Applied Mathematics, 2010, 1, 425-430
doi:10.4236/am.2010.15056 Published Online November 2010 (http://www.SciRP.org/journal/am)
Note on Gradient Estimate of Heat Kernel for
Schrödinger Operators
Shijun Zheng
Department of Mathematical Sciences, Georgia Southern University, Statesboro, USA
Received August 27, 2010; revised September 24, 2010; accepted September 27, 2010
Abstract
Let =
H
V  be a Schrödinger operator on n
. We show that gradient estimates for the heat kernel of
H
with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral
operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory
for
p
L and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent
proofs. We give a counterexample in one dimension to show that there exists V in the Schwartz class such
that the long time gradient heat kernel estimate fails.
Keywords: Heat Kernel, Schrödinger Operator, Functional Calculus
1. Introduction
Consider a Schrödinger operator =
H
V  on n ,
where V is a real-valued potential in )(
1n
loc
L. It is
noted in [1,2] that for positive V, if
H
admits the
following gradient estimates for its heat kernel
),(=),(yxeyxp tH
t
: for all n
yx , and 0>t,
,|),(| /
2
||/2 tyxcn
nt etcyxp 
 (1)
,|),(| /
2
||1)/2( tyxcn
ntxetcyxp 
 (2)
then the kernel of )(H
j
 and its derivatives satisfy a
polynomial decay as in (4), where j
 is a function in
certain Sobolev space with support in ],22[ jj
. As is
well-known, the decay estimate in (4) implies the
Littlewood-Paley inequality for )( np
L [3-6].
For positive V, based on heat kernel estimates one
can show (4) by a scaling argument [2]. In this paper we
will prove the general case, namely Theorem 1, by
modifying the proofs in [7,8] and [2].
However, in general the gradient estimates (1), (2) do
not hold for all
t
. This situation may occur when
H
is
a Schrödinger operator with negative potential, or the
sub-Laplacian on a Lie group of polynomial growth, cf.
[9-12]. A second part of this paper is to show such a
counterexample, based on Theorem 1.
Recall that for a Borel measurable function
 :

, one can define the spectral operator )(H

by functional calculus ()= ()
H
dE




, where

dE
is the spectral measure of
H
. The kernel of )(H

is
denoted ),)(( yxH

in the following sense. Let
A
be
an operator on a measure space ),(

dM ,

d being a
Borel measure on
M
. If there exists a locally integrable
function A
K: 

MM such that
,=()
=(, ) ()()()()
M
A
MM
Af gAf gd
K
xy fygxdxdy






for all gf , in )(
0MC with fsupp and
g
supp
being disjoint, where )(
0MC is the set of continuous
functions on
M
with compact supports, then
A
is
said to have the kernel ),(:=),( yxKyxA A. Throughout
this paper, c or C will denote an absolute positive
constant.
The main result is the following theorem for
)(
1n
loc
LV .
Theorem 1 Suppose that the kernel of tH e satisfies
the upper Gaussian bound for 0,1=

0.>,|),(| /
2
||)/2(tetcyxe tyxcn
n
tH
x 

(3)
Let

be supported in 1,1][ and belong to
)(
2
1

 
N n
H for some fixed 0N and 0>

. Then
for each 0N, there exists a constant N
c independent
of

such that for all 

j
1
( )/2/2
2
|()(,)|
2(12||) ,
xj
n
jnjN N
NH
Hxy
cxy








(4)

S. J. ZHENG 426
where )(2=)( xx j
j

 and )(:=
ss HH denotes
the usual Sobolev space with norm
22/2
2
=(1/ )
s
s
H
L
f
ddx f.
When V is positive, a result of the above type was
proved and applied to the cases for the Hermite and
Laguerre operators [1]. The observation was that if
0V, then the constants corresponding to Lemma 2 do
not change for


