Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 1, Article 22, 2003
COMPARISON OF GREEN FUNCTIONS FOR GENERALIZED SCHRÖDINGER
OPERATORS ON C 1,1 -DOMAINS
LOTFI RIAHI
D EPARTMENT OF M ATHEMATICS ,
FACULTY OF S CIENCES OF T UNIS ,
C AMPUS U NIVERSITAIRE 1060
T UNIS , T UNISIA .
Lotfi.Riahi@fst.rnu.tn
Received 16 September, 2002; accepted 10 February, 2003
Communicated by A.M. Fink
1,1
A BSTRACT. We establish some inequalities on the 12 ∆-Green function G
on bounded C 1
domain. We use these inequalities to prove the existence of the 2 ∆ − µ -Green function Gµ
and its comparability to G, where µ is in some general class of signed Radon measures. Finally
we prove that the choice of this class is essentially optimal.
Key words and phrases: Green function, Schrödinger operator, 3G-Theorem.
2000 Mathematics Subject Classification. 34B27, 35J10.
1. I NTRODUCTION
The first aim of this paper is to prove some inequalities on the Green function G of 21 ∆ on
bounded C 1,1 -domain Ω in Rn , n ≥ 3, where ∆ is the Laplacian operator. In particular we
give an alternative and shorter proof of the 3G-Theorem established in [9] using long and sharp
discussions. The 3G-Theorem includes the usual one proved in [11], [4] and [3], which was
very useful to obtain some potential theoretic results. The second is to prove a comparison
theorem between the Green function G and the Green function Gµ of the Schrödinger operator
1
∆ − µ on Ω, where µ is allowed to be in some class of signed Radon measures. In contrast
2
to [9], there is no restriction on the sign of µ in this work. This comparison theorem is very
important in the sense that it enables us to deduce some potential theoretic results for 12 ∆ − µ
which are known to hold for 12 ∆. This is stated at the end of the paper. Moreover our result
covers the case of signed Radon measures with bounded Newtonian potentials i.e,
Z
1
sup
|µ|(dy) < +∞.
n−2
x∈Ω Ω |x − y|
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
100-02
2
L OTFI R IAHI
The Schrödinger operator 12 ∆ − f , with f belonging to the Kato class Knloc which is studied by
several authors (see [1], [3], [4], [11]) is just the special case where µ has the density f with
respect to the Lebesgue measure. In particular our results cover the ones proved by Zhao [11].
Finally we show that the choice of this class is essentially optimal.
Our paper is organized as follows.
In Section 2, we give some notations and recall some known results. In Section 3, we prove
some inequalities on the Green function of 12 ∆ on bounded C 1,1 -domain. A new and a shorter
proof of the 3G-Theorem established in [9] is given. In Section 4, we introduce a general class
of signed Radon measures on Ω denoted by K(Ω) that will be considered in this work. We give
some examples and we study some properties of this class. In Section 5, we prove a comparison
theorem between the Green functions of 12 ∆ and the Schrödinger operator 12 ∆−µ, where µ is in
the class K(Ω). We also show that when µ is nonnegative the condition µ ∈ K(Ω) is necessary
for the comparison theorem to hold.
Throughout the paper the letter C will denote a generic positive constant which may vary in
value from line to line.
2. P RELIMINARIES AND N OTATIONS
Throughout the paper Ω denotes a bounded C 1,1 -domain in Rn , n ≥ 3. This means that for
each z ∈ ∂Ω there exists a ball B(z, R0 ), R0 > 0 and a coordinate system of Rn such that in
these coordinates,
B(z, R0 ) ∩ Ω = B(z, R0 ) ∩ {(x0 , xn )/x0 ∈ Rn−1 , xn > f (x0 )},
and
B(z, R0 ) ∩ ∂Ω = B(z, R0 ) ∩ {(x0 , f (x0 ))/x0 ∈ Rn−1 },
where f is a C 1,1 -function.
∆ denotes the Laplacian operator on Rn and G
its Green function on Ω. For a signed Radon
1
measure µ on Ω, we denote by Gµ the 2 ∆ − µ -Green function on Ω, when it exists.
