Journal of Inequalities in Pure and
Applied Mathematics
COMPARISON OF GREEN FUNCTIONS FOR GENERALIZED
SCHRÖDINGER OPERATORS ON C 1,1 -DOMAINS
volume 4, issue 1, article 22,
2003.
LOTFI RIAHI
Department of Mathematics,
Faculty of Sciences of Tunis,
Campus Universitaire 1060
Tunis, TUNISIA.
EMail: Lotfi.Riahi@fst.rnu.tn
Received 16 September, 2002;
accepted 10 February, 2003.
Communicated by: A.M. Fink
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
We establish some inequalities on the 12 ∆-Green function G on bounded C 1,1domain. We use these inequalities to prove the existence of the 12 ∆ − 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.
2000 Mathematics Subject Classification: 34B27, 35J10.
Key words: Green function, Schrödinger operator, 3G-Theorem.
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
Lotfi Riahi
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Inequalities on the Green Function G . . . . . . . . . . . . . . . . . . . . 7
4
The Class K (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5
The Green Function for 12 ∆ − µ . . . . . . . . . . . . . . . . . . . . . . . . 21
References
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1.
Introduction
The first aim of this paper is to prove some inequalities on the Green function G
of 12 ∆ 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 12 ∆ − µ on Ω, where µ is allowed to be in some class of signed Radon
measures. In contrast 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
1
∆. This is stated at the end of the paper. Moreover our result covers the case
2
of signed Radon measures with bounded Newtonian potentials i.e,
Z
1
sup
|µ|(dy) < +∞.
n−2
x∈Ω Ω |x − y|
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
Comparison of Green Functions
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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.
Comparison of Green Functions
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2.
Preliminaries and Notations
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
1
signed Radon 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,
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on the curvature of ∂Ω and on the dimension n such that
d(x)d(y)
1
−1
C min 1,
≤ G(x, y)
2
|x − y|
|x − y|n−2
d(x)d(y)
1
≤ C min 1,
.
|x − y|2 |x − y|n−2
for all x, y ∈ Ω.
Comparison of Green Functions
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3.
Inequalities on the Green Function 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.
Theorem 3.1 (3G-Theorem). There exists a constant C = C(Ω, n) > 0 such
that for x, y, z ∈ Ω, we have
G(x, z)G(z, y)
d(z)
d(z)
≤ 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
(3.1)
≤C
+
.
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
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then
d(x)d(y)
d(x)d(y)
d(x)d(y)
≤ min 1,
≤2
,
2
2
|x − y| + d(x)d(y)
|x − y|
|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),
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where
N (x, y) =
|x −
Therefore (3.1) is equivalent to
d(x)d(y)
.
− y|2 + d(x)d(y))
y|n−2 (|x
(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
(3.3)
|x − y|n−2 ≤ (|x − z| + |z − y|)n−2
≤ 2n−2 |z − y|n−2 ,
and
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|x − y|2 + d(x)d(y) ≤ (|x − z| + |z − y|)2 + (|x − z| + d(z))d(y)
(3.4)
≤ 4|z − y|2 + |z − y|d(y) + d(z)d(y).
If |z − y| ≤ d(z), then
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|z − y|d(y) ≤ d(z)d(y).
(3.5)
If |z − y| ≥ d(z), then
(3.6)
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|z − y|d(y) ≤ |z − y|(d(z) + |z − y|)
≤ 2|z − y|2 .
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From (3.4), (3.5) and (3.6), we obtain
|x − y|2 + d(x)d(y) ≤ 6(|z − y|2 + d(y)d(z)).
(3.7)
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 .
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
d(y)
C
G(x, y) ≤
min
d(x)
|x − y|n−2
d(y)
d(x)
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Proof. By Theorem 2.1, we have
Put t =
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
d(y) d(y)2
,
d(x) |x − y|2
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.
> 0. From the inequality |x − y| ≥ |d(y) − d(x)|, it follows
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min
2
d(y) d(y)
,
d(x) |x − y|2
2
d(y)
d(y)
,
d(x) |d(y) − d(x)|2
t2
.
= min t,
(t − 1)2
≤ min
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t2
Since min t, (t−1)
≤ 4, for all t > 0, then we obtain
2
d(y)
4C
G(x, y) ≤
.
d(x)
|x − y|n−2
By symmetry we also have
4C
d(x)
G(x, y) ≤
.
d(y)
|x − y|n−2
This ends the proof.