VH = with )(=)( 2xVxV 

(called scaling-invariance in what follows), according
to the Feynman-Kac path integral formula [13]
(())
0
()=(()) ,
t
Vsds
tH
x
efx Efte








here x
E is the integral over the path space

with
respect to the Wiener measure x

, n x and )(t

stands for a brownian motion (generic path).
For general V the technical difficulty is that we do
not have such a scaling-invariance. We are able to
overcome this difficulty by establishing Lemma 6, a
scaling version of the weighted 1
L inequality for
),)(( yxH
j
 with s
H
 , for which we directly use
the scaling information indicated by the time variable
appearing in Lemma 2. Thus this leads to the proof of the
main theorem by combining methods of Hebisch and the
author’s in [7,2].
In Section 3 we give a counterexample to show that
for 2
)cosh1)((=)(   xxV 

, 

, the estimates
in (1) and (2) fail for 
t
.
Note that under the condition in Theorem 1, (4) is
valid for all )(
0

C
j
 , j satisfying (i)
}2|:|{supp j
jxx 
 and (ii) ,2|)(| )( kj
k
k
jcx 
 

0=0k
. A corollary is that (3) implies the
Littlewood-Paley inequality
21/2
()
()
(|()()|),1<<
pnj
Lpn
jL
fHf p   



(5)
for both homogeneous and inhomogeneous systems

j

, according to [1, Theorem 1.5], see also [2,3].
2. Heat Kernel Having Upper Gaussian
Bound Implies Rapid Decay for Spectral
Kernels
In this section we prove Theorem 1. Following [7] we
begin with a simple lemma.
Lemma 2 Suppose that (1) holds. Then
2/2
|(,)| =:()
tH n
exydxct Ct



).,(=:|),(| 2
||tsCcedxeyxe tcsyxstH 


The next lemma can be easily proved by a duality
argument and we omit the details.
Lemma 3 Let L be a selfadjoint operator on
)(
2n
L and )(, 

L  . Then for each y,
22
22
()()(,)() ()(,).
L
L
LyLLy  

 
If in addition )( L

is unitary, then the equality
holds.
Let w be a submultiplicative weight on nn

,
i.e., ),(),(),(0 yzwzxwyxw


, n
zyx  ,, . For
simplicity we also assume ),(=),(xywyxw. Define the
norm for )( 21n
loc
Lk  as follows:
(, )=|(, )|(,).
sup n
wn
y
kxykxy wxydx
 
Then given two operators 21,LL , it holds that
121 2
()(,)(,)(,).
www
LLxyL xyLxy (6)
The following lemma is a scaling version of [8,
Theorem 3.1] for )(
1n
loc
LV .
Lemma 4 Suppose that (1) holds. Let H
j
jeL 
2
=.
Then for each 0a, there exists a constant ),(= ancc
depending on an, only such that for all 

j and


k,
/2
/2
|()(,)|(12||)
( ,)(1||).
ikL ja
j
j
na
eLxyxydx
cnak


 (7)
Proof. (a) First we show the case 0=j. For nota-
tional convenience write 0 =LL , then by Lemma 2 we
have, with 1=t,
2
(, )(1)= sup Lx
y
LxyCc
.=)(=:,1)(|),(| sup 2
||csyxs
y
cesCsCdxeyxL 


Let ayx yxeyx |)|(1=),( ||
 
 , s<<0

. Then
dxyxeeyxLyxL ayxsyxs |)|(1|),(|=),( ||)(|| 





.)(=:|)|(1 sup )( ,,
||)(
,as
ayxs
yx
csCyxesC


 
In view of Lemma 3, setting 1 =||(,)kLxy



,
we have
  |>|||
=|)|(1|),)((| yxyx
aikLdxyxyxLe
|| (,)
/2 2
(1)(, )(, )kLxy
na
Lx
LyLxye e




  
12
,, ,,
(1)(1||())(),
na
s
asa
CkCscCsc




 

where we note that by (6)

S. J. ZHENG 427
=0
()(,)
(, )!
n
ikL
n
ikLx y
exy n




||(, )
=0
|| (,)=.
!
nkLxy
n
n
kLxy e
n




It is easy to calculate that

,,
10
=>.








