s)\|_{r(s)} = 1$. For the derivative at $\tau=s$ we obtain therefore

%

$$

\eqalign{{d\over ds}\,\ln  \|u(s)\|_{r(s)}

 = & \,{\dot r(s)\over  r(s)^2}\int | u(s,x) |^{r(s)}

\ln  | u(s,x) |^{r(s)}\,d^2\!x \cr

& + {1 \over 2}\int |u(s,x)|^{(r(s)-2)}\, {d\over ds}|u(s,x)|^2 \,d^2\!x \ .\cr}

\eqno(3.4)

$$

%

The integrals have to be taken over ${\Bbb R}^2$.

The formal computation can be easily justified by an approximation argument.

For simplicity, the arguments $s$ and $x$ in the integrand on the right side

will be omitted from now on.



Using (3.1) we obtain, after a partial integration

%

$$

\eqalign{{1\over 2}\int |u|^{(r-2)}\,{d\over ds} |u|^2

          = & -{1 \over 2}(r-2)\int|u|^{(r-2)}\,(\nabla |u|)^2\cr

           & -{1\over 2}\int |u|^{(r-2)}\,|(\nabla + \rmi A)\,u|^2 \ .}

\eqno(3.5)

$$

%

The integration by parts can be justified as follows. Since $u \in

\Dom(H)$, by the Leinfelder--Simader Theorem [LS81] we can pick a sequence

$u_n \in C^{\infty}_0

({\Bbb R}^2)$ such that $u_n \to u, Hu_n \to Hu$ in $L^2$. Inspecting

the proof of the Leinfelder--Simader Theorem one sees that the sequence

$u_n$ can be chosen to converge to $u$ in $L^1$ and to have a uniform bound

on the  $L^{\infty}$ norm.

Thus $u_n$ converges to $u$ in $L^p$ for all $1\leq p <\infty$.

In particular, all the following computations can be justified in the same

fashion and we can assume without restriction that

$u \in C^{\infty}_0({\Bbb R}^2)$.





If we set $u=f+\rmi g$ then $|u|= \sqrt {f^2+g^2}$.

We find

%

$$

|(\nabla + \rmi A)\,u|^2= (\nabla |u|)^2 + |A+\nabla S|^2\,|u|^2 = X^2+Y^2

\eqno(3.6)

$$

%

where we have introduced two

real vector fields $X$ and $Y$ over ${\Bbb R}^2$,

%

$$

X = \nabla^\bot |u| =

\Bigl({\partial\over \partial x_2},-{\partial\over \partial x_1}\Bigr)|u|,

$$

%

$$

Y = (A+\nabla S)\,|u|.

$$

%

Here the symbol $\nabla S$ denotes the expression

%

$$

{f \nabla g - g \nabla f \over f^2+g^2} ,

$$

%

which is defined wherever $f^2+g^2 > 0$.



For any $c>0$ we may estimate 

%

$$

c^2X^2 + Y^2 \ge 2c \, X \cdot Y

\eqno(3.7)

$$

%

with equality if and only if $cX=Y$.

If $B=B_0$ is a constant magnetic field we can choose a Gaussian

function

%

$$

u=N\,\exp\left\{-{B_0\over 4c}\,x^2\right\}

\eqno(3.8)

$$

%

such that equality holds everywhere in (3.7), because

(with $A$ as in Eq.~(1.2))

%

$$

cX = c\,\nabla^\bot u = {B_0\over 2}\,(-x_2,x_1)\,u = Au = Y \ .

$$

%

Let us now insert (3.6) and (3.7) into (3.5).