For x ∈ Ω let d(x) = d(x, ∂Ω), the distance from x to the boundary of Ω. We denote by d(Ω)
the diameter of Ω.
Since Ω is a bounded C 1,1 -domain, then it has the following geometrical property:
There exists r0 > 0 depending only on Ω such that for any z ∈ ∂Ω and 0 < r ≤ r0
there exist two balls B1z (r) and B2z (r) of radius r such that B1z (r) ⊂ Ω, B2z (r) ⊂ Rn \ Ω and
{z} = ∂B1z (r) ∩ ∂B2z (r).
We recall the following interesting estimates on the Green function G which are due to Grüter
and Widman [5], Zhao [11] and Hueber [6].
Theorem 2.1. There exists a constant C > 0 depending on the diameter of Ω, on the curvature
of ∂Ω and on the dimension n such that
d(x)d(y)
1
d(x)d(y)
1
−1
C min 1,
≤
G(x,
y)
≤
C
min
1,
.
|x − y|2 |x − y|n−2
|x − y|2 |x − y|n−2
for all x, y ∈ Ω.
3. I NEQUALITIES ON THE G REEN F UNCTION G
In this section we first give a new and a simple proof of the 3G-Theorem established in [9].
We also derive other inequalities on the Green function G that will be used in the next sections.
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C OMPARISON OF G REEN F UNCTIONS FOR G ENERALIZED S CHRÖDINGER O PERATORS ON C 1,1 -D OMAINS
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Theorem 3.1 (3G-Theorem). There exists a constant C = C(Ω, n) > 0 such that for x, y, z ∈
Ω, we have
d(z)
d(z)
G(x, z)G(z, y)
≤ C(
G(x, z) +
G(z, y)).
G(x, y)
d(x)
d(y)
Proof. The inequality of the theorem is equivalent to
1
1
1
≤C
+
.
(3.1)
d(z)G(x, y)
d(x)G(z, y) d(y)G(x, z)
On the other hand, since for a > 0, b > 0,
ab
ab
≤ min(a, b) ≤ 2
,
a+b
a+b
then
d(x)d(y)
d(x)d(y)
d(x)d(y)
≤
min
1,
,
≤
2
|x − y|2 + d(x)d(y)
|x − y|2
|x − y|2 + d(x)d(y)
and hence, from Theorem 2.1, we obtain
C −1 N (x, y) ≤ G(x, y) ≤ CN (x, y),
where
N (x, y) =
|x −
d(x)d(y)
.
− y|2 + d(x)d(y))
y|n−2 (|x
Therefore (3.1) is equivalent to
(3.2) |x − y|n−2 (|x − y|2 + d(x)d(y)) ≤ C(|z − y|n−2 (|z − y|2 + d(z)d(y))
+ |x − z|n−2 (|x − z|2 + d(x)d(z))).
Then, we shall prove (3.2). By symmetry we may assume that |x − z| ≤ |y − z|. We have
|x − y|n−2 ≤ (|x − z| + |z − y|)n−2
≤ 2n−2 |z − y|n−2 ,
(3.3)
and
|x − y|2 + d(x)d(y) ≤ (|x − z| + |z − y|)2 + (|x − z| + d(z))d(y)
≤ 4|z − y|2 + |z − y|d(y) + d(z)d(y).
(3.4)
If |z − y| ≤ d(z), then
|z − y|d(y) ≤ d(z)d(y).
(3.5)
If |z − y| ≥ d(z), then
|z − y|d(y) ≤ |z − y|(d(z) + |z − y|)
≤ 2|z − y|2 .
(3.6)
From (3.4), (3.5) and (3.6), we obtain
(3.7)
|x − y|2 + d(x)d(y) ≤ 6(|z − y|2 + d(y)d(z)).
From (3.3) and (3.7), we obtain
|x − y|n−2 (|x − y|2 + d(x)d(y)) ≤ 2n+1 |z − y|n−2 (|z − y|2 + d(z)d(y)).