 
 


a
sa as
as
ca
eas
s
Hence taking /2= s

and fixing 0>= 0 ss give
that
.|)|)(1,,(|)|(1|),)((|2
0
a
n
aikLkanscdxyxyxLe 


(b) Similarly we show the case for all 

j. If
H
j
eL
2
=, then Lemma 2 tells that with j t 2=
/222 2|),(|jnH
jcdxyxe  


.|),(|2
2||2 s
j
cyxsH
jcedxeyxe 




For j let ajyx
j
jyxeyx |)|2(1=),( /2||
/2
2  
 ,
s<<0

. Then similar to (a) we obtain
/2
2| |/2
(, )=|(, )|(12||)
jxy ja
jj
j
LxyLxy exydx




.)(=|)|(1 sup )( ,,
||)(
as
axs
x
csCxesC


 
It follows that, with /2= s

and 0>= 0 ss fixed,
   |>|||
/2 =|)|2(1|),)((| yxyx aj
j
j
ikL dxyxyxLe
/2 /2
2
|| (,)
/2
2
(12)(,)
(,)
nja
nj
Lx
kL xy
j
j
j
jj
cLy
Lxy ee




 


,|)|)(1,( /2 an kanc  
where we set /2 1 =2||( ,)
j
j
j
kLxy



.
We also need a basic property on the weighted 2

norm of Fourier coefficients of a compactly supported
function in Sobolev space, which can be proved by
elementary Fourier expansions.
Lemma 5 Let 0s, 0>T and
])([0,=])([0,00 TCTH s denote the subspace of Sobolev
space )( 
s
H. Then we have for all ])([0,
0THg s
,
20
ˆ() ,
s
H
s
Tgn cg
 (8)
where 1/222
2)/|)(|(=}{ s
n
s
nTnn 




 and )(
ˆng
are the Fourier coefficents of g over the interval ][0,T.
The inequality in (8) can be replaced by equality
(however we will not use this improvement), which is a
special case of the general norm characterization for
periodic functions in ])([0,TH s, see e.g. [14].
It follows from Lemma 4 and Lemma 5 the following
weighted 1
L estimates for ),)(( yxH
j
, which is an
improved version of [2, Lemma 3.1], where the
restriction 0V is removed.
Lemma 6 Suppose )(
1n
loc
LV and the kernel of
tH
e satisfies for all 0>t
.|),(| /
2
||/2 tyxcn
n
tH etcyxe   (9)
If Nns


1)/2(> , 0N and 10,10][supp

,
then
/2
1()
()
,
(2)( ,)2(), sup jjN
s
n nH
L
n
jy
Hy yc






here ||1:=xx



.
Proof. Let 1,1][



. If ][0,2:=supp

Ig , then
g
has the Fourier series expansion on
I
,)(
ˆ
=)( ikx
k
ekgxg

where dxexgkg ikx
)(
2
1
=)(
ˆ2
0


. Let



eeg )(=)( and


ik
kef =)( . Then
yyyg )/log(=)(


with ],[supp 1eeg  , and so
).()(
ˆ
=)(
ˆ
=)(2 22
2H
j
k
k
H
j
H
j
ike
k
jefkgeekgH  






(10)
It follows from Lemma 4, (10) and Lemma 5 that for
each y
Nn
k
Nj
jkkgcdxyxyxH
 
/2/2 |)|(1|)(
ˆ
|)(2|),)((|
1)/2()/2(1/2 |)|(1|)|(1|)(
ˆ
|=  


kkkgc Nn
k
1/21
1/2
122)|)|(1(|)|(1|)(
ˆ
|  




 

kkkgc
k
Nn
k
1/2
/2(1 )/2([0,2])
0
nN
H
cg





1/2
([ 1,1])
0,
s
H
c 



where 1)/2(=



nNs

and the last inequality
follows from a change of variable and interpolation.
Remark 7 Let 

