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Vol. 5 (2000) Paper no. 5, pages 1–47.
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol5/paper5.abs.html
THE NORM ESTIMATE OF THE DIFFERENCE BETWEEN
¨
THE KAC OPERATOR AND SCHRODINGER
SEMIGROUP II:
THE GENERAL CASE INCLUDING THE RELATIVISTIC CASE
Takashi Ichinose1
Department of Mathematics, Kanazawa University, Kanazawa, 920-1192, Japan
ichinose@kappa.s.kanazawa-u.ac.jp
Satoshi Takanobu2
Department of Mathematics, Kanazawa University, Kanazawa, 920-1192, Japan
takanobu@kappa.s.kanazawa-u.ac.jp

Abstract More thorough results than in our previous paper in Nagoya Math. J. are given
on the Lp -operator norm estimates for the Kac operator e−tV /2 e−tH0 e−tV /2 compared with the
Schr¨
odinger semigroup e−t(H0 +V ) . The Schr¨
odinger operators H0 + V to be treated in this paper
are more general ones associated with the L´evy process, including the relativistic Schr¨
odinger
operator. The method of proof is probabilistic based on the Feynman-Kac formula. It differs
from our previous work in the point of using the Feynman-Kac formula not directly for these
operators, but instead through subordination from the Brownian motion, which enables us to
deal with all these operators in a unified way. As an application of such estimates the Trotter
product formula in the Lp -operator norm, with error bounds, for these Schr¨
odinger semigroups
is also derived.
Keywords Schr¨
odinger operator, Schr¨
odinger semigroup, relativistic Schr¨
odinger operator,
Trotter product formula, Lie-Trotter-Kato product formula, Feynman-Kac formula, subordination of Brownian motion, Kato’s inequality
AMS subject classification 47D07, 35J10, 47F05, 60J65, 60J35

Submitted to EJP on October 21, 1999. Final version accepted on January 26, 2000.
1
Partially supported by Grant-in-Aid for Scientific Research No. 11440040, Ministry of Education, Science and
Culture, Japan
2
Partially supported by Grant-in-Aid for Scientific Research No. 10440030, Ministry of Education, Science and
Culture, Japan.

1. Introduction
By the Kac operator we mean an operator of the kind K(t) = e−tV /2 e−tH0 e−tV /2 , where H =
H0 + V ≡ −∆/2 + V (x) is the nonrelativistic Schr¨
odinger operator in L2 (R d ) with mass 1 with
scalar potential V (x) bounded from below. This K(t) may correspond to the transfer operator for
a lattice model in statistical mechanics studied by M. Kac [Ka]. There it is one of the important
problems to know asymptotic spectral properties of K(t) for t ↓ 0. To this end, in [H1, H2] Helffer
estimated the L2 -operator norm of the difference between K(t) and the Schr¨
odinger semigroup
e−tH to be of order O(t2 ) for small t > 0, if V (x) satisfies |∂ α V (x)| ≤ Cα (1 + |x|2 )(2−|α|)+ /2
for every multi-index α with a constant Cα . Then such norm estimates may be applied to get
spectral properties of K(t) in comparison with those of H.

In [I-Tak1] and [I-Tak2] we have extended his result to the case of more general scalar potentials V (x) even in the Lp -operator norm, 1 ≤ p ≤ ∞, making a probabilistic approach based
on the Feynman-Kac formula. In [I-Tak2] we have also considered this problem for both the
nonrelativistic Schr¨
odinger operator
√odinger operator H = H0 + V and the relativistic Schr¨
H r = H0r + V ≡ −∆ + 1 − 1 + V (x) with light velocity 1. The Lp -operator norm of this
difference is estimated to be of order O(ta ) of small t > 0 with a ≥ 1, though the relativistic
case shows for small t > 0 a slightly different behavior from the nonrelativistic case. As another
application of these results the Trotter product formula for the nonrelativistic and relativistic
Schr¨
odinger operators in the Lp -operator norm with error bounds is obtained. There are also
related L2 results with operator-theoretic methods, for which we refer to [D-I-Tam].
The aim of this paper is to generalize and refine the result of [I-Tak2] in the relativistic case,
odinger operator
√ admitting of more general operators than the free relativistic Schr¨
H0r = −∆ + 1 − 1 as well as relaxing the conditions for the potentials V (x). We use the
probabilistic method with Feynman-Kac formula, though observing everything in a unified way
through subordination from the Brownian motion. In this respect the present method differs
from that in [I-Tak2] used for the relativistic Schr¨
odinger operator H r , which made the best of

r
the explicit expression of the integral kernel of e−tH0 .
The more general operator we have in mind is the following operator
H0ψ = ψ( 21 (−∆ + 1)) − ψ( 21 ),

(1.1)

which will play the same role as the relativistic Schr¨
odinger operator
√
H0r = −∆ + 1 − 1

(1.2)

in [I-Tak2]. Obviously, H0ψ is a selfadjoint operator in L2 (R d ). Here ψ(λ) is a continuous
increasing function on [0, ∞) with ψ(0) = 0 and ψ(∞) = ∞ expressed as
Z
ψ(λ) =
(1 − e−λl )n(dl), λ ≥ 0,
(1.3)

(0,∞)

where n(dl) is a L´evy measure on (0, ∞) (i.e. a measure on (0, ∞) such that
with n((0, ∞)) = ∞. It is clear that
Z
ψ(λ + 12 ) − ψ( 12 ) =
(1 − e−λl )e−l/2 n(dl).
(0,∞)

2

R

(0,∞) l∧1n(dl)

< ∞)
(1.4)

As a special case of H0ψ we have for ψ(λ) = (2λ)α , 0 < α < 1, the operator
(α)

H0

= (−∆ + 1)α − 1,

(1.5)
(1/2)

which reduces to the relativistic Schr¨
odinger operator when α = 1/2: H0
α
the L´evy measure is n(dl) = {2 α/Γ(1 − α)}l−1−α dl.

= H0r . In this case

To formulate our results we are going to describe what kind of function V (x) is. Let 0 < γ, δ ≤ 1,
0 ≤ κ ≤ 1, 0 ≤ µ, ν, ρ < ∞, 0 ≤ C1 , C2 , c1 , c2 < ∞ and 0 < c < ∞. Let V : R d → [0, ∞) be a
continuous function satisfying one of the following five conditions:
(A)0

|V (x) − V (y)| ≤ C1 |x − y|γ ;

(A)1

V is a C 1 -function such that
(i) |∇V (z)| ≤ C1 (1 + V (z)1−δ ), (ii) |∇V (x) − ∇V (y)| ≤ C2 |x − y|κ ;

(A)2

V is a C 1 -function such that
(i) |∇V (z)| ≤ C1 (1 + V (z)1−δ ),
(ii) |∇V (x) − ∇V (y)|
o
n
≤ C2 V (x)(1−2δ)+ (1 + |x − y|µ ) + 1 + |x − y|ν |x − y|;

(V)1

V is a C 1 -function such that
(i) V (z) ≥ chziρ , (ii) |∇V (z)| ≤ c1 hzi(ρ−1)+ ;

(V)2

V is a C 2 -function such that
(i) V (z) ≥ chziρ , (ii) |∇V (z)| ≤ c1 hzi(ρ−1)+ ,
(iii) |∇2 V (z)| ≤ c2 hzi(ρ−2)+ .

Here hzi :=

p
1 + |z|2 .

Conditions (A)0 , (A)1 and (A)2 on V (x) are used in [Tak] and are more general than in [ITak1,2], while conditions (V)1 and (V)2 are used in [D-I-Tam]. But these conditions may not be
best possible. A simple example of a function which has property (A)0 , (A)1 or (A)2 is, needless
to say, V (x) = |x|r (0 < r < ∞), and a slightly complicated one V (x) = |x|r (2 + sin log |x|),
according as 0 < r ≤ 1, 1 < r < 2 or r ≥ 2. Also V (x) = 1 + |x1 − x2 |r (x = (x1 , x2 , . . . , xd ))
satisfies (A)0 , (A)1 or (A)2 with the same r as above, but neither (V)1 nor (V)2 . To the contrary
R |x|
V (x) = 1 + |x| 0 (1 + sin(θ 2 ))dθ satisfies (V)1 , but neither (V)2 , (A)0 , (A)1 nor (A)2 .
The operator H0ψ +V is essentially selfadjoint on C0∞ (R d ), and so its unique selfadjoint extension

ψ
is also denoted by the same H0ψ + V . The semigroup e−t(H0 +V ) on L2 (R d ) is extended to a
strongly continuous semigroup on Lp (R d ) (1 ≤ p < ∞) and C∞ (R d ), to be denoted by the same
ψ
e−t(H0 +V ) . Here C∞ (R d ) is the Banach space of the continuous functions on R d vanishing at
infinity. To be complete, these and further facts are proved in Appendix.
3

As for the L´evy measure n(dl) introduced in (1.3) and (1.4), we make the following assumption:
(L)

For some α ∈ [0, 1], n((·, ∞)) is regularly varying at zero with exponent −α, i.e., there
exists a slowly varying function L(λ) at infinity such that
n((t, ∞)) ∼ t−α L( 1t )

as t ↓ 0.