This proves (3.2) with C = 2n+1 .
J. Inequal. Pure and Appl. Math., 4(1) Art. 22, 2003
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4
L OTFI R IAHI
Lemma 3.2. There exists a constant C = C(Ω, n) > 0 such that for all x, y ∈ Ω, we have
d(y)
C
G(x, y) ≤
.
d(x)
|x − y|n−2
Proof. By Theorem 2.1, we have
d(y)
C
G(x, y) ≤
min
d(x)
|x − y|n−2
d(y) d(y)2
,
d(x) |x − y|2
.
d(y)
d(x)
> 0. From the inequality |x − y| ≥ |d(y) − d(x)|, it follows
d(y) d(y)2
d(y)
d(y)2
min
,
≤ min
,
d(x) |x − y|2
d(x) |d(y) − d(x)|2
t2
= min t,
.
(t − 1)2
t2
Since min t, (t−1)
≤ 4, for all t > 0, then we obtain
2
Put t =
d(y)
4C
G(x, y) ≤
.
d(x)
|x − y|n−2
By symmetry we also have
d(x)
4C
G(x, y) ≤
.
d(y)
|x − y|n−2
This ends the proof.
The usual 3G-Theorem proved in [3, 4, 11] is well known under the following form which is
a simple consequence of Theorem 3.1 and Lemma 3.2.
Corollary 3.3. There exists a constant C = C(Ω, n) > 0 such that for x, y, z ∈ Ω, we have
G(x, z)G(z, y)
1
1
≤C
+
.
G(x, y)
|x − z|n−2 |z − y|n−2
4. T HE C LASS K (Ω)
Definition 4.1. Let µ be a signed Radon measure on Ω. We say that µ is in the class K(Ω) if it
satisfies
Z
d(y)
||µ|| ≡ sup
G(x, y)|µ|(dy) < +∞,
x∈Ω Ω d(x)
where |µ| is the total variation of µ.
In the following we study some properties of the class K(Ω) and to this end we first need to
prove the following lemma.
Lemma 4.1. For x, y ∈ Ω, we have
If d(x)d(y) ≥ |x − y|2 , then
1
d(y) ≤ d(x) ≤ 3d(y).
3
If d(x)d(y) ≤ |x − y|2 , then
max(d(x), d(y)) ≤ 2|x − y|.
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C OMPARISON OF G REEN F UNCTIONS FOR G ENERALIZED S CHRÖDINGER O PERATORS ON C 1,1 -D OMAINS
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Proof. If d(x)d(y) ≥ |x − y|2 , then in view of the inequality |x − y| ≥ |d(x) − d(y)|, we obtain
d(x)d(y) ≥ |d(x) − d(y)|2 ,
which implies
3 d(x)d(y) ≥ d(x)2 + d(y)2 ,
and then
1
d(y) ≤ d(x) ≤ 3 d(y).
3
If d(x)d(y) ≤ |x − y|2 , then in view of the inequality d(x) ≥ d(y) − |x − y|, we obtain
d(y)(d(y) − |x − y|) ≤ |x − y|2 ,
which gives
d(y)2 ≤ |x − y|2 + d(y)|x − y|
2
1
≤ |x − y| + d(y) .
2
The last inequality yields
1
d(y) ≤ |x − y|.
2
Similarly, we have
1
d(x) ≤ |x − y|.
2
The following proposition provides some interesting examples of measures in the class K(Ω).
Proposition 4.2. For α ∈ R, the measure
1
dy
d(y)α
is in the class K(Ω) if and only if α < 2.
Proof. We first assume α < 2 and we will prove that
Z
d(y)
1
sup
G(x, y)
dy < +∞.
d(y)α
x∈Ω Ω d(x)
By Theorem 2.1, we have
Z
Z
d(y)
1
1
1
d(y)2−α
(4.1) sup
G(x, y)
dy
≤
C
sup
min
,
dy.
d(y)α
d(x)d(y) |x − y|2 |x − y|n−2
x∈Ω Ω d(x)
x∈Ω Ω
On the other hand
Z
min
Ω
1
1
,
d(x)d(y) |x − y|2
Z
=
d(y)2−α
dy
|x − y|n−2
Ω∩(d(x)d(y)≥|x−y|2 )
(4.2)
Z
· · · dy +
· · · dy
Ω∩(d(x)d(y)≤|x−y|2 )
≡ I1 + I2 .