Comparison of Green Functions
for Generalized Schrödinger
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Lotfi Riahi
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
1
1
G(x, z)G(z, y)
≤C
+
.
G(x, y)
|x − z|n−2 |z − y|n−2
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4.
The Class 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.
Comparison of Green Functions
for Generalized Schrödinger
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Lotfi Riahi
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
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max(d(x), d(y)) ≤ 2|x − y|.
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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 ,
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which implies
3 d(x)d(y) ≥ d(x)2 + d(y)2 ,
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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
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The last inequality yields
1
d(y) ≤ |x − y|.
2
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Similarly, we have
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d(x) ≤ |x − y|.
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The following proposition provides some interesting examples of measures
in the class K(Ω).
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Proposition 4.2. For α ∈ R, the measure
only if α < 2.
1
dy
d(y)α
is in the class K(Ω) if and
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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
(4.1)
sup
x∈Ω
Ω
1
d(y)
G(x, y)
dy
d(x)
d(y)α
Z
≤ C sup min
x∈Ω
Ω
1
1
,
d(x)d(y) |x − y|2
d(y)2−α
dy.
|x − y|n−2
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
Lotfi Riahi
On the other hand
Z
1
1
d(y)2−α
min
,
dy
d(x)d(y) |x − y|2 |x − y|n−2
Ω
Z
Z
=
· · · dy +
Ω∩(d(x)d(y)≥|x−y|2 )
(4.2)
Ω∩(d(x)d(y)≤|x−y|2 )
≡ I1 + I2 .
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|
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· · · dy
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√
−α
Z
3 d(x)
≤ Cd(x)
rdr
0
(4.3)
≤ Cd(x)2−α
≤ 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
Comparison of Green Functions
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2−α
(4.4)
=
2
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
d(y)
1
G(x, y)
dy ≤ Cd(Ω)2−α < +∞.
sup
α
d(y)
x∈Ω Ω d(x)
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)
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√
We first remark that when d(x) ≤
5−1
|x
2
− y|, we have
√
d(y) ≤ d(x) + |x − y| ≤
5+1
|x − y|
2
and then d(x)d(y) ≤ |x − y|2 . By Theorem 2.1, we have
Z
d(y)
1
sup
G(x, y)
dy
d(y)α
x∈Ω Ω d(x)
Z
1
1
d(y)2−α
−1
≥ C sup min
,
dy
d(x)d(y) |x − y|2 |x − y|n−2
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
∂Ω and put x0 the center of B1z0 (r0 ).
Let z0 ∈
Ω. For x ∈]z0 , x0 ], we have
This means B1z0 (r0 )
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= B(x0 , r0 ) ⊂
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|y − x| ≤ |y − z0 | + d(x),
and
(
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√
3− 5
y ∈ D : d(x) ≤
|y − z0 |
2
)
(
⊂
y ∈ D : d(x) ≤
√
)
5−1
|y − x| .
2
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Hence for x ∈]z0 , x0 ], we have
Z
d(y)2−α
(4.6)
dy
√
n
|x
−
y|
Ω∩(d(x)≤ 5−1
|x−y|)
2
Z
≥
√
Ω∩(d(x)≤ 3−2
5
|y−z0 |)
|y − z0 |2−α
dy.
(|y − z0 | + d(x))n
From (4.5) and (4.6), we obtain
Z
Z
d(y)
1
1
−1
sup
G(x, y)
dy ≥ C
dy
α
d(y)
|y − z0 |n+α−2
x∈Ω Ω d(x)
ZΩ
1
≥ C −1
dy
n+α−2
z0
B1 (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
1
dy
(4.8)
n+α−2
|y|<r0 |y − ξ|
Z r0
Z π
(sin θ1 )n−2
n−1
= wn−2
r
dθ1 dr.
n+α
−1
0
0 (r 2 + r 2 − 2rr0 cos θ1 ) 2
0
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By making the change of variables t = tan θ21 , we obtain
Z π
(sin θ1 )n−2
dθ1
n+α
−1
0 (r 2 + r 2 − 2rr0 cos θ1 ) 2
0
Z +∞
α−n
tn−2 (1 + t2 ) 2
n−1
=2
dt
n+α
0
((r + r0 )2 t2 + (r0 − r)2 ) 2 −1
α−n
2
r0 −r 2 2
n−2
1
+
(
)
s
n−1
1−n Z +∞ s
r0 +r
2 (r0 + r)
ds
=
n+α
(r0 − r)α−1
(s2 + 1) 2 −1
0
k
≥
,
(r0 − r)α−1
where k = k(r0 , α, n) > 0.