(1.6)

Here a positive function L(·) is called slowly varying at infinity if for any c > 0,

lim

λ↑∞

L(cλ)
= 1.
L(λ)

Let φ−1 (·) be the inverse function of φ(λ) := ψ(λ + 1/2) − ψ(1/2). (Note that φ is strictly
increasing.) Under the above assumption, set
L1 (λ) :=

(

Γ(1 − α)L(λ)
if 0 ≤ α < 1
R 1/λ
n((s, ∞))ds if α = 1,
0

L2 (x) := L1 (φ−1 (x))−1/α

if

0 < α ≤ 1.

These two functions are slowly varying at infinity, and we have φ(λ) ∼ λα L1 (λ)Ras λ → ∞ and
φ−1 (x) ∼ x1/α L2 (x) as x → ∞, as will be seen from Fact in Section 6, so that 0· (φ−1 (θ))−α dθ
(0 < α < 1) is also slowly varying at infinity.
Now we state the main results of this paper, which generalize the results in [I-Tak2]. In the
following k · kp→p stands for the Lp -operator norm for 1 ≤ p < ∞ and the supremum norm on
C∞ (R d ) for p = ∞.
Theorem 1. Suppose assumption (L) and let 1 ≤ p ≤ ∞. Then the following estimates (i), (ii)
and (iii) hold for small t > 0.
(i) Under (A)0 ,
ψ

ψ

ke−tV /2 e−tH0 e−tV /2 − e−t(H0 +V ) kp→p ,
ψ

ψ

ke−tV e−tH0 − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

ke−tH0 /2 e−tV e−tH0 /2 − e−t(H0 +V ) kp→p

O(t2 )
if α < γ/2





R 1/t
=
O(t2 0 (φ−1 (θ))−α dθ) if α = γ/2





if γ/2 < α.
O(t1+ γ/2α L2 ( 1t )−γ/2 )

(ii) Under (A)1 ,
ψ

ψ

ke−tV /2 e−tH0 e−tV /2 − e−t(H0 +V ) kp→p
4


if
O(t1+1∧2δ )





R 1/t
=
O(t1+2δ ) + O(t2 0 (φ−1 (θ))−α dθ)
if





O(t1+2δ ) + O(t1+ (1+κ)/2α L2 ( 1t )−(1+κ)/2 ) if
ψ

α < (1 + κ)/2 or κ = 1
α = (1 + κ)/2 < 1
(1 + κ)/2 < α,

ψ

ke−tV e−tH0 − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

ke−tH0 /2 e−tV e−tH0 /2 − e−t(H0 +V ) kp→p

O(t1+δ )
if α < 1/2





R 1/t
=
O(t1+δ 0 (φ−1 (θ))−α dθ) if α = 1/2





O(tδ+ 1/2α L2 ( 1t )−1/2 )
if 1/2 < α.

(iii) Under (A)2 ,

ψ

ψ

= O(t1+1∧2δ ),

ke−tV /2 e−tH0 e−tV /2 − e−t(H0 +V ) kp→p
ψ

ψ

ke−tV e−tH0 − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

ke−tH0 /2 e−tV e−tH0 /2 − e−t(H0 +V ) kp→p

if α < 1/2
O(t1+δ )





R 1/t
=
O(t1+δ 0 (φ−1 (θ))−α dθ) if α = 1/2





if 1/2 < α.
O(tδ+ 1/2α L2 ( 1t )−1/2 )

In fact, the first estimate in (iii) holds independent of (L).

A consequence of Theorem 1 is the following Trotter product formula in the Lp -operator norm
with error bounds.
Theorem 2. Suppose assumption (L) and let 1 ≤ p ≤ ∞. Then the following estimates (i),
(ii), (iii) and (iv) hold uniformly on each finite t-interval on [0, ∞).

(i) Under (A)0 ,

ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p

if α < γ/2
O(n−1 )





Rn
O(n−1 0 (φ−1 (θ))−α dθ) if α = γ/2
=





O(n−γ/2α L2 (n)−γ/2 )
if γ/2 < α.
5

(ii) Under (A)1 ,
ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p

if
O(n−1∧2δ )





Rn
O(n−2δ ) + O(n−1 0 (φ−1 (θ))−α dθ)
if
=





O(n−2δ ) + O(n−(1+κ)/2α L2 (n)−(1+κ)/2 ) if

α < (1 + κ)/2 or κ = 1
α = (1 + κ)/2 < 1
(1 + κ)/2 < α.

(iii) Under (A)2 ,

ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p
= O(n−1∧2δ ).
(iv) Under (V)i (i = 1, 2),
ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p
= O(n−i/2∨ρ ).
In fact, the asymptotic estimates (iii) and (iv) hold independent of (L).
ψ

Notice here that though the estimates with small t, in Theorem 1, for e−tV e−tH0 and
ψ
ψ
ψ
e−tH0 /2 e−tV e−tH0 /2 are of worse order than that for e−tV /2 e−tH0 e−tV /2 , one has, in Theorem 2,
the same error bounds with large n for these three products.
Finally we give a comment on what kind of operators are to be covered by our H0ψ +V . To this end
(α)
we briefly illustrate how our result reads on the Trotter product formula in the case H0 +V with
(α)
H0 = (−∆ + 1)α − 1, 0 < α < 1, in (1.5). In this case, we have n((t, ∞)) = (2α /Γ(1 − α)) t−α ,
or L2 (·) ≡ 2−1 , so that
Z x
(φ−1 (θ))−α dθ ∼ 2α log x as x → ∞.
0

Therefore Theorem 2 says that for 1 ≤ p ≤ ∞ and uniformly on each finite t-interval in [0, ∞),
(α)

k(e−tV /2n e−tH0

(α)
−tH0 /n

k(e−tV /n e

(α)
−tH0 /2n

k(e

(α)

/n −tV /2n n

e

) − e−t(H0

(α)
−t(H0 +V

)n − e

(α)
−tH0 /2n

e−tV /n e

)

+V )

kp→p ,

kp→p ,

(α)

)n − e−t(H0

6

+V ) k
p→p


if α < γ/2
O(n−1 )





O(n−1 log n) if α = γ/2
=





O(n−γ/2α )
if γ/2 < α

=



O(n−1∧2δ )







 O(n−1 log n)

under (A)0 ,

if

α < (1 + κ)/2

if

α = (1 + κ)/2 and 1/2 ≤ δ ≤ 1



O(n−2δ )
if






 O(n−2δ∧ (1+κ)/2α ) if

α = (1 + κ)/2 and 0 < δ < 1/2

under (A)1 .

(1 + κ)/2 < α

An important remark is the following. In the above example, the case α = 1 is missing. This is
equivalent to the nonrelativistic case H0 + V = −∆/2 + V (x), treated in [Tak] (cf. [I-Tak1,2]).
However we may think that this case is also
in our results, Theorems 1 and
√ implicitly contained
2
r
2
4
2, for α = 1/2. Indeed, by using H0 (c) = −c ∆ + c − c with light velocity c restored in place
of H0r in (1.2), we can obtain the case α = 1/2 so as to involve the parameter c (light velocity).
r
Since, in the nonrelativistic limit c → ∞, the relativistic Schr¨
odinger semigroup e−t(H0 (c)+V ) is
strongly convergent to the nonrelativistic Schr¨
odinger semigroup e−t(H0 +V ) uniformly on each
finite t-interval in [0, ∞) (e.g. [I2]), we can reproduce the nonrelativistic result in [Tak] (cf.
Remark following Theorem 2.3).
In Section 2, we state our results in more general form: we generalize Theorems 1 and 2 to
Theorems 2.1 and 2.2 / 2.3 by introducing the subordinator σt , namely, a time-homogeneous
L´evy process associated with the L´evy measure e−l/2 n(dl). Moreover we state Theorem 2.4 on
asymptotics of the moments of the process σt . Once we know these asymptotics, we can obtain
Theorems 1 and 2 from Theorems 2.1 and 2.2 / 2.3. These four theorems are proved in Sections
3 – 6.
In Appendix, we give a full study of the semigroups e−t(H0 +V ) , t ≥ 0, on Lp (R d ), 1 ≤ p < ∞
and C∞ (R d ) defined through the Feynman-Kac formula. We show they constitute a strongly
continuous contraction semigroup there. It is also shown that its infinitesimal generator Gψ,V
p
ψ
∞
d
has C0 (R ) as a core, by establishing Kato’s inequality for the operator H0 . Some of these
results seem to be new.
ψ

The authors would like to thank the referee for his / her careful reading of the manuscript and
for a number of comments.
2. General results
In this section we shall prove the theorems in a little more general setting based on probability
theory. To describe it we introduce some notations and notions. For a continuous function
V : R d → [0, ∞), set
ψ

K(t) := e−tV /2 e−tH0 e−tV /2 ,
ψ

G(t) := e−tV e−tH0 ,
7

ψ

ψ

R(t) := e−tH0 /2 e−tV e−tH0 /2
and
ψ

QK (t) := K(t) − e−t(H0 +V ) ,
ψ

QG (t) := G(t) − e−t(H0 +V ) ,
ψ

QR (t) := R(t) − e−t(H0 +V ) .
Suppose we are given the independent random objects N (·) and B(·) on some probability space
(Ω, F, P):
(i) N (dsdl) is a Poisson random measure on [0, ∞)×(0, ∞) such that E [N (dsdl)] = dse−l/2 n(dl);
(ii) (B(t))t≥0 is a d-dimensional Brownian motion starting at 0.
Set
σt :=

Z

0

t+Z

l N (dsdl).