J. Inequal. Pure and Appl. Math., 4(1) Art. 22, 2003
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6
L OTFI R IAHI
We estimate I1 . From Lemma 4.1, we have
Z
d(y)1−α
I1 =
dy
n−2
Ω∩(d(x)d(y)≥|x−y|2 ) d(x)|x − y|
Z
1
−α
≤ Cd(x)
dy
√
n−2
|x−y|≤ 3 d(x) |x − y|
Z √3 d(x)
−α
≤ Cd(x)
rdr
0
2−α
≤ Cd(x)
(4.3)
≤ Cd(Ω)2−α .
Now we estimate I2 . From Lemma 4.1, we have
Z
d(y)2−α
I2 =
dy
n
Ω∩(d(x)d(y)≤|x−y|2 ) |x − y|
Z
1
2−α
≤2
dy
n−2+α
Ω |x − y|
Z d(Ω)
2−α
≤ 2 wn−1
r1−α dr
0
2−α
wn−1
d(Ω)2−α ,
2−α
where wn−1 is the area of the unit sphere Sn−1 in Rn .
Combining (4.1), (4.2), (4.3) and (4.4), we obtain
Z
1
d(y)
sup
G(x, y)
dy ≤ Cd(Ω)2−α < +∞.
α
d(x)
d(y)
x∈Ω Ω
(4.4)
2
=
Now we assume α ≥ 2 and we will prove that
Z
d(y)
1
sup
G(x, y)
dy = +∞.
d(y)α
x∈Ω Ω d(x)
√
We first remark that when d(x) ≤
5−1
|x
2
− y|, we have
√
5+1
|x − y|
d(y) ≤ d(x) + |x − y| ≤
2
and then d(x)d(y) ≤ |x − y|2 . By Theorem 2.1, we have
Z
Z
d(y)
1
1
1
d(y)2−α
−1
sup
G(x, y)
dy
≥
C
sup
min
,
dy
d(y)α
d(x)d(y) |x − y|2 |x − y|n−2
x∈Ω Ω d(x)
x∈Ω Ω
Z
d(y)2−α
−1
(4.5)
≥ C sup
dy.
√
n
x∈Ω Ω∩(d(x)≤ 5−1 |x−y|) |x − y|
2
Let z0 ∈ ∂Ω and put x0 the center of B1z0 (r0 ). This means B1z0 (r0 ) = B(x0 , r0 ) ⊂ Ω. For
x ∈]z0 , x0 ], we have
|y − x| ≤ |y − z0 | + d(x),
and
(
) (
)
√
√
3− 5
5−1
y ∈ D : d(x) ≤
|y − z0 | ⊂ y ∈ D : d(x) ≤
|y − x| .
2
2
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C OMPARISON OF G REEN F UNCTIONS FOR G ENERALIZED S CHRÖDINGER O PERATORS ON C 1,1 -D OMAINS
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Hence for x ∈]z0 , x0 ], we have
Z
Z
d(y)2−α
|y − z0 |2−α
(4.6)
dy
≥
dy.
√
√
n
n
Ω∩(d(x)≤ 5−1
|x−y|) |x − y|
Ω∩(d(x)≤ 3−2 5 |y−z0 |) (|y − z0 | + d(x))
2
From (4.5) and (4.6), we obtain
Z
Z
d(y)
1
1
−1
sup
G(x, y)
dy ≥ C
dy
α
n+α−2
d(y)
x∈Ω Ω d(x)
Ω |y − z0 |
Z
1
−1
≥C
dy
n+α−2
z
B1 0 (r0 ) |y − z0 |
Z
1
−1
=C
dy
n+α−2
|y−x0 |<r0 |y − z0 |
Z
1
−1
(4.7)
=C
dy,
n+α−2
|y|<r0 |y − ξ|
where ξ = z0 − x0 with |ξ| = r0 .