VVV =, 0

V on n , 3n.
Then the heat kernel estimate in (9) holds if 
V is in
Kato class and
K
V, the global Kato norm of  V, is
less than 1)
2
(/:= /2 n
n
n
 , see [15]. Also (9) holds

S. J. ZHENG 428
whenever 0V is locally integrable on n
, 1n.
2.1. Proof of Theorem 1
With (3) and Lemma 6 we are in a position to prove (4).
The proof is similar to that of Proposition 3.3 in [2] in
the case of positive V. For completeness, we present
the details here.
.),))(()(,(=),)(( dzyzHezxeyxH j
tHtH
x
z
jx  


By (3) we have
|),)((| yxH
jx 

2
()/2 ||/
()/()/
()/
ncxzt NN
n
N
ctex ztx zt
zy t
  


 

dzyzHetyz j
tHN|),))(((|)/( 
()/2
()/ ()/
|((2))(,)|.
nNN
n
tH j
ctxy tzy t
eHzydz
  




Applying Lemma 6 with j
t 2=, we obtain
|),)((| yxH
jx 


 
/2 1
2
/N
nnN
nH
ctx yte




 


1
/2
2
()/ ,>0.
n
nNN
nH
ctx yt



 

Remark 8 In the following section we will show that
there exists

V, the Schwartz class, such that (4)
does not hold for j. By Theorem 1, this means
that for such V the gradient upper Gaussian bound (3)
does not hold for all
t
.
3. A Counterexample to the Gradient Heat
Kernel Estimate
Consider the solvable model

VdxdH 22 /= ,


, where
.sech1)(=)( 2xxV  

We know from [5] that solving the Helmholtz equation
for {0}\k
),,(=),( 2kxekkxeH

yields the following formula for the continuum eigen-
functions:
,),(
||
1
))(sign(=),(
1=
ikx
j
ekxP
kij
kkxe













where ),tanh(=),( ikxpkxP

is defined by the
recursion formula
11
(tanh ,)
=((tanh ,))(tanh)(tanh ,),
pxik
dpx ikikxpx ik
dx





with 1
0

p. Note that ),(=),( kxekxe  and the
function
()
22
=1
(, ,)(,)(,)
1
=(,)(,)
ik xy
j
xyk exkeyk
PxkPy ke
jk



 





(11)
is real analytic on 3 . Moreover,

H has only
absolutely continuous spectrum )[0,= 
ac

and point
spectrum .},4,1,{=2
  
pp The corresponding
eigenfunctions

1=
}{ nn e in 2 L are Schwartz functions
that are linear combinations of xx
mtanh
sech , 

m,
0


. Let acac EHH

= denote the absolutely
continuous part of

H and )[0, = EEac the
corresponding orthogonal projection. If )(
0C

, then
we have for all 21 LLf  ,
,),)(()(),(=)()( 2
1=
nn
n
eefndyyfyxKxfH  

 

where dxxexfef nn)()(=),(  and
dkkyekxekyxK ),(),()()(2=),( 21 

 (12)
is the kernel of acacEHH )(=)(


, cf. [16]. Since

H
has eigenfunctions in )(

and pp

is finite, from
now on it is essential to check the kernel
),(=),)(( yxKyxHac

instead of the kernel of )(


H.
3.1. Decay for the Kernel of acjEH )(
Let )(}{ 0= 

 C jj
 satisfy
}2||2:{supp)i( 2jj
jxx 

 and

jcx kj
k
k
j,2|)(|)i(i )(
 , 0 k. Let
),)((=),( yxHyx acjj


. In [5] we showed that for each
N
,,|)|2(12|),(| /2/2  jyxcyx Njj
Nj
 (13)
but (with 1=

)
Njj
Njxyxcyx   |)|2(12|),(| /2|)|/2(1

 (14)
only holds for 0j and does not hold for all 0<j.
This suggests that (3) fails for 1=

and 1>t (or
more precisely


t
), according to Theorem 1.
Now consider the system  jj}{ which satisfy (i),
(ii) as in Section 1. We may assume )(2=)( xx j
j


for a fixed

in 1,1])([

C with 1=)(x on
]
2
1
,
2
1
[. Let  f and  f be the Fourier transform
and its inverse of f on . The following lemma