(2.1)

(0,∞)

Then (σt )t≥0 is a time-homogeneous L´evy process with increasing paths such that
E [e−λσt ] = e−t(ψ(λ+ 1/2)−ψ(1/2))

(2.2)

(e.g. Note 1.7.1 in [It-MK]). Note that σt has moments of all order (cf. (6.1)), which is to be
seen at the beginning of Section 6. We use a subordination of B(·) by a subordinator σ· , i.e., a
process (B(σt ))t≥0 on R d . This is a L´evy process such that
E [e

√

−1hp,B(σt )i

] = e−t(ψ((|p|

),

2 +1)/2)−ψ(1/2)

ψ

which corresponds to the semigroup {e−tH0 }t≥0 with generator H0ψ in (1.1).
We prove the following generalization of Theorems 1 and 2.
Theorem 2.1. Let 1 ≤ p ≤ ∞ and t ≥ 0.

(i) Under (A)0 ,

γ/2

kQK (t)kp→p , kQG (t)kp→p , kQR (t)kp→p ≤ const(γ, d) C1 t E [σt

].

(ii) Under (A)1 ,
i
h
2
P
j(1+κ)/2
kQK (t)kp→p ≤ const(δ, κ, d) C12 (t2 + t2δ )E [σt ] +
] ,
(C2 t)j E [σt
j=1

kQG (t)kp→p , kQR (t)kp→p ≤ const(δ, κ, d)

2 n
P

j=1

8

j/2

j(1+κ)/2

C1j (tj + tjδ )E [σt ] + (C2 t)j E [σt

o
] .

(iii) Under (A)2 ,
h
2 n
P
(C2 t)j E [σtj ]
kQK (t)kp→p ≤ const(δ, µ, ν, d) C12 (t2 + t2δ )E [σt ] +
j=1

j(1+ ν/2)

+ (C2 t)j E [σt

j(1+ µ/2)

] + (C2 t1∧2δ )j E [σtj ] + (C2 t1∧2δ )j E [σt

kQG (t)kp→p , kQR (t)kp→p ≤ const(δ, µ, ν, d)

oi
] ,

2 n
P
j/2
C1j (tj + tjδ )E [σt ] + (C2 t)j E [σtj ]

j=1

j(1+ ν/2)

] + (C2 t1∧2δ )j E [σtj ]
o
j(1+ µ/2)
+ (C2 t1∧2δ )j E [σt
] .
+ (C2 t)j E [σt

Theorem 2.2. Let 1 ≤ p ≤ ∞, t ≥ 0 and n ∈ N .

(i) Under (A)0 ,

ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p
γ/2

≤ const(γ, d) C1 t E [σt/n ].
(ii) Under (A)1 ,
ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p
h 
i

2
P
j(1+κ)/2
≤ const(δ, κ, d) C12 ( nt )2 + ( nt )2δ nE [σt/n ] +
] ,
(C2 nt )j nE [σt/n
j=1

ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p

h 
1/2
(1+κ)/2
]
≤ const(δ, κ, d) n1 C1 (t + tδ )E [σt ] + C1 tE [σt
+ C1



t
n


i


2
P
1/2
j(1+κ)/2
] .
(C2 nt )j nE [σt/n
+ ( nt )δ E [σt/n ] + C12 ( nt )2 + ( nt )2δ nE [σt/n ] +
j=1

(iii) Under (A)2 ,
ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p
h 

2 n
P
j
(C2 nt )j nE [σt/n
≤ const(δ, µ, ν, d) C12 ( nt )2 + ( nt )2δ nE [σt/n ] +
]
j=1

j(1+ ν/2)

+ (C2 nt )j nE [σt/n

j(1+ µ/2)

j
] + (C2 ( nt )1∧2δ )j nE [σt/n
] + (C2 ( nt )1∧2δ )j nE [σt/n

9

oi
] ,

ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p
h 
1/2
1+ µ/2
])
≤ const(δ, µ, ν, d) n1 C1 (t + tδ )E [σt ] + C2 t1∧2δ (E [σt ] + E [σt





1+ ν/2
1/2
+ C2 t(E [σt ] + E [σt
]) + C1 nt + ( nt )δ E [σt/n ] + C12 ( nt )2 + ( nt )2δ nE [σt/n ]
+

2 n
P
j(1+ ν/2)
j
j
(C2 nt )j nE [σt/n
] + (C2 nt )j nE [σt/n
] + (C2 ( nt )1∧2δ )j nE [σt/n
]

j=1

j(1+ µ/2)

+ (C2 ( nt )1∧2δ )j nE [σt/n

oi
] .

Theorem 2.3. Let 1 ≤ p ≤ ∞ and t ≥ 0.

(i) Under (V)1 for n ≥ 22(2∨ρ) ,
ψ

ψ

k(e−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p
h
≤ const(ρ, c, c1 , d) n−1/2∨ρ t2/(ρ∧2)∨1 −1 + (t2 + t2(1∧((ρ∧2)∨1)/2ρ) )nE [σt/n ]
+

2 
P

j=1

j(2∨ρ)/2

j
(tj + tj2/2∨ρ )nE [σt/n
] + tj nE [σt/n
ψ

i
] ,

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p
h
1/2
≤ const(ρ, c, c1 , d) n−1/2∨ρ t2/(ρ∧2)∨1 −1 + (t + t1∧((ρ∧2)∨1)/2ρ )E [σt ]
(2∨ρ)/2

+ t2/2∨ρ E [σt ] + t(E [σt ] + E [σt
]) + (t2 + t2(1∧((ρ∧2)∨1)/2ρ) )nE [σt/n ]
oi
2 n
P
j(2∨ρ)/2
j
(tj + tj2/2∨ρ )nE [σt/n
+
] + tj nE [σt/n
] .
j=1

(ii) Under (V)2 for n ≥ 1,
ψ

ψ

ke−tV /2n e−tH0 /n e−tV /2n )n − e−t(H0 +V ) kp→p
h
≤ const(ρ, c, c1 , c2 , d) n−2/2∨ρ (t2 + t2/1∨ρ )nE [σt/n ]
+

i
2 
P
j(2∨ρ)/2
j
] ,
] + tj nE [σt/n
(tj + tj2/2∨ρ )nE [σt/n

j=1

ψ

ψ

k(e−tV /n e−tH0 /n )n − e−t(H0 +V ) kp→p ,
ψ

ψ

ψ

k(e−tH0 /2n e−tV /n e−tH0 /2n )n − e−t(H0 +V ) kp→p

h
1/2
(2∨ρ)/2
]
≤ const(ρ, c, c1 , c2 , d) n−2/2∨ρ (t + t1/1∨ρ )E [σt ] + (t + t2/2∨ρ )E [σt ] + tE [σt
+ (t2 + t2/1∨ρ )nE [σt/n ] +