We take a spherical coordinate system (r, θ1 , . . . , θn−1 ) such that ξ = (|ξ|, 0, . . . , 0). Then,
we have
Z
Z r0
Z π
1
(sin θ1 )n−2
n−1
(4.8)
dy
=
w
r
dθ1 dr.
n−2
n+α
n+α−2
−1
|y|<r0 |y − ξ|
0
0 (r 2 + r 2 − 2rr0 cos θ1 ) 2
0
By making the change of variables t = tan
Z π
(sin θ1 )n−2
n+α
0
(r2 + r02 − 2rr0 cos θ1 ) 2
Z +∞
n−1
=2
θ1
,
2
−1
we obtain
dθ1
tn−2 (1 + t2 )
α−n
2
dt
n+α
((r + r0 )2 t2 + (r0 − r)2 ) 2 −1
α−n
2
r0 −r 2 2
n−2
Z
1 + ( r0 +r ) s
2n−1 (r0 + r)1−n +∞ s
=
ds
n+α
(r0 − r)α−1
(s2 + 1) 2 −1
0
k
≥
,
(r0 − r)α−1
0
where k = k(r0 , α, n) > 0.
This implies
Z r0
Z π
Z r0
(sin θ1 )n−2
rn−1
n−1
(4.9)
r
dθ
dr
≥
k
dr = +∞.
1
n+α
−1
2
(r0 − r)α−1
0
0 (r 2 + r0 − 2rr0 cos θ1 ) 2
0
From (4.7), (4.8) and (4.9), we obtain
Z
d(y)
1
sup
G(x, y)
dy = +∞.
d(y)α
x∈Ω Ω d(x)
This ends the proof.
Now we compare the class K(Ω) with the class of signed Radon measures with bounded
NewtonianR potentials. A signed Radon measure µ is said to be of bounded Newtonian potential
if supx∈Ω Ω |x−y|1 n−2 |µ|(dy) < +∞.
Proposition 4.3. The class K(Ω) properly contains the class of signed Radon measures with
bounded Newtonian potentials.
J. Inequal. Pure and Appl. Math., 4(1) Art. 22, 2003
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L OTFI R IAHI
Proof. From definitions and Lemma 3.2, it is clear that the class of signed Radon measures with
bounded Newtonian potentials is contained in K(Ω). In the sequel we will prove that for 1 ≤ α,
Z
1
dy = +∞
α
Ω d(y)
and then
Z
1
1
dy = +∞.
sup
n−2
d(y)α
x∈Ω Ω |x − y|
1
In particular for 1 ≤ α < 2, d(y)
α dy does not define a bounded Newtonian potential and by
1
Proposition 4.2, we know that d(y)α dy ∈ K(Ω).
Without loss of generality we assume that 0 ∈ ∂Ω. We know that there exists R0 > 0 such
that
B(0, R0 ) ∩ Ω = B(0, R0 ) ∩ {(x0 , xn )/x0 ∈ Rn−1 , xn > f (x0 )},
and
B(0, R0 ) ∩ ∂Ω = B(0, R0 ) ∩ {(x0 , f (x0 ))/x0 ∈ Rn−1 },
where f is a C 1,1 -function.
By the continuity of f , there exists ρ0 ∈ 0, R40 such that for |y 0 | < ρ0 , we have |f (y 0 )| < R20 .
Hence for all y = (y 0 , yn ) such that |y 0 | < ρ0 and 0 < yn − f (y 0 ) < R40 , we have (y 0 , f (y 0 )) ∈
∂Ω and y ∈ B(0, R0 ) ∩ Ω which give d(y) ≤ yn − f (y 0 ).