Since $r-1>0$ we can add and subtract a positive constant $c^2$ with

$0<c<\sqrt{r-1}$ in order to obtain

%

$$

\eqalign{{1\over 2}\int &|u|^{(r-2)}\,{d\over ds} |u|^2

          = -{1\over 2} (r-1-c^2) \int |u|^{(r-2)}\,(\nabla |u|)^2   \cr

           &-{1\over 2}\int |u|^{(r-2)}\,c^2(\nabla |u|)^2

             -{1\over 2} \int |u|^{(r-2)}\,(A+\nabla S)^2\,|u|^2                \cr

       \le &-{1\over 2} (r-1-c^2) \int |u|^{(r-2)}\,(\nabla |u|)^2

             -c\int |u|^{(r-2)}\,(\nabla^\bot |u|)\cdot (A+\nabla S)\,|u|}

$$

%

The last term can be rewritten as

%

$$

\eqalign{

-{c \over r}\int &(\nabla^\bot |u|^r)\cdot (A+\nabla S)    \cr

&=-{c \over r}\int \{\nabla^\bot \cdot (|u|^r (A+\nabla S))+B|u|^r\}  \cr

&=-{c \over r}\int B|u|^r  .  }

$$

%

The only thing to observe is that $\nabla^\bot \cdot \nabla S =0$ wherever

$|u| > 0$.



Using $B(x)\ge B_0$ and $\int |u|^{r}=1$ these estimates lead to the following

family of inequalities for each $c\in (0,\sqrt{r-1})$

%

$$

\eqalign{{d\over ds}\,\ln \|u\|_{r}

 &\le {\dot r\over  r^2} \int

 |u|^{r} \ln   |u|^{r} \cr

&- {1\over 2} (r-1-c^2)\int |u|^{(r-2)} (\nabla|u|)^2 -{c B_0\over r} \ .\cr}

\eqno(3.9)

$$

%

Equality holds in case of a constant magnetic field $B=B_0$ and $u$ a Gaussian

function given by Eq.~(3.8).

We also note that in the special case $\dot r(s) = 0$ the above calculation can

be simplified to give

%

$$

{d\over ds}\,\ln \|u\|_{r} \le -\sqrt{r-1}\,{B_0\over r} \ .

\eqno(3.10)

$$



Now we apply the following family of logarithmic Sobolev

inequalities

%

$$

\int g^2\ln g^2 - {\lambda\over \pi}\int(\nabla g)^2 \le -2-\ln\lambda \ ,\qquad

\hbox{all $\lambda > 0$},

\eqno(3.11)

$$

%

where we assumed that $\int g^2 = 1$. Equality holds if $g$ is a normalized

Gaussian,

%

$$

g(x) = {1\over\sqrt{\lambda}}\,\exp\left\{ -{\pi\over 2\lambda}\,x^2

\right\} \ .

$$

%

This family is essentially Gross' logarithmic Sobolev inequality rewritten

as an inequality on ${\Bbb R}^2$ (see, e.g., [CL96]).

We insert (3.11) into (3.9) with

%

$$

\lambda = {2\pi\over \dot r} (r-1-c^2),\qquad g = |u|^{r/2} \ .

\eqno(3.12)

$$

%

Here we have to assume that $\dot r>0$.

With this choice of $\lambda$ we obtain

%

$$

{d\over ds}\,\ln \|u\|_{r} \le

-{\dot r\over  r^2} \left\{

2+\ln{2\pi\over\dot r}(r-1-c^2)

\right\} -{cB_0\over r} \ .

\eqno(3.13)

$$

%

This result holds for all $0<c<\sqrt{r-1}$.

The expression on the right side has a minimum for $c=a$ where



%

$$

a = {1\over B_0 r}\bigl( \sqrt{\dot r^2 + B_0^2 r^2 (r-1)}

- \dot r \bigr) = a(r,\dot r) \ .

\eqno(3.14)

$$

%

Note that for $\dot r = 0$ this formula is consistent with Eq.~(3.10).

In order to determine the case of equality in (3.13) we determine $\lambda$

according to (3.12) for the optimal value $a$ of $c$.

This gives $\lambda = 4\pi a/B_0 r$ so that the Gaussian function which

optimizes

the logarithmic Sobolev inequality is given by

%

$$

g = u^{r/2} = N \exp\left\{ - {B_0 r\over 8a}\,x^2\right\}.

$$

%

As shown above, the corresponding function $u$ at the same time optimizes

Eq.~(3.7) in case of a constant magnetic field.