2 n
P

j=1

j(2∨ρ)/2

j
] + tj nE [σt/n
(tj + tj2/2∨ρ )nE [σt/n

10

o
]

i
1/2
+ n−1/1∨ρ E [σt/n ](t + t1/1∨ρ ) .
Remark. As noted at the end of Section 1, the nonrelativistic case for H0 + V = −∆/2 + V ,
being equivalent to the case α = 1 which Theorems 1 and 2 fail to cover, can be thought
to be implicitly contained in the relativistic
√ case, of the above three theorems, for the relativistic Schr¨
odinger operator H0r (c) ≡ −c2 ∆ + c4 − c2 with the light velocity c ≥ 1 restored. We have √
H0ψ = H0r (c), where
√ this ψ(λ) is a c-dependent function (1.3) given by
ψ(λ) := ψ(λ; c) = 2c2 λ + c4 − c2 − c4 − c2 associated with the c-dependent L´evy measure
2
e−l/2 n(dl; c) = (2π)−1/2 ce−c l/2 l−3/2 dl. In this case, Theorem 2.1 and Theorems 2.2 / 2.3 hold
with the corresponding c-dependent subordinator σt (c), just as they stand, namely, only with
E [σsa ] replaced by E [σs (c)a ] for each respective s > 0 and a > 0. Then the nonrelativistic case
in question is obtained as the nonrelativistic limit c → ∞ of this c-dependent relativistic case,
turning out to be just Theorems 2.1 and 2.2 / 2.3 with E [σsa ] replaced by sa . This is because
r
one can show that, as c → ∞, the relativistic Schr¨
odinger semigroup e−t(H0 (c)+V ) on the LHS
converges strongly to the nonrelativistic Schr¨
odinger semigroup e−t(H0 +V ) uniformly on each
a
finite t-interval in [0, ∞) (cf. [I2]), and E [σt (c) ] on the RHS tends to ta . Then taking the most
dominant contribution on the RHS for small t or large n reproduces the same nonrelativistic
result as in [Tak].
Theorems 1 and 2 follow immediately from Theorems 2.1 and 2.2 / 2.3, if one knows the asymptotics for t ↓ 0 of the moments of σt to investigate which of the terms on the RHS makes a
dominant contribution for small t or large n. These asymptotics are given by the following
theorem.
Theorem 2.4. Suppose assumption (L). Let a > 0.
R
(i) If α < a or a ≥ 1, then (0,∞) la e−l/2 n(dl) < ∞ and
E [σta ]

∼ t

Z

la e−l/2 n(dl)

(0,∞)

as t ↓ 0.

In fact, for a ≥ 1 this always holds independent of (L).

(ii) If α = a and a < 1, then

E [σta ] ∼

1
t
Γ(1 − α)

Z

1/t

(φ−1 (θ))−α dθ

0

as t ↓ 0.

(iii) If 0 < a < α, then
E [σta ] ∼

Γ(1 − αa ) a/α
t L2 ( 1t )−a
Γ(1 − a)

as t ↓ 0.

The proofs of Theorems 2.1, 2.2, 2.3 and 2.4 are given in Sections 3, 4, 5 and 6, respectively. To
show Theorem 2.1, in fact, we prove estimates of the integral kernels of QK (t), QG (t) and QR (t)
by a finite positive linear combination of tc E [|x − y|a σtb p(σt , x − y)], where p(t, x − y) is the heat
kernel (see (A.2)). Such estimates of the integral kernels of the three operators of difference in
Theorems 2.2 / 2.3 also can be obtained (cf. [Tak]), but are omitted.
11

3. Proof of Theorem 2.1
It is easily seen (see (A.6)) that for f ∈ C0∞ (R d )
h


i
K(t)f (x) = E exp − 2t (V (x) + V (x + Xt )) f (x + Xt ) ,
h


i
G(t)f (x) = E exp −tV (x) f (x + Xt ) ,
h


i
R(t)f (x) = E exp −tV (x + Xt/2 ) f (x + Xt )

(3.1)
(3.2)
(3.3)

and generally

h


i
n
t P
K( nt )n f (x) = E exp − 2n
(V (x + X(k−1)t/n ) + V (x + Xkt/n )) f (x + Xt ) ,

(3.4)

k=1

G( nt )n f (x)

h



i
n
t P
V (x + X(k−1)t/n ) f (x + Xt ) ,
= E exp − n

(3.5)

k=1

h


i
n
P
R( nt )n f (x) = E exp − nt
V (x + X(2k−1)t/2n ) f (x + Xt ) .

(3.6)

k=1

Further, for f ∈ C0∞ (R d ) we have (see (A.13))
Z
h
i
QK (t)f (x) =
dyf (y)E σ E B [vK (t, x, y; σ)]p(σt , x − y) ,

Rd

QG (t)f (x) =
QR (t)f (x) =

Z

Rd

Z

Rd

h
i
dyf (y)E σ E B [vG (t, x, y; σ)]p(σt , x − y) ,
h
i
f (y)dy E σ E B [vR (t, x, y; σ)]p(σt , x − y) ,

where E σ and E B are the expectations with respect to σ· and B· , respectively,



 Z t
σt ,y
vK (t, x, y; σ) := exp − 2t (V (x) + V (y)) − exp −
V (B0,x
(σs ))ds ,

(3.7)
(3.8)
(3.9)

(3.10)

0




 Z t
σt ,y
vG (t, x, y; σ) := exp −tV (x) − exp −
V (B0,x
(σs ))ds ,

(3.11)

0




 Z t
σt ,y
σt ,y
vR (t, x, y; σ) := exp −tV (B0,x (σt/2 )) − exp −
V (B0,x
(σs ))ds ,

(3.12)

0

and, for τ > 0, x, y ∈ R d and 0 ≤ θ ≤ τ
τ,y
B0,x
(θ) := x + τθ (y − x) + B0τ (θ)

B0τ (θ)

(3.13)

:= B(θ) − τθ B(τ ).

Since
a

b

b

2

e − e = (a − b)e + (a − b)

Z

1

0

12

(1 − θ)eθa e(1−θ)b dθ,

a, b ∈ R,

we have



vK (t, x, y; σ) = wK (t, x, y; σ) exp − 2t (V (x) + V (y))
Z 1

 Z t
σt ,y
2
V (B0,x
(σs ))ds
− wK (t, x, y; σ)
(1 − θ) exp −θ
0
0


t
× exp −(1 − θ) 2 (V (x) + V (y)) dθ
=: vK1 (t, x, y; σ) + vK2 (t, x, y; σ),


vG (t, x, y; σ) = wG (t, x, y; σ) exp −tV (x)
Z 1

 Z t
σt ,y
2
V (B0,x
(σs ))ds
− wG (t, x, y; σ)
(1 − θ) exp −θ
0
0


× exp −(1 − θ)tV (x) dθ
=: vG1 (t, x, y; σ) + vG2 (t, x, y; σ),


σt ,y
vR (t, x, y; σ) = wR (t, x, y; σ) exp −tV (B0,x
(σt/2 ))
Z 1

 Z t
σt ,y
2
V (B0,x
(σs ))ds
(1 − θ) exp −θ
− wR (t, x, y; σ)
0
0


σt ,y
× exp −(1 − θ)tV (B0,x
(σt/2 )) dθ
=: vR1 (t, x, y; σ) + vR2 (t, x, y; σ),

(3.14)

(3.15)

(3.16)

where
wK (t, x, y; σ) :=

− 2t (V

(x) + V (y)) +
Z

wG (t, x, y; σ) := −tV (x) +
wR (t, x, y; σ) :=

t
0

Z

t
0

σt ,y
V (B0,x
(σs ))ds,

σt ,y
V (B0,x
(σs ))ds,

σt ,y
−tV (B0,x
(σt/2 ))

+

Z

t

0

(3.18)

σt ,y
V (B0,x
(σs ))ds.

When V is further a C 1 -function, since
V (z) − V (w) = h∇V (w), z − wi +

Z

1

h∇V (w + θ(z − w)) − ∇V (w), z − widθ,

0

we have
wK (t, x, y; σ) =

1
2 h∇V

(x) − ∇V (y), y − xi

Z
1
+ 2 h∇V (y), y − xi

t
0

σs
σt ds

Z

t
0

−

σs
σt ds

Z

0

t

σt −σs
σt ds

Z t
D
E
1
+ 2 ∇V (x) + ∇V (y),
B0σt (σs )ds
0

13

(3.17)



(3.19)

Z

Z 1D
∇V (x + θ( σσst (y − x) + B0σt (σs ))) − ∇V (x),
0
0
E
σt
σs
σt (y − x) + B0 (σs ) dθ
Z t Z 1D
1
s
+ 2
∇V (y + θ( σtσ−σ
ds
(x − y) + B0σt (σs ))) − ∇V (y),
t
0
0
E
σt
σt −σs
(x
−
y)
+
B
(σ
)
s dθ
0
σt
t

+

1
2

=:

5
X

ds

wKj (t, x, y; σ),

(3.20)

j=1

wG (t, x, y; σ) =
+

D
Z

∇V (x),
t

t
0

Z 1D

( σσst (y − x) + B0σt (σs ))ds

E

∇V (x + θ( σσst (y − x) + B0σt (σs ))) − ∇V (x),
0
E
σt
σs
(y
−
x)
+
B
(σ
)
s dθ
0
σt
ds

0

Z

=: wG1 (t, x, y; σ) + wG2 (t, x, y; σ),
Z t
E
D
σt ,y
σt ,y
σt ,y
wR (t, x, y; σ) = ∇V (B0,x
(σs ) − B0,x
(σt/2 ))ds
(σt/2 )), (B0,x
+

Z

t

0

σt ,y
σt ,y
σt ,y
∇V (B0,x
(σt/2 ) + θ(B0,x
(σs ) − B0,x
(σt/2 )))
0
E
σt ,y
σt ,y
σt ,y
− ∇V (B0,x
(σt/2 )), B0,x
(σs ) − B0,x
(σt/2 ) dθ

ds
0

Z 1D

(3.21)

=: wR1 (t, x, y; σ) + wR2 (t, x, y; σ).