Using these observations we have
Z
Z
1
1
dy ≥
dy
α
α
Ω d(y)
Ω∩B(0,R0 ) d(y)
Z
Z
1
≥
dyn dy 0
R0 (y − f (y 0 ))α
0
0
n
|y |<ρ0 0<yn −f (y )< 4
Z
Z R0
4
1
=
dy 0
dr = +∞.
rα
|y 0 |<ρ0
0
We next prove that the Kato class Knloc is properly contained in K(Ω). For the reader’s
convenience we recall the definition of the Kato class Knloc .
Definition 4.2. A Borel measurable function f on Ω is in the Kato class Knloc if it satisfies
Z
|f (y)|
lim sup
dy = 0.
r→0 x∈Ω (|x−y|<r)∩Ω |x − y|n−2
Proposition 4.4. The class K(Ω) properly contains the Kato class Knloc .
Proof. Let f be in Knloc . We have
Z
lim sup
r→0 x∈Ω
(|x−y|<r)∩Ω
|f (y)|
dy = 0.
|x − y|n−2
Then, there exists r > 0 such that
Z
(4.10)
sup
x∈Ω
(|x−y|<r)∩Ω
|f (y)|
dy ≤ 1.
|x − y|n−2
This yields
Z
sup
x∈Ω
J. Inequal. Pure and Appl. Math., 4(1) Art. 22, 2003
|f (y)|dy ≤ rn−2 .
(|x−y|<r)∩Ω
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On the other hand, since Ω is a compact subset then there are x1 , . . . , xp ∈ Ω, p ∈ N∗ such that
Ω = ∪pi=1 B(xi , r) ∩ Ω. Hence the last inequality gives
Z
|f (y)|dy ≤ prn−2 .
Ω
It follows that
Z
sup
(4.11)
x∈Ω
(|x−y|≥r)∩Ω
From (4.10) and (4.11) we obtain
Z
sup
x∈Ω
Ω
|f (y)|
dy ≤ p.
|x − y|n−2
|f (y)|
dy ≤ p + 1 < +∞.
|x − y|n−2
This means that f (y)dy defines a bounded Newtonian potential and the result holds from Proposition 4.3.
5. T HE G REEN F UNCTION
FOR 12 ∆ − µ
In this section we prove that when µ ∈ K(Ω) the Green function Gµ of the Schrödinger
operator 12 ∆ − µ exists and it is comparable to G. We first prove the following result.
Theorem 5.1. There exists a constant C = C(Ω, n) > 0 such that for all µ ∈ K(Ω) and all
nonnegative superharmonic function h on Ω, we have
Z
G(x, y)h(y)|µ|(dy) ≤ C||µ||h(x),
Ω
for all x ∈ Ω.
Proof. By the 3G-Theorem, we have
Z
(5.1)
G(x, y)G(y, z)|µ|(dy) ≤ 2C||µ||G(x, z),
Ω
for all x, z ∈ Ω.
Now let h be a nonnegative superharmonic function on Ω; there is an increasing sequence
(hn )n of nonnegative measurable functions on Ω such that
Z
h(x) = sup G(x, z)hn (z)dz,
n
Ω
for all x ∈ Ω.
From (5.1), we have
Z Z
Z
G(x, y)G(y, z)|µ|(dy)hn (z)dz ≤ 2C||µ|| G(x, z)hn (z)dz,
Ω
Ω
Ω
for all x ∈ Ω.
By the Fubini’s theorem, we obtain
Z
Z
Z
G(x, y) G(y, z)hn (z)dz|µ|(dy) ≤ 2C||µ|| G(x, z)hn (z)dz,
Ω
Ω
Ω
for all x ∈ Ω.
When n tends to +∞, we obtain
Z
G(x, y)h(y)|µ|(dy) ≤ 2C||µ||h(x),
Ω
for all x ∈ Ω.
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L OTFI R IAHI
Corollary 5.2. Let µ ∈ K(Ω). Then
Z
sup
x∈Ω
G(x, y)|µ|(dy) < +∞.