Hence we find that equality holds in the estimate (3.13) (with $c=a$)

provided that

$B=B_0$ is constant and $u$ is the Gaussian function defined above.



For the optimal choice of $c$ the inequality (3.13) becomes

%

$$

{d\over ds}\,\ln \|u(s)\|_{r(s)} \le - L(r(s),\dot r(s)) \ ,

$$

%

where (with $a=a(r,\dot r)$ as defined in (3.14))

%

$$

L(r, \dot r) =  {\dot r\over r^2}\,\left\{

2+\ln{4\pi a\over B_0 r}

\right\}  + {a B_0\over r} \ .

$$

%

This estimate remains true for $\dot r(s) = 0$ and $a=\sqrt{r-1}$

with $L(r,0) = aB_0/r$, cf. Eq~(3.10).







\medskip\noindent{\bf Remark:}

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

We note that if for some $s\in [0,t]$, the derivative $\dot r$ is negative,

then $d/ds\,\,\ln \|u\|_{r}$ is unbounded.

This can be seen by choosing a constant magnetic field $A(x,y)=(-y,x)$

and

%

$$

u(x,y) = {\rm e}^{\rmi xy}\,g(x)\,h(y) \ ,

$$

%

where $g$ and $h$ are positive with $\int g^{r} = \int h^{r} = 1$.

The integrals in (3.4) and (3.5) can be evaluated explicitly for

Gaussian functions. Setting

%

$$

g(x) = \left({1\over \pi}\right)^{1/2r}\,\exp\left(- {x^2\over r}\right)\ ,

\qquad

h(y) = \left({\alpha\over \pi}\right)^{1/2r}\,\exp\left(-\alpha {y^2\over

r}\right) \ ,

$$

%

we find

%

$$

{d\over ds}\,\ln \|u\|_{r} = {\dot r\over 2r^2}\,(\ln \alpha -2 \ln \pi - 2)

-{r-1 \over r^2}(1+\alpha)-1 \ ,

$$

%

which for $\dot r<0$ tends to $+\infty$, as $\alpha\to 0$.







\vfill\eject

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

\noindent{\bf 4. PROOFS OF THEOREM 1.2 AND 1.3}

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

\medskip





\noindent

{\it Proof of Theorem 1.2:} By duality, the semigroup $e^{Ht}:L^p \to L^q$ has

the same norm as the one from $L^{q'} \to L^{p'}$ where $p',q'$ are the dual

indices. Thus c) follows from b).



To prove b) for $q<\infty$ we integrate the inequality (3.2)

and obtain

%

$$

{\Vert u(t) \Vert_q \over \Vert u(0) \Vert_p}

\leq \exp{\Bigl\{-\int_0^t L(r(s),\dot r(s))\,ds \Bigr\}}\ .

\eqno(4.1)

$$

%

Recall that the only assumption on $r(s)$ is that $r(0)=p$, $r(t)=q$ and that

$\dot r(s)$ be nonnegative. Thus, we can optimize the right side of (4.1)

over $r(s)$ which leads to an interesting problem in the calculus of variations.

The associated differential equation is



%

$$

\ddot r = 2\,{\dot r^2\over r} + {1 \over 2}B_0^2 r (r-2).

\eqno(4.2)

$$

%

It is easily verified that $r(s)$ defined by Eq~(1.11) is a solution of (4.2)

and satisfies $r(0)=p$.

Moreover, if we choose $a_0$ as in (1.7) we see that $r(t)=q$.

Using the function $r(s)$ as a trial function in (4.1) we obtain b)

after an exceedingly tedious but straightforward computation.



Next we consider b) with $q = \infty$. We remark that the definition (1.11) of

the trial function $r(s)$ makes still sense in this case, the condition for

monotonicity follows from (1.12).

We have $\lim_{s\to t} r(s) = \infty$ and $2<r(s)<\infty$ for $s<t$.