(3.22)

In the following we shall prove Theorem 2.1 only in Cases (A)2 and (A)0 . The proof of Case
(A)1 is omitted; it is similar to that of (A)2 .
3.1. Case (A)2
In this subsection, we suppose condition (A)2 on V (x).
Claim 3.1.
 h
i


E σ E B [vK1 (t, x, y; σ)]p(σt , x − y) 

h
≤ const(δ, µ, ν, d) C2 t1∧2δ E σ [|x − y|2 p(σt , x − y)] + E σ [σt p(σt , x − y)]

1+ µ/2
+ E σ [|x − y|2+µ p(σt , x − y)] + E σ [σt
p(σt , x − y)]

+ t E σ [|x − y|2 p(σt , x − y)] + E σ [σt p(σt , x − y)] + E σ [|x − y|2+ν p(σt , x − y)]
i
1+ ν/2
p(σt , x − y)] .
+ E σ [σt
14

Proof. In view of (3.14) and (3.20), we set
5
X

vK1 (t, x, y; σ) =

wKj (t, x, y; σ)e−t(V (x)+V (y))/2

j=1

=:

5
X

vK1j (t, x, y; σ).

(3.23)

j=1

Clearly
E B [wK3 (t, x, y; σ)] =

1
2

D

∇V (x) + ∇V (y),

Z

t
0

E B [B0σt (σs )]ds

E

= 0,

h
i
L
and hence E B vK13 (t, x, y; σ) = 0. By the fact (σt − σt−s )0≤s≤t ∼ (σs )0≤s≤t ,
h
i
E σ wK2 (t, x, y; σ)p(σt , x − y)
=

1
2 h∇V

= 0,

 hZ
(y), y − xi E σ

t

0

σs
σt ds p(σt , x

hZ
i
− y) − E σ

t
0

σt −σt−s
σt −σt−t ds p(σt

i
− σt−t , x − y)

h
h
i
i
and hence E σ E B [vK12 (t, x, y; σ)]p(σt , x − y) = E σ vK12 (t, x, y; σ)p(σt , x − y) = 0. By (A)2 (ii)
|vK11 (t, x, y; σ)| = |wK1 (t, x, y; σ)|e−t( V (x)+V (y))/2
≤

1
2 |∇V

≤

C2
2

≤

C2
2

n

(x) − ∇V (y)||x − y| t e−t(V (x)+V (y))/2

o
V (x)(1−2δ)+ (1 + |x − y|µ ) + 1 + |x − y|ν |x − y|2 t e−tV (x)/2

o
V (x)(1−2δ)+ e−tV (x)/2 t(|x − y|2 + |x − y|2+µ ) + t(|x − y|2 + |x − y|2+ν )
o
n
2
2+µ
2
2+ν
+ (1−2δ)+ 1∧2δ
≤ C22 ( 2(1−2δ)
)
t
(|x
−
y|
+
|x
−
y|
)
+
t(|x
−
y|
+
|x
−
y|
)
.
e
n

(3.24)

Here (and hereafter) the following inequality has been (will be) used:
tb e−t ≤ ( eb )b ,

t ≥ 0, b ≥ 0,

(3.25)

where for b = 0 we understand (0/e)0 := 1. By (A)2 (ii) and (3.25) again
|vK14 (t, x, y; σ)| = |wK4 (t, x, y; σ)|e−t(V (x)+V (y))/2
Z t Z 1
1
≤ 2
|∇V (x + θ( σσst (y − x) + B0σt (σs ))) − ∇V (x)|
ds
0

0

≤

C2
2

Z tn
0

× | σσst (y − x) + B0σt (σs )|dθ e−tV (x)/2



V (x)(1−2δ)+ e−tV (x)/2 | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+µ
15

o
+ | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+ν ds

≤

Similarly

C2
2

Z tn
0

+ (1−2δ)+ −(1−2δ)+
( 2(1−2δ)
)
t
e


× | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+µ
o
+ | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+ν ds.

|vK15 (t, x, y; σ)|
Z tn
C2
+ (1−2δ)+ −(1−2δ)+
≤ 2
( 2(1−2δ)
)
t
e
0


σt
σt
2
2+µ
σt −σs
s
(x
−
y)
+
B
(x
−
y)
+
B
(σ
)|
+
|
(σ
)|
× | σtσ−σ
s
s
0
0
σt
t
o
σt
σt
2
2+ν
σt −σs
s
+ | σtσ−σ
ds.
(x
−
y)
+
B
(x
−
y)
+
B
(σ
)|
+
|
(σ
)|
s
s
0
0
σt
t

Note that for a > 0 and 0 ≤ θ ≤ τ (τ > 0)
h


i
E B | τθ z + B0τ (θ)|a ≤ 3(a−1)+ |z|a + 2C(a, d)τ a/2 ,
h


i
a
a/2
(a−1)+
τ
a
|z|
≤
3
E B | τ −θ
+
2C(a,
d)τ
z
+
B
(θ)|
0
τ

(3.26)

(3.27)

(3.28)

R
where C(a, d) := E B [|B(1)|a ] = Rd |y|a p(1, y)dy. Thus, taking expectation E B in (3.26) and
(3.27), we have
h
i
h
i
E B |vK14 (t, x, y; σ)| + E B |vK15 (t, x, y; σ)|
n
+ (1−2δ)+ 1∧2δ
≤ C2 ( 2(1−2δ)
)
t
e


1+ µ/2
× 3|x − y|2 + 6C(2, d)σt + 31+µ |x − y|2+µ + 31+µ 2C(2 + µ, d)σt
o

1+ ν/2
.
+ t 3|x − y|2 + 6C(2, d)σt + 31+ν |x − y|2+ν + 31+ν 2C(2 + ν, d)σt
Collecting all the above into (3.23) yields the estimate in Claim 3.1 and the proof is complete.



Claim 3.2.
h
i
E σ E B [|vK2 (t, x, y; σ)|]p(σt , x − y)
h


≤ const(δ, µ, ν, d) C12 (t2 + t2δ ) E σ [|x − y|2 p(σt , x − y)] + E σ [σt p(σt , x − y)]

+ C22 t2(1∧2δ) E σ [|x − y|4 p(σt , x − y)] + E σ [σt2 p(σt , x − y)]

2+µ
4+2µ
p(σt , x − y)] + E σ [σt p(σt , x − y)]
+ E σ [|x − y|
16


+ C22 t2 E σ [|x − y|4 p(σt , x − y)] + E σ [σt2 p(σt , x − y)]
i
+ E σ [|x − y|4+2ν p(σt , x − y)] + E σ [σt2+ν p(σt , x − y)] .
Proof. By (A)2 (i)
Z t
P
 D

E
 3
1

( σσst (y − x) + B0σt (σs ))ds
wKj (t, x, y; σ) =  2 ∇V (x),

j=1

0

Z t
D
E

σt
1
s
( σtσ−σ
(σ
))ds
(x
−
y)
+
B
+ 2 ∇V (y),

s
0
t
0

≤

C1
2

n

(1 + V (x)1−δ )

Z

+ (1 + V (y)1−δ )

t

| σσst (y − x) + B0σt (σs )|ds

0

Z

t
0

o
σt
s
| σtσ−σ
(x
−
y)
+
B
(σ
)|ds
.
s
0
t

This estimate together with (3.26) and (3.27) gives us that
|wK (t, x, y; σ)|e−θt(V (x)+V (y))/4
≤ |

3
P

wKj (t, x, y; σ)|e−θt(V (x)+V (y))/4 +

j=1

5
P

j=4

|wKj (t, x, y; σ)|e−θt(V (x)+V (y))/4



1−δ −1+δ −1+δ
)
θ
t
≤ C21 1 + ( 4(1−δ)
e
Z t

σt
s
×
| σσst (y − x) + B0σt (σs )| + | σtσ−σ
(x
−
y)
+
B
(σ
)|
ds
s
0
t
0

+

C2
2

Z

0

t

n
+ (1−2δ)+
ds θ −(1−2δ)+ t−(1−2δ)+ ( 4(1−2δ)
)
e

× | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+µ

s
s
(x − y) + B0σt (σs )|2 + | σtσ−σ
(x − y) + B0σt (σs )|2+µ
+ | σtσ−σ
t
t

+ | σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+ν

o
σt
σt
2
2+ν
σt −σs
s
.
(σ
)|
+
|
(σ
)|
(x
−
y)
+
B
(x
−
y)
+
B
+ | σtσ−σ
s
s
0
0
σt
t

By the Schwarz inequality, it follows that

2
|wK (t, x, y; σ)|e−θt(V (x)+V (y))/4
h
≤ 12 ( C21 )2 (t + ( 4(1−δ)
)2(1−δ) θ −2+2δ t−1+2δ )
e
Z t
Z t