S. J. ZHENG 429
shows that (13) does not hold for ),)(( yxHacj  when
j.
Lemma 9 Let ),( yxK j be the kernel of acj EH)(.
a) For each 0 N there exists N c such that for all
0j,
.|)|2(12|),(| /2/2 Njj
Nj yxcyxK 
 (15)
b) For each 0 N there exists N c such that for all
0>j, precisely
.|)|2(12|),(| 1/2/2   yxcyxK jj
Nj (16)
In particular, the decay in (15) does not hold for all
0>j with 1>N.
c) There exist positive constants C and c such that
for all j,
,|)(||)(||),(| || dueuCyxCyxK uyxc
jjj





where )(=)( 2 kkjj and it is easily to see that for
each N, there exists N c such that for all j
.|)|2(12|)(| /2/2 Njj
Nj yxcyx 

Proof. (a) Let /2 2=j  . By (12), (11) and integration
by parts we have
  
()
2221
=1
2,=(1)
[()() (,)(,)],
ik xy NN
j
N
kj
j
ix yKxye
kjkPxkPykdk






 


which can be written as a finite sum of
2()221()()
2
=1
(tanh) (tanh)
[(())(())(())] ()
m
irs
j
xy
kkqkxy





 


(17)

 m,0 , Nsri = , )(
2kq 
are polynomials of
degree

2. We obtain for each N and all 0

j
11/2(1)
|()(,)|=() =()=(2),
NiNjN
j
xyKxyOOO 


using













).(=
)(=))((
)(=))((
2)(
2
2)(122
1=
)(2
ss
rr
ii
j
kOq
kOk
Ok







This proves (15) for 0j.
In order to show part (c) for j, using partial
fractions we write ),( yxK j as a finite sum of
2221
2
=1
(tanh) (tanh)()()()(),







 



m
j
xyk kqkxy
(18)
which is bounded by (up to a constant multiple)
22
=1
|[()]( )||[()]()|,
jj
abk
kxyk xy
k





 


where 


ba,. The general term in the sum is
estimated by
,|)(||)(])([| ||
22 dueuCyx
k
kba
kuyxc
jj
 





in terms of the identities
2
||
1
2
=)()( k ke x

 (19)
.
1
2
=)())(sign( 2
||
k
ik
kex x


 (20)
(b) Finally we prove the sharp estimate in (16). For
0>j, (15) does not hold for 2N, instead we have
only, with 0,1=N,
,|)|2(12|),(| /2/2Njj
jyxcyxK 

by using similar argument and noting (17), (19), (20).
Indeed, let 0>J, 2N. It is easy to find }{ j

satisfying (i’) and (ii’) such that
).(1=)( xxj
J
J




We have by (13)
.22|),)((|)( 1)/2(1)/2( 
   NJNj
J
Nacj
N
J
cyxHyx  
On the other hand, from (18) and the relevant steps in
part (c) we observe that if 1>N,

[0, )
|| ||
() ()(,)
=()(, )(, )
=finite sum of()
sign( ),
N
N
N
cx ycx y
xyH xy
xy exkeykdk
xy
exye









 






1
(21)
where 0,




are of the form tanhtanh m cx y
.
This shows that the term ),)(()( yxHyxacJ
N cannot
admit a decay of 1)/2( 2 NJ for all 0>J, otherwise one
would have ,2|),)(()(| 1)/2(
)[0,


NJN yxHyx 

1 which
leads to a contradiction that the sum of those functions in
(21) must vanish, by letting J.
Remark 10 The argument in the proof of part (b) can
be made rigorous by replacing )(
][0, 
H