Choose $T$ with $0<T<t$ and write $u(T) = e^{T H}u_0$. Then

%

$$

{\| e^{tH}u_0\|_\infty \over \|u_0\|_p} =

{\| e^{T H} u_0 \|_{r(T)} \over \|u_0\|_p}\,

{\| e^{(t-T)H}u(T) \|_\infty \over \|u(T)\|_{r(T)} }\ .

$$

%

Here the second factor is bounded by some constant $K(T)$ which tends

to $1$, as $T \to t$, because the semigroup $e^{s H}$ is

bounded as an operator from $L^r$ to $L^\infty$ for all $r\ge 1$.

The first factor can be estimated as above by integrating $(3.2)$, so that for

all $0<T<t$,

%

$$

{\| e^{tH}u_0\|_\infty \over \|u_0\|_p} \le K(T)\,

\exp\left\{

- \int_0^T L\bigl(r(s),\dot r(s)\bigr)\, ds

\right\} \ .

$$

%

For the trial function $r(s)$ with $q=\infty$ one can see that $\dot

r(s)/r(s)^2$ as well as

$a(r(s),\dot r(s))$ remain bounded for $s \to t$.

Therefore the function

$L(r(s),\dot r(s))$ given by (3.3) has only a logarithmic and hence integrable

singularity, as $s\to t$. Hence the estimate above makes sense for $T\to t$

and we find that (4.1) remains valid even for $q=\infty$.





To prove a) we again consider the function $r(s)$ defined by Eq~(1.11). Again

$r(0)=p \leq 2$ and if we choose $a_0$ as in (1.7), $r(t)=q \geq 2$. Denote

by $\theta$ the time where $r(\theta)=2$. Such a time exists since $r(s)$ is

strictly increasing. Now

%

$$

{\Vert u(t) \Vert_q \over \Vert u(0) \Vert_p}

={\Vert u(t) \Vert_q \over \Vert u(\theta) \Vert_2}{\Vert u(\theta) \Vert_2

\over \Vert u(0) \Vert_p}\ ,

$$

%

and the second factor on the right can be estimated according to part b)

by the constant $C(\theta;p,2)$

whose value is given by the right side of (1.8) with $q=2$ and $t= \theta$.



The

first factor is estimated by writing

%

$$

{\Vert u(t) \Vert_q \over \Vert u(\theta) \Vert_2}=

{\Vert e^{H(t-\theta)}u(\theta) \Vert_q \over \Vert u(\theta) \Vert_2}

\leq \sup{{\Vert u(t-\theta) \Vert_q \over \Vert u(0) \Vert_2}}

$$

%

which is bounded by $C(t-\theta;2,q)$.

Combining these estimates  and using formula (1.10) yields the result.



\bigskip

\noindent

{\it Proof of Theorem 1.3:}

Following [D89] we consider the semigroup given by the kernel

%

$$

Q_t(x,y)=e^{-\alpha x}e^{tH(B)}(x,y)e^{\alpha y} \ ,

$$

%

which is generated by the operator

%

$$

{1 \over 2}(\nabla+\alpha+\rmi A)^2 \ .

$$

%

Retracing the steps of the proof of Theorem 3.1 for $u(t)(\cdot)=\int

Q_t(\cdot,y)u_0(y)

{\rm d}y$ yields the estimate

%

$$

{d\over ds}\,\ln \|u(s)\|_{r(s)} \le - L(r(s),\dot r(s)) + \alpha^2 \ ,

$$

%

with  $L(r(s),\dot r(s))$ given by (3.3). From the proof of Theorem 1.2

it now follows that

%

$$

|Q_t(x,y)| \leq {B_0 \over 4 \pi \sinh({B_0 t \over 2})}\, e^{\alpha^2 t} \ ,

$$

%

or

%

$$

|e^{H(B)t}(x,y)| \leq {B_0 \over 4 \pi

\sinh({B_0 t \over2})} \,e^{\alpha (x-y) + \alpha^2 t}  .

$$

%

Optimizing with respect to $\alpha$ yields the result.





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



\bigskip



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

\noindent{\bf References}

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

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Ultracontractivity and the heat kernel for Schr\"odinger operators and Dirichlet

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\end