σt
2
s
×
| σσst (y − x) + B0σt (σs )|2 ds +
| σtσ−σ
(σ
)|
ds
(x
−
y)
+
B
s
0
t
0
0
n
+ 2(1−2δ)+ −2(1−2δ)+ 2(1∧2δ)−1
)
θ
t
+ ( C22 )2 ( 4(1−2δ)
e
17



(3.29)

Z

×

| σσst (y − x) + B0σt (σs )|4 ds +

0

+
+t

t

Z

t

0

+

Z

Z

t
0

s
| σtσ−σ
(x
t

| σσst (y
t
0

− x) +

s
| σtσ−σ
(x
t

− y)

+

Z

t
0

B0σt (σs )|4 ds

t

| σσst (y − x) + B0σt (σs )|4+2µ ds

0

+ B0σt (σs )|4 ds

B0σt (σs )|4 ds

− y) +

Z

+

Z

0

t

s
| σtσ−σ
(x
t

− y) +

B0σt (σs )|4+2µ ds

| σσst (y − x) + B0σt (σs )|4+2ν ds

+

Z

0

t

s
| σtσ−σ
(x − y) + B0σt (σs )|4+2ν ds
t

Take expectation E B above, and integrate in θ. Then
E B [|vK2 (t, x, y; σ)|]
Z
h
2
≤ E B wK (t, x, y; σ)

=

Z

1

0

θE B

1

θe−θt(V (x)+V (y))/2 dθ

0



oi

.

i

2 i
h
dθ
|wK (t, x, y; σ)|e−θt(V (x)+V (y))/4

h
≤ 12 ( C21 )2 3(t2 + ( 4(1−δ)
)2(1−δ) 1δ t2δ )(|x − y|2 + 2C(2, d)σt )
e
n
2(1∧2δ)
+ 2(1−2δ)+
1
+ ( C22 )2 ( 4(1−2δ)
)
e
1∧2δ t

× [33 (|x − y|4 + 2C(4, d)σt2 ) + 33+2µ (|x − y|4+2µ + 2C(4 + 2µ, d)σt2+µ )]
oi
+ t2 [33 (|x − y|4 + 2C(4, d)σt2 ) + 33+2ν (|x − y|4+2ν + 2C(4 + 2ν, d)σt2+ν )] ,

whence follows immediately the estimate in Claim 3.2.



Claim 3.3.
h
h
i
i
E σ E B [|vG (t, x, y; σ)|]p(σt , x − y) , E σ E B [|vR (t, x, y; σ)|]p(σt , x − y)
≤ const(δ, µ, ν, d)


2 h
P

j=1



j/2
C1j (tj + tjδ ) E σ [|x − y|j p(σt , x − y)] + E σ [σt p(σt , x − y)]

+ C2j tj(1∧2δ) E σ [|x − y|2j p(σt , x − y)] + E σ [σtj p(σt , x − y)]

j(1+ µ/2)
+ E σ [|x − y|j(2+µ) p(σt , x − y)] + E σ [σt
p(σt , x − y)]

+ C2j tj E σ [|x − y|2j p(σt , x − y)] + E σ [σtj p(σt , x − y)]
i
j(1+ ν/2)
+ E σ [|x − y|j(2+ν) p(σt , x − y)] + E σ [σt
p(σt , x − y)] .
Proof. Similarly to what is done in (3.29), (3.26) and (3.27), we have
|wG1 (t, x, y; σ)|e−rtV (x)
≤ C1 (1 +

1−δ
( 1−δ
(rt)−1+δ )
e )

Z

t
0

| σσst (y − x) + B0σt (σs )|ds,
18

(3.30)

|wG2 (t, x, y; σ)|e−rtV (x)
h
+ (1−2δ)+
≤ C2 ( (1−2δ)
)
(rt)−(1−2δ)+
e
Z t

| σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+µ ds
×
0

Z t
 i
| σσst (y − x) + B0σt (σs )|2 + | σσst (y − x) + B0σt (σs )|2+ν ds ,
+

(3.31)

0

σt ,y

|wR1 (t, x, y; σ)|e−rtV (B0,x
≤ C1 (1 +

(σt/2 ))

1−δ
( 1−δ
(rt)−1+δ )
e )

Z

t
0

σt ,y
σt ,y
|B0,x
(σs ) − B0,x
(σt/2 )|ds,

(3.32)

σt ,y

|wR2 (t, x, y; σ)|e−rtV (B0,x (σt/2 ))
h
+ (1−2δ)+
≤ C2 ( (1−2δ)
)
(rt)−(1−2δ)+
e
Z t

σt ,y
σt ,y
σt ,y
σt ,y
|B0,x
(σs ) − B0,x
(σt/2 )|2 + |B0,x
(σs ) − B0,x
(σt/2 )|2+µ ds
×
0

Z t
 i
σt ,y
σt ,y
σt ,y
σt ,y
|B0,x
+
(σs ) − B0,x
(σt/2 )|2 + |B0,x
(σs ) − B0,x
(σt/2 )|2+ν ds .

(3.33)

0

By (3.15), (3.16), (3.21) and (3.22), note that
|vG (t, x, y; σ)|

≤ |wG1 (t, x, y; σ)|e−tV (x) + |wG2 (t, x, y; σ)|e−tV (x)
Z 1 
2
θ |wG1 (t, x, y; σ)|e−θtV (x)/2 + |wG2 (t, x, y; σ)|e−θtV (x)/2 dθ,
+

(3.34)

0

|vR (t, x, y; σ)|

σt ,y
σt ,y
≤ |wR1 (t, x, y; σ)|e−tV (B0,x (σt/2 )) + |wR2 (t, x, y; σ)|e−tV (B0,x (σt/2 ))
Z 1 
σt ,y
+
θ |wR1 (t, x, y; σ)|e−θtV (B0,x (σt/2 ))/2

0

σt ,y

+ |wR2 (t, x, y; σ)|e−θtV (B0,x

(σt/2 ))/2

2

dθ.

(3.35)

Also note that for a > 0 and 0 ≤ θ1 , θ2 ≤ τ (τ > 0) (cf. (3.28))
i
h
τ,y
τ,y
E B |B0,x
(θ1 ) − B0,x
(θ2 )|a ≤ 3(a−1)+ (|x − y|a + 2C(a, d)τ a/2 ).

(3.36)

Collecting all the above yields the estimate in Claim 3.3 immediately.

We are now in a position to prove Theorem 2.1(iii). To do so, we need the following lemma.
R
Lemma 3.1. Let 1 ≤ p ≤ ∞. Then, for a, b ≥ 0 with C(a, d) = Rd |y|a p(1, y)dy,

Z


|f (y)| E σ [| · −y|a σtb p(σt , · − y)]dy
fa,b (t) :=

Rd

p

19



a/2 +b

≤ C(a, d)E σ [σt

] kf kp ,

f ∈ Lp (R d ).

Proof. For p = ∞, the described estimate is obvious. So let 1 ≤ p < ∞. First we note the
Minkowski inequality for integrals: If h(x, y) is a measurable function on a σ-finite product
measure space (X × Y, α(dx) × β(dy)), then
Z Z
Z Z
1/p
p
1/p
|h(x, y)|p β(dy)
α(dx).
|h(x, y)|α(dx) β(dy)
≤
Y

X

X

Note also that for c ≥ 0
Z



Rd

Y



|f (y)|| · −y|c p(τ, · − y)dy
≤ C(c, d)τ c/2 kf kp .
p

By these inequalities, the estimate is obtained as follows:
Z

i
h
Z




a b
|f (y)| E σ [| · −y| σt p(σt , · − y)]dy
≤ E σ
|f (y)|| · −y|a σtb p(σt , · − y)dy


Rd

Rd

p

≤

p

a/2 +b
C(a, d)E σ [σt
] kf kp .

Proof of Theorem 2.1(iii). By Claims 3.1, 3.2 with (3.7)
Z

kQK (t)f kp ≤
|f (y)| |E σ [E B [vK1 (t, ·, y; σ)]p(σt , · − y)]|dy
Rd
Z


+
|f (y)| E σ [E B [|vK2 (t, ·, y; σ)|]p(σt , · − y)]dy

Rd



p

h

≤ const(δ, µ, ν, d) C12 (t2 + t2δ )(f2,0 (t) + f0,1 (t))
+

2 n
X

C2j tj(1∧2δ) (f2j,0 (t) + f0,j (t) + fj(2+µ),0 (t) + f0,j(1+ µ/2) (t))

j=1

oi
+ C2j tj (f2j,0 (t) + f0,j (t) + fj(2+ν),0 (t) + f0,j(1+ ν/2) (t)) .