Ω
Let µ be a signed Radon measure in the class K(Ω), i.e. ||µ|| < +∞. The Jordan decomposition into positive and negative parts says that µ = µ+ − µ− and |µ| = µ+ + µ− . From Corollary
5.2, the functions
Z
Z
+
x→
G(x, y)µ (dy) and x →
G(x, y)µ− (dy)
Ω
Ω
are two continuous potentials on Ω, and the real continuous function
Z
x→
G(x, y)µ(dy)
Ω
corresponds to the difference of these two potentials. Hence from the perturbed theory studied
in [2], it follows that there exists a Green function Gµ for the Schrödinger operator 21 ∆ − µ on
Ω satisfying the resolvent equation:
Z
G(x, y) = Gµ (x, y) + G(x, z)Gµ (z, y)dµ(z),
Ω
for all x, y ∈ Ω.
Our main result is the following.
Theorem 5.3. Assume that µ ∈ K(Ω) with ||µ|| sufficiently small. Then the Green functions G
and Gµ are comparable, i.e. there is a constant C = C(Ω, n, ||µ||) > 0 such that
C −1 G ≤ Gµ ≤ CG.
Proof. We have the resolvent equation:
Z
G(x, y) = Gµ (x, y) +
G(x, z)Gµ (z, y)dµ(z)
Ω
≡ Gµ (x, y) + G ∗ Gµ (x, y).
Then
Gµ = G − G ∗ Gµ .
By iteration we obtain
Gµ = G +
(5.2)
X
(−1)m G∗m+1 ,
m≥1
where
∗2
Z
G (x, y) ≡ G ∗ G(x, y) =
G(x, z)G(z, y)dµ(z),
Ω
and
G∗m+1 = G∗m ∗ G.
From the 3G-Theorem, we have
Z
1
G(x, z)G(z, y)|µ|(dz)
G(x, y) Ω
Z
Z
d(z)
d(z)
≤C
G(x, z)|µ|(dz) +
G(z, y)|µ|(dz)
Ω d(x)
Ω d(y)
≤ 2C||µ||.
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C OMPARISON OF G REEN F UNCTIONS FOR G ENERALIZED S CHRÖDINGER O PERATORS ON C 1,1 -D OMAINS
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In particular, we have
|G∗2 | ≤ 2C||µ||G.
By recurrence, we obtain
|G∗m+1 | ≤ (2C||µ||)m G.
(5.3)
When ||µ|| is sufficiently small so that 2C||µ|| < 21 , we obtain, from (5.2) and (5.3),
X
|Gµ − G| ≤
(2C||µ||)m G
m≥1
=
2C||µ||
G,
1 − 2C||µ||
which yields
1 − 4C||µ||
1 − 2C||µ||
G ≤ Gµ ≤
1
G.
1 − 2C||µ||
Recall that when µ is a nonnegative Radon measure, we know by [8] that the Green function
Gµ of 21 ∆ − µ exists and satisfies the resolvent equation:
Z
G(x, y) = Gµ (x, y) + G(x, z)Gµ (z, y)dµ(z),
Ω
for all x, y ∈ Ω.
Next we show that in this case, the condition µ ∈ K(Ω) is necessary and sufficient for the
comparability result.
Lemma 5.4. There exists a constant C = C(Ω, n) > 0 such that
Z
−1
C d(x) ≤
G(x, y)dy ≤ Cd(x),
Ω
for all x ∈ Ω.
Proof. From Theorem 2.1, we have
C
d(x)d(y)
G(x, y) ≤
min 1,
,
|x − y|n−2
|x − y|2
for all x, y ∈ Ω.
If d(y) ≤ 2|x − y|, then
(5.4)
G(x, y) ≤ 2C
d(x)
.
|x − y|n−1
If d(y) ≥ 2|x − y|, then d(x) ≥ d(y) − |x − y| ≥ |x − y|, which implies
(5.5)
G(x, y) ≤
C
d(x)
≤C
.
n−2
|x − y|
|x − y|n−1
Combining (5.4) and (5.5), we obtain
G(x, y) ≤ 2C
d(x)
,
|x − y|n−1
for all x, y ∈ Ω.