This implies
Z
(4.9)
0
r0
rn−1
Z
0
Contents
(sin θ1 )n−2
+
r02
− 2rr0 cos θ1 )
dθ1 dr
n+α
−1
2
Z
≥k
0
r0
rn−1
dr = +∞.
(r0 − r)α−1
From (4.7), (4.8) and (4.9), we obtain
Z
d(y)
1
G(x, y)
dy = +∞.
sup
d(y)α
x∈Ω Ω d(x)
This ends the proof.
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(r2
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Now we compare the class K(Ω) with the class of signed Radon measures
with bounded Newtonian potentials. A signed
Radon measure µ is said to be of
R
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.
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
sup
x∈Ω
Ω
1
1
dy = +∞.
|x − y|n−2 d(y)α
1
In particular for 1 ≤ α < 2, d(y)
α dy does not define a bounded Newtonian
1
potential and by 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 },
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
Lotfi Riahi
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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
0 ))α
R
(y
−
f
(y
0
0
0
n
|y |<ρ0 0<yn −f (y )< 4
Z
Z R0
4
1
0
dr = +∞.
=
dy
rα
|y 0 |<ρ0
0
Comparison of Green Functions
for Generalized Schrödinger
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Lotfi Riahi
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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 .
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Proof. Let f be in Knloc . We have
Z
lim sup
r→0 x∈Ω
(|x−y|<r)∩Ω
Then, there exists r > 0 such that
Z
(4.10)
sup
x∈Ω
(|x−y|<r)∩Ω
|f (y)|
dy = 0.
|x − y|n−2
|f (y)|
dy ≤ 1.
|x − y|n−2
This yields
Z
|f (y)|dy ≤ r
sup
x∈Ω
n−2
.
(|x−y|<r)∩Ω
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
(4.11)
sup
x∈Ω
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
(|x−y|≥r)∩Ω
|f (y)|
dy ≤ p.
|x − y|n−2
From (4.10) and (4.11) we obtain
Z
|f (y)|
dy ≤ p + 1 < +∞.
sup
n−2
x∈Ω Ω |x − y|
This means that f (y)dy defines a bounded Newtonian potential and the result
holds from Proposition 4.3.
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5.
The Green Function 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
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
for all x ∈ Ω.
Lotfi Riahi
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
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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
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
for all x ∈ Ω.
Lotfi Riahi
Corollary 5.2. Let µ ∈ K(Ω). Then
Z
sup G(x, y)|µ|(dy) < +∞.
x∈Ω
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
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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
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Gµ = G − G ∗ Gµ .
By iteration we obtain
(5.2)
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Gµ = G +
X
m≥1
(−1)m G∗m+1 ,
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where
Z
∗2
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(y)
Ω d(x)
≤ 2C||µ||.
Comparison of Green Functions
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Operators on C 1,1 -Domains
Lotfi Riahi
Title Page
In particular, we have
∗2
|G | ≤ 2C||µ||G.
By recurrence, we obtain
(5.3)
|G
∗m+1
m
| ≤ (2C||µ||) G.
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When ||µ|| is sufficiently small so that 2C||µ|| < 12 , we obtain, from (5.2) and
(5.3),
X
|Gµ − G| ≤
(2C||µ||)m G
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2C||µ||
=
G,
1 − 2C||µ||
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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 12 ∆ − 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
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J
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
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for all x, y ∈ Ω.
If d(y) ≤ 2|x − y|, then
(5.4)
Comparison of Green Functions
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Page 25 of 31
G(x, y) ≤ 2C
d(x)
.
|x − y|n−1
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If d(y) ≥ 2|x − y|, then d(x) ≥ d(y) − |x − y| ≥ |x − y|, which implies
G(x, y) ≤
(5.5)
d(x)
C
≤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
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
for all x, y ∈ Ω.
This yields
Z
Lotfi Riahi
Z
G(x, y)dy ≤ 2Cd(x)
Ω
ZΩ
1
dy
|x − y|n−1
≤ 2Cd(x)
0≤|x−y|≤d(Ω)
Z d(Ω)
= 2Cwn−1 d(x)
1
dy
|x − y|n−1
dr
0
= C1 d(x).