By Claim 3.3 with (3.8), (3.9)
Z



kQ G (t)f kp ≤
|f (y)| E σ [E B [|v G (t, ·, y; σ)|]p(σt , · − y)]dy
R

Rd

≤ const(δ, µ, ν, d)

R

p

2 h
X
C1j (tj + tjδ )(fj,0 (t) + f0,j/2 (t))
j=1

+ C2j tj(1∧2δ) (f2j,0 (t) + f0,j (t) + fj(2+µ),0 (t) + f0,j(1+ µ/2) (t))
i
+ C2j tj (f2j,0 (t) + f0,j (t) + fj(2+ν),0 (t) + f0,j(1+ ν/2) (t)) .

Combining these with Lemma 3.1 we have the assertion of Theorem 2.1(iii).

20



3.2. Case (A)0
In this subsection, we suppose condition (A)0 on V (x). In this case
|vK (t, x, y; σ)| ≤ |wK (t, x, y; σ)|
Z t
C1
| σσst (y − x) + B0σt (σs )|γ ds +
≤ 2
0

C1
2

Z

t

0

s
| σtσ−σ
(x − y) + B0σt (σs )|γ ds,
t

|vG (t, x, y; σ)| ≤ |wG (t, x, y; σ)|
Z t
≤ C1
| σσst (y − x) + B0σt (σs )|γ ds,
0

|vR (t, x, y; σ)| ≤ |wR (t, x, y; σ)|
Z t
σt ,y
σt ,y
|B0,x
(σs ) − B0,x
(σt/2 )|γ ds.
≤ C1
0

Here taking expectation E B , we have by (3.28) or (3.36),
E B [|vK (t, x, y; σ)|], E B [|vG (t, x, y; σ)|], E B [|vR (t, x, y; σ)|]
γ/2

≤ C1 t(|x − y|γ + 2C(γ, d)σt

)

and hence, by (3.7), (3.8) and (3.9)
|QK (t)f (x)|, |QG (t)f (x)|, |QR (t)f (x)|
h
i
nZ
|f (y)| E σ |x − y|γ p(σt , x − y) dy
≤ C1 t
Rd
Z
i o
h
γ/2
+ 2C(γ, d)
|f (y)| E σ σt p(σt , x − y) dy .

Rd

From this and Lemma 3.1 the assertion of Theorem 2.1(i) follows immediately.
4. Proof of Theorem 2.2
For notational simplicity we set H0 := H0ψ and H := H0 + V , in the following, so that K(t) =
e−tV /2 e−tH0 e−tV /2 , G(t) = e−tV e−tH0 and R(t) = e−tH0 /2 e−tV e−tH0 /2 .
4.1. Proof of Theorem 2.2 for K(t)
Since K(t) and e−sH are contractions, we have
kK( nt )n − e−tH kp→p = k
≤

n−1
X
k=0

n−1
X
k=0

K( nt )n−1−k (K( nt ) − e−tH/n )e−ktH/n kp→p

kK( nt ) − e−tH/n kp→p

= nkQK ( nt )kp→p .
Combined with the estimates for QK (t) in Theorem 2.1, the desired bound for K(t/n)n − e−tH
in Case (A)0 , (A)1 or (A)2 is obtained immediately.
21

4.2. Proof of Theorem 2.2 for G(t) and R(t) in Case (A)0
In the same way as above
kG( nt )n − e−tH kp→p ≤ nkQG ( nt )kp→p ,
kR( nt )n − e−tH kp→p ≤ nkQR ( nt )kp→p ,
from which together with Theorem 2.1(i), the desired bounds follow immediately.
4.3. Proof of Theorem 2.2 for G(t) and R(t) in Case (A)1 or (A)2
In this subsection we suppose that V (x) satisfies (A)1 or (A)2 .
We first observe that for t ≥ 0 and n ∈ N
1 n−1
G( nt )n − e−tH = e−tV /2n (K( n−1
− e−(n−1)tH/n )e−tV /2n e−tH0 /n
n t n−1 )

+ [e−tV /2n , e−(n−1)tH/n ]e−tV /2n e−tH0 /n + e−(n−1)tH/n QG ( nt ),
1 n−1
− e−(n−1)tH/n )e−tV /2n e−tH0 /2n
R( nt )n − e−tH = e−tH0 /2n e−tV /2n (K( n−1
n t n−1 )

+ e−tH0 /2n [e−tV /2n , e−(n−1)tH/n ]e−tV /2n e−tH0 /2n
+ [e−tH0 /2n , e−(n−1)tH/n ]e−tV /n e−tH0 /2n + e−(n−1)tH/n QR ( nt ),
where [A, B] = AB − BA. Hence
1 n−1
kG( nt )n − e−tH kp→p ≤ kK( n−1
− e−(n−1)tH/n kp→p
n t n−1 )

+ k[e−tV /2n , e−(n−1)tH/n ]kp→p + kQG ( nt )kp→p ,

(4.1)

1 n−1
kR( nt )n − e−tH kp→p ≤ kK( n−1
− e−(n−1)tH/n kp→p
n t n−1 )

+ k[e−tV /2n , e−(n−1)tH/n ]kp→p + k[e−tH0 /2n , e−(n−1)tH/n ]kp→p
+ kQR ( nt )kp→p .

(4.2)

As for the first term on the RHS of (4.1) and (4.2), we see by Theorem 2.2 which was proved in
Section 4.1
1 n−1
kK( n−1
− e−(n−1)tH/n kp→p
n t n−1 )

i

h 
2
P

j(1+κ)/2
2
t 2
t j
t 2δ

] ,
(C2 n ) (n − 1)E [σt/n
 const(δ, κ, d) C1 ( n ) + ( n ) (n − 1)E [σt/n ] +


j=1






in Case (A)1 ,




h 


2 n
≤
P
j
2
t j
t 2
t 2δ

const(δ,
µ,
ν,
d)
C
(C
]
E
[σ
]
+
(
(n
−
1)
(n − 1)E [σt/n
)
+
(
)
)
2

t/n
1
n
n
n


j=1




oi


j(1+ ν/2)
j(1+ µ/2)
j

+ (n − 1)E [σt/n
] + (C2 ( nt )1∧2δ )j (n − 1)E [σt/n
] + (n − 1)E [σt/n
]
,






in Case (A)2 .

22

As for the third term on the RHS of (4.1) and the fourth term of (4.2), we see by Theorem 2.1
kQG ( nt )kp→p , kQR ( nt )kp→p

o
2 n
P
j/2
j(1+κ)/2

j t j
t jδ
t j

C
((
)
+
(
)
)
)
const
(δ,
κ,
d)
]
+
(C
]
,
E
[σ
E
[σ

2n
1 n
n
t/n
t/n


j=1







2 n
P
j/2
C1j (( nt )j + ( nt )jδ )E [σt/n ]
const
(δ,
µ,
ν,
d)
≤

j=1



j(1+ µ/2)
j


+ (C2 ( nt )1∧2δ )j (E [σt/n
] + E [σt/n
])



o


j(1+ ν/2)

+ (C t )j (E [σ j ] + E [σ
]) ,
2n

t/n

t/n

in Case (A)1 ,

in Case (A)2 .

Therefore we need to estimate the middle terms of (4.1) and (4.2).
Claim 4.1. Let s ≥ 0 and t > 0. Then

k[e−sV , e−tH ]kp→p , k[e−sH0 , e−tH ]kp→p
h
i

−1+δ )E [σ 1/2 ] + C E [σ (1+κ)/2 ] ,

(1
+
t
const
(δ,
κ,
d)s
C
in Case (A)1 ,

1
2
t
t




h
1/2
1+ µ/2
≤
const (δ, µ, ν, d)s C1 (1 + t−1+δ )E [σt ] + C2 t−(1−2δ)+ (E [σt ] + E [σt
])



i


1+ ν/2

+ C2 (E [σt ] + E [σt
]) ,
in Case (A)2 .

Proof. First we estimate the Lp -operator norm of [e−sV , e−tH ]. We have (by (A.13)) that for
f ∈ C0 (R d )
[e−sV , e−tH ]f (x)
Z

i
h
 Z t
σt ,y
−sV (x)
−sV (y)
=
V (B0,x
(σr ))dr p(σt , x − y) dy.
f (y)(e
−e
)E exp −

Rd

0

Hence we have
|[e−sV , e−tH ]f (x)|
Z
h
 Z t

i
σt ,y
≤ s
V (B0,x
(σr ))dr p(σt , x − y) dy.
|f (y)|E |V (y) − V (x)| exp −

Rd

0

To estimate the integrand in (4.3), note by Taylor’s theorem that
V (y) − V (x) =

Z

t
0

+

σt ,y
h∇V (B0,x
(σr )), y − xi drt

Z

0

1

dθ

Z

0

t

σt ,y
h∇V (x + θ(y − x)) − ∇V (B0,x
(σr )), y − xi dr
t .