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12
L OTFI R IAHI
This yields
Z
Z
G(x, y)dy ≤ 2Cd(x)
Ω
ZΩ
1
dy
|x − y|n−1
≤ 2Cd(x)
0≤|x−y|≤d(Ω)
Z
= 2Cwn−1 d(x)
1
dy
|x − y|n−1
d(Ω)
dr
0
= C1 d(x).
From Theorem 2.1, we also have
d(x)d(y)
C −1
min 1,
≤ G(x, y),
|x − y|n−2
|x − y|2
for all x, y ∈ Ω.
This implies
d(x)d(y)
C −1
≤ G(x, y),
d(Ω)n
for all x, y ∈ Ω.
Hence
Z
Z
−1
−n
C d(Ω) d(x) d(y)dy ≤
G(x, y)dy,
Ω
which means
Ω
Z
C2 d(x) ≤
G(x, y)dy,
Ω
for all x ∈ Ω.
Theorem 5.5. Let µ be a nonnegative Radon measure. Then, the Green function Gµ of 21 ∆ − µ
on Ω is comparable to G if and only if µ ∈ K(Ω).
Proof. We have the integral equation:
Z
G(x, y) = Gµ (x, y) +
G(x, z)Gµ (z, y)dµ(z),
Ω
for all x, y ∈ Ω.
We first assume that Gµ and G are comparable which means that there exists a constant C ≥ 1
such that
C −1 G ≤ Gµ ≤ G.
Hence
Z
G(x, z)G(z, y)dµ(z) ≤ (C − 1)G(x, y),
Ω
for all x, y ∈ Ω.
This implies
Z Z
Z
G(x, z)G(z, y)dµ(z)dy ≤ (C − 1)
Ω
Ω
G(x, y)dy.
Ω
for all x ∈ Ω.
Using the Fubini’s theorem, it follows that
Z
Z
Z
G(x, z) G(z, y)dydµ(z) ≤ (C − 1) G(x, y)dy.
Ω
Ω
Ω
for all x ∈ Ω.
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C OMPARISON OF G REEN F UNCTIONS FOR G ENERALIZED S CHRÖDINGER O PERATORS ON C 1,1 -D OMAINS
13
From Lemma 5.4, we deduce that
Z
d(z)G(x, z)dµ(z) ≤ C 0 d(x),
Ω
for all x ∈ Ω.
This means that
Z
sup
x∈Ω
Ω
d(z)
G(x, z)dµ(z) ≤ C 0 ,
d(x)
and then µ ∈ K(Ω).
µ
Now let µ ∈ K(Ω), which means ||µ|| < +∞. By Theorem 5.3 the Green function G 8C||µ||
µ
of the Schrödinger operator ∆ − 8C||µ|| is comparable to G. This means that there exists C > 1
such that
µ
C −1 G ≤ G 8C||µ||
≤ G.
By Theorem 1 in [10], it follows that
C −8C||µ|| G ≤ Gµ ≤ G,
which ends the proof.
Remark 5.6. In view of the paper [7], our results hold also when we replace the Laplace operator by an elliptic operator
n
X
n
X ∂
∂2
+
bi
L=
ai,j
∂xi ∂xj
∂xi
i,j=1
i=1
which is uniformly elliptic with bounded Hölder continuous coefficients ai,j , bi .
Remark 5.7. The comparison theorem serves as a main tool
to obtain some potential-theoretic
results. For example it implies the equivalence of 12 ∆ − µ -potential and 12∆-potential of any
measure with support contained in Ω and then the equivalence of 21 ∆ − µ -capacity and 12 ∆capacity of any set in Ω. These equivalences say that the fine topology, polar sets, etc. are the
same for 12 ∆ and 12 ∆ − µ. Following
the argument in [7], the comparison theorem also implies
1
the equivalence of 2 ∆ − µ -harmonic measure and 12 ∆-harmonic measure on ∂Ω. This gives
rise to a boundary Harnack principle and a comparison theorem for nonnegative 21 ∆ − µ solutions and nonnegative 12 ∆-solutions vanishing continuously on a part of ∂Ω (see [4]).
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