1 with
)(

H
L

, and then let L to get the same
contradiction.
3.2. The Derivative of the Kernel of


j
ac
ΦHE
Similar argument shows that

S. J. ZHENG 430
/2(2)
|(,)|2 /||
j
NN
xj N
K
xycx y

 
holds for all 0>j but does not hold for all 0<j.
Therefore the inequality in (4) does not hold for general
1
loc
VL.
4. Acknowledgements
The author gratefully thanks the hospitality and support
of Department of Mathematics, University of California,
Riverside during his visit in November 2008. In parti-
cular he would like to thank Professor Qi S. Zhang for
the kind invitation and discussions. The author also
wishes to thank the referee for careful reading of the
original manuscript, whose comments have improved the
presentation of this article.
5. References
[1] G. Ólafsson and S. Zheng, “Harmonic Analysis Related
to Schrödinger Operators,” Contemporary Mathematics,
Vol. 464, 2008, pp. 213-230.
[2] S. Zheng, “Littlewood-Paley Theorem for Schrödinger
Operators,” Analysis in Theory and Applications, Vol. 22,
No. 4, 2006, pp. 353-361.
[3] J. Epperson, “Triebel-Lizorkin Spaces for Hermite Ex-
pansions,” Studia Mathematica, Vol. 114, No. 1, 1995, pp.
87-103.
[4] J. Dziubański, “Triebel-Lizorkin Spaces Associated with
Laguerre and Hermite Expansions,” Proceedings of the
American Mathematical Society, Vol. 125, No. 12, 1997,
pp. 3547-3554.
[5] G. Ólafsson and S. Zheng, “Function Spaces Associated
with Schrödinger Operators: The Pöschl-Teller Poten-
tial,” Journal of Fourier Analysis and Application, Vol.
12, No. 6, 2006, pp. 653-674.
[6] J. Benedetto and S. Zheng, “Besov Spaces for the
Schrödinger Operator with Barrier Potential,” to appear
in Complex Analysis and Operator Theory, Birkhäuser.
[7] W. Hebisch, “A Multiplier Theorem for Schrödinger
Operators,” Colloquium Mathematicum, Vol. 60-61, No.
2, 1990, pp. 659-664.
[8] W. Hebisch, “Almost Everywhere Summability of Ei-
genfunction Expansions Associated to Elliptic Opera-
tors,” Studia Mathematica, Vol. 96, No. 3, 1990, pp.
263-275.
[9] E. Ouhabaz, “Sharp Gaussian Bounds and p
L-Growth
of Semigroups Associated with Elliptic and Schrödinger
Operators,” Proceedings of the American Mathematical
Society, Vol. 134, No. 12, 2006, pp. 3567-3575.
[10] G. Furioli, C. Melzi and A. Veneruso, “Littlewood-Paley
Decompositions and Besov Spaces on Lie Groups of
Polynomial Growth,” Mathematische Nachrichten, Vol.
279, No. 9-10, 2006, pp. 1028-1040.
[11] A. Grigor’yan, “Upper Bounds of Derivatives of the Heat
Kernel on an Arbitrary Complete Manifold,” Journal of
Functional Analysis, Vol. 127, No. 2, 1995, pp. 363-389.
[12] Q. Zhang, “Global Bounds of Schrödinger Heat Kernels
with Negative Potentials,” Journal of Functional Analysis,
Vol. 182, No. 2, 2001, pp. 344-370.
[13] B. Simon, “Schrödinger Semigroups,” Bulletin of the
American Mathematical Society, Vol. 7, No. 3, 1982, pp.
447-526.
[14] H.-J. Schmeisser and H. Triebel, “Topics in Fourier
Analysis and Function Spaces,” Wiley-Interscience,
Chichester, 1987.
[15] P. D’Ancona and V. Pierfelice, “On the Wave Equation
with a Large Rough Potential,” Journal of Functional
Analysis, Vol. 227, No. 1, 2005, pp. 30-77.
[16] S. Zheng, “A Representation Formula Related to Schrö-
dinger Operators,” Analysis in Theory and Applications,
Vol. 20, No. 3, 2004, pp. 294-296.