23

(4.3)

In Case (A)1 , it follows that
|V (y) − V (x)| ≤

Z

t

0

+

σt ,y
C1 (1 + V (B0,x
(σr ))1−δ ) drt |x − y|

Z

1

dθ

0

Z

t
0

C2 |( σσrt − θ)(y − x) + B0σt (σr )|κ dr |x−y|
t

Z t


σt ,y
−1+δ
( V (B0,x
(σr ))dr)1−δ |x − y|
≤ C1 1 + t
+ C2 1t

Z

0

1

dθ

Z

0

t

0

|( σσrt − θ)(y − x) + B0σt (σr )|κ dr|x − y|

(4.4)

where the last inequality is due to Jensen’s inequality. In Case (A)2
|V (y) − V (x)|
Z t
σt ,y
≤
C1 (1 + V (B0,x
(σr ))1−δ ) dr
t |x − y|
0

+

Z

1

0

Z

n
σt ,y
(σr ))(1−2δ)+ (1 + |( σσrt − θ)(y − x) + B0σt (σr )|µ )
C2 V (B0,x
0
o
+ 1 + |( σσrt − θ)(y − x) + B0σt (σr )|ν |( σσrt − θ)(y − x) + B0σt (σr )| drt |x − y|
dθ

t

Z t


σt ,y
−1+δ
≤ C1 1 + t
V (B0,x
(σr ))dr)1−δ |x − y|
(
0

+ C2 t

−(1−2δ)+

Z

0

t

V

σt ,y
(B0,x
(σr ))dr

(1−2δ)+ Z 1 
0

max |( σσt − θ)(y − x) + B0σt (σ)|

0≤σ≤σt


+ max |( σσt − θ)(y − x) + B0σt (σ)|1+µ dθ|x − y|
0≤σ≤σt

+ C2 1t

Z

Z t
|( σσrt − θ)(y − x) + B0σt (σr )|
0

σr
+ |( σt − θ)(y − x) + B0σt (σr )|1+ν dr|x − y|.

1

dθ

0

By (3.25), (4.4) and (4.5) imply the desired estimate:
 Z t

σt ,y
|V (y) − V (x)| exp −
V (B0,x
(σr ))dr
0

24

(4.5)
























1−δ t−1+δ )|x − y|
C1 (1 + ( 1−δ
e )
R
R
1
t
+ C2 1t 0 dθ 0 |( σσrt − θ)(y − x) + B0σt (σr )|κ dr|x − y|,

in Case (A)1 ,

1−δ t−1+δ )|x − y|
C1 (1 + ( 1−δ
e )

+ (1−2δ)+ −(1−2δ)+
+ C2 ( (1−2δ)
)
t
e

R
1
≤
× 0 max |( σσt − θ)(y − x) + B0σt (σ)|


0≤σ≤σt





σt
σ
1+µ dθ|x − y|

+
max
(σ)|
−
θ)(y
−
x)
+
B
|(

0

0≤σ≤σt σt



R
R 

1

 + C2 1t 0 dθ 0t |( σσrt − θ)(y − x) + B0σt (σr )|






σt
σr
1+ν

+ |( σt − θ)(y − x) + B0 (σr )|
dr|x − y|,
in Case (A)2 .

We take expectation E B in the above. This time we use the following moment estimate: For
a > 0, τ > 0, 0 ≤ θ ≤ 1 and z ∈ Rd
h
i
E B |( τt − θ)z + B0τ (t)|a ≤ 3(a−1)+ (|z|a + 2C(a, d)τ a/2 ),
h
i
e d)τ a/2 )
E B max |( τt − θ)z + B0τ (t)|a ≤ 3(a−1)+ (|z|a + 2C(a,
(4.6)
0≤t≤τ

e d) = E B [ max |B(t)|a ], and thereby we have
where C(a, d) = E B [|B(1)|a ] and C(a,
0≤t≤1

i
h
 Z t
σt ,y
V (B0,x
(σr ))dr
E B |V (y) − V (x)| exp −
0

≤


κ/2
1−δ −1+δ

C1 (1 + ( 1−δ
t
)|x − y| + C2 (|x − y|1+κ + 2C(κ, d)σt |x − y|),

e )





in Case (A)1 ,







1−δ 1−δ −1+δ

)|x − y|

 C1 (1 + ( e ) t

+ (1−2δ)+ −(1−2δ)+
(4.7)
)
t
+ C2 ( (1−2δ)
e






e + µ, d)σ (1+µ)/2 ) |x − y|
e d)σ 1/2 + 3µ (|x − y|1+µ + 2C(1

× |x − y| + 2C(1,
t
t







1/2
ν (|x − y|1+ν + 2C(1 + ν, d)σ (1+ν)/2 ) |x − y|,

+
3
|x
−
y|
+
2C(1,
d)σ
+
C

2
t
t





in Case (A)2 .

Hence follows the desired bound for [e−sV , e−tH ] by Lemma 3.1 with (4.3).
Next we estimate the Lp -operator norm of [e−sH0 , e−tH ].

First we suppose that V : R d → [0, ∞) is in C ∞ and all its derivatives have polynomial growth.
Then it is easily verified that (cf. Claim A.2 and its Remark)
(i) e−tH (S(R d )) ⊂ S(Rd ), in particular, e−tH0 (S(R d )) ⊂ S(R d ), and
T
T
ψ,V
(ii) S(R d ) ⊂
( ψ,V
( ψ,0
= ψ,0
− V on S(R d ).
p ) ∩
p ) and
p
p

G

1≤p≤∞

DG

1≤p≤∞

DG

G

G

G

Here ψ,V
(1 ≤ p < ∞) is the infinitesimal generator of {e−t(H0 +V ) } on Lp (R d ) and ψ,V
p
∞ the
one on C∞ (R d ). By these facts the following formula holds in Lp (R d ) (1 ≤ p < ∞) and C∞ (R d ):
25

For each f ∈ S(R d )
−sH0

[e

−tH

,e

]f =

Z

s

e−uH0 [V, e−tH ]e−(s−u)H0 f du.

0

Hence, taking Lp -norm in the above yields that for each f ∈ S(R d )
Z s
−sH0 −tH
k[e
k[V, e−tH ]e−(s−u)H0 f kp du.
,e
]f kp ≤

(4.8)

0

Now let V satisfy (A)1 or (A)2 . In this case V is not necessarily
smooth. So, take a nonnegative
R
h ∈ C0∞ with support in {x ∈ R d ; |x| ≤ 1} and Rd h(x)dx = 1. Set V ε = V ∗ hε with
hε (x) = (1/ε)d h(x/ε). Then V ε is in C ∞ (R d → [0, ∞)), and satisfies condition (A)1 or (A)2
with the same const’s as V does. Further, by (A)1 (i) or (A)2 (ii) all the derivatives of V ε have
polynomial growth. Hence, by (4.7) and Lemma 3.1 it holds that for g ∈ S(R d )
ε

k[V ε , e−t(H0 +V ) ]gkp
i
h

−1+δ )E [σ 1/2 ] + C E [σ (1+κ)/2 ] kgk ,

in Case (A)1 ,
const
(δ,
κ,
d)
C
(1
+
t

2
p
1
t
t




h
1/2
1+ µ/2
≤
const
(δ,
µ,
ν,
d)
C1 (1 + t−1+δ )E [σt ] + C2 t−(1−2δ)+ (E [σt ] + E [σt
])



i


1+ ν/2

]) kgkp ,
in Case (A)2 .
+ C2 (E [σt ] + E [σt

Since (4.8) holds with V = V ε , by combining this with the above we have
ε

k[e−sH0 , e−t(H0 +V ) ]f kp
h
i

−1+δ )E [σ 1/2 ] + C E [σ (1+κ)/2 ] kf k ,

const
(δ,
κ,
d)s
C
(1
+
t
in Case (A)1 ,

1
2
p
t
t




h
1/2
1+ µ/2
≤
const (δ, µ, ν, d)s C1 (1 + t−1+δ )E [σt ] + C2 t−(1−2δ)+ (E [σt ] + E [σt
])



i


1+ ν/2

+ C2 (E [σt ] + E [σt
]) kf kp ,
in Case (A)2 .

Finally let ε ↓ 0. Since V ε → V compact uniformly, we see by the Feynman-Kac formula
ε
ε
(A.6) that e−t(H0 +V ) f → e−t(H0 +V ) f boundedly pointwise, so that [e−sH0 , e−t(H0 +V ) ]f →
[e−sH0 , e−t(H0 +V ) ]f pointwise. Hence the desired bound for [e−sH0 , e−t(H0 +V ) ] follows immediately by the Fatou inequality.

We return to estimate G(t/n)n − e−tH and R(t/n)n − e−tH . By Claim 4.1

k[e−tV /2n , e−(n−1)tH/n ]kp→p , k[e−tH0 /2n , e−(n−1)tH/n ]kp→p
h
i

1
δ )E [σ 1/2 ] + C tE [σ (1+κ)/2 ] ,

C
in Case (A)1 ,
(t
+
t
const
(δ,
κ,
d)

2
1
t
t
n




h
1/2
1+ µ/2
≤
const (