From Theorem 2.1, we also have
C −1
d(x)d(y)
min 1,
≤ G(x, y),
|x − y|n−2
|x − y|2
for all x, y ∈ Ω.
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This implies
C −1
d(x)d(y)
≤ G(x, y),
d(Ω)n
for all x, y ∈ Ω.
Hence
C
−1
−n
d(Ω)
Z
Z
d(y)dy ≤
d(x)
Ω
which means
G(x, y)dy,
Ω
Z
C2 d(x) ≤
G(x, y)dy,
Ω
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
for all x ∈ Ω.
Lotfi Riahi
Theorem 5.5. Let µ be a nonnegative Radon measure. Then, the Green function
Gµ of 12 ∆ − µ on Ω is comparable to G if and only if µ ∈ K(Ω).
Title Page
Proof. We have the integral equation:
Contents
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
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Z
G(x, z)G(z, y)dµ(z) ≤ (C − 1)G(x, y),
Ω
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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.
From Lemma 5.4, we deduce that
Z
d(z)G(x, z)dµ(z) ≤ C 0 d(x),
Ω
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for all x ∈ Ω.
This means that
Z
sup
x∈Ω
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
Ω
d(z)
G(x, z)dµ(z) ≤ C 0 ,
d(x)
and then µ ∈ K(Ω).
Now let µ ∈ K(Ω), which means ||µ|| < +∞. By Theorem 5.3 the Green
µ
µ
of the Schrödinger operator ∆ − 8C||µ||
is comparable to G.
function G 8C||µ||
This means that there exists C > 1 such that
µ
C −1 G ≤ G 8C||µ||
≤ G.
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By Theorem 1 in [10], it follows that
C −8C||µ|| G ≤ Gµ ≤ G,
which ends the proof.
Remark 5.1. 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
L=
ai,j
+
bi
∂x
∂x
∂xi
i
j
i,j=1
i=1
Comparison of Green Functions
for Generalized Schrödinger
Operators on C 1,1 -Domains
Lotfi Riahi
which is uniformly elliptic with bounded Hölder continuous coefficients ai,j , bi .
Remark 5.2. 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 12 ∆ − 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
1
∆ − µ. Following the argument
in [7], the comparison theorem also implies
2
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 21 ∆-solutions vanishing
continuously on a part of ∂Ω (see [4]).
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References
[1] A. AIZENMAN AND B. SIMON, Brownian motion and Harnack’s inequality for Schrödinger’s operators, Comm. Pure Appl. Math., 35 (1982),
209–273.
[2] A. BOUKRICHA, W. HANSEN AND H. HUEBER, Continuous solutions
of the generalized Schrödinger equation and perturbation of harmonic
spaces, Expositiones Math., 5 (1987), 97–135.
[3] K.L. CHUNG AND Z. ZHAO, From Brownian motion to Schrödinger’s
equation, Springer Verlag, New York, 1995.
[4] M. CRANSTON, E.B. FABES AND Z. ZHAO, Conditional gauge and
potential theory for the Schrödinger operator, Trans. Amer. Math. Soc.,
307(1) (1988), 171–194.
[5] M. GRÜTER AND K.O. WIDMAN, The Green function for uniformly
elliptic equations, Manuscripta Math., 37 (1982), 303–342.
[6] H. HUEBER, A uniform estimate for the Green functions on C
domains, Bibos publication Universität Bielefeld, (1986).
1,1
-
[7] H. HUEBER AND M. SIEVEKING, Uniform bounds for quotients of
Green functions on C 1,1 -domains, Ann. Inst. Fourier, 32(1) (1982) 105–
117.
[8] H. MAAGLI AND M. SELMI, Perturbation des résolvantes et des semigroupes par une mesure de Radon positive, Mathematische Zeitschrift, 205
(1990), 379–393.
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[9] M. SELMI, Comparaison des noyaux de Green sur les domaines C 1,1 ,
Revue Roumaine de Mathématiques Pures et Appliquées, 36 (1991), 91–
100.
[10] M. SELMI, Critere de comparaison de certains noyaux de Green, Lecture
Notes in Math., 1235 (1987), 172–193.
[11] Z. ZHAO, Green function for Schrödinger operator and condioned Feynmann Kac Gauge, Jour. Math. Anal. Appl., 116 (1986), 309–334.
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Operators on C 1,1 -Domains
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