ON THE ITERATED GREEN FUNCTIONS ON A
BOUNDED DOMAIN AND THEIR RELATED KATO
CLASS OF POTENTIALS
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
HABIB MÂAGLI AND NOUREDDINE ZEDDINI
Département de Mathématiques,
Faculté des Sciences de Tunis,
Campus Universitaire, 2092 Tunis, Tunisia.
EMail: habib.maagli@fst.rnu.tn and noureddine.zeddini@ipein.rnu.tn
Received:
30 May, 2006
Accepted:
18 February, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
34B27.
Key words:
Green function, Gauss semigroup, Kato class.
Abstract:
We use the results of Zhang [15, 16] and Davies [7] on the behavior of
the heat kernel p(t, x, y) on a bounded C 1,1 domain D to find again
the result of Grunau-Sweers [9] concerning the estimates of the iterated
Greens functions Gm,n (D). Next, we use these estimates to characterize, by means of p(t, x, y), the Kato class Km,n (D) and we give new
examples of functions belonging to this class.
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Contents
1
Introduction
3
2
Proof of Theorem 1.1
9
3
The Kato Class Km,n (D)
18
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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1.
Introduction
Let D be a bounded C 1,1 domain in Rn , n ≥ 3 and p (t, x, y) be the density of the
Gauss semigroup on D. Combining the results of Zhang [15], [16] and those of
Davies or Davies-Simon [7], [8] a qualitatively sharp understanding of the boundary
behaviour of p (t, x, y) is given as follows: There exist positive constants c1 , c2 and
λ0 depending only on D such that for all t > 0 and x , y ∈ D,
(1.1)
|x−y|2
−λ0 t−c2 t
δ(y)
c1 e
δ(x)
√
√
∧1
∧1
n
t2
t∧1
t∧1
2
−λ0 t− |x−y|
c2 t
δ(y)
e
δ(x)
√
≤ p (t, x, y) ≤ √
∧1
∧1
,
n
c1 t 2
t∧1
t∧1
where δ(x) denotes the Euclidean distance from x to the boundary of D.
Let G(x, y) be the Green’s function of the laplacien ∆ in D with a Dirichlet
condition on ∂D. Then G is given by
Z ∞
(1.2)
G(x, y) =
p (t, x, y) dt, for x , y ∈ D.
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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0
For a positive integer m, we denote by Gm,n the Green’s function of the operator
u 7→ (−∆)m u on D with Navier boundary conditions ∆j u = 0 on ∂D for 0 ≤ j ≤
m − 1. Then G1,n = G and Gm,n satisfies for m ≥ 2
Z Z
Gm,n (x, y) =
G(x, z)Gm−1,n (z, y) dz.
D
D
Using the Fubini theorem and the Chapman-Kolmogorov identity, we show by in-
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duction that for each m ≥ 1 and x , y ∈ D we have
Z ∞
1
(1.3)
Gm,n (x, y) =
tm−1 p (t, x, y) dt.
(m − 1)! 0
In this paper we will use (1.1) and (1.3) to find again the result of Grunau and Sweers
in [9] concerning the sharp estimates of Gm,n . More precisely we will give another
proof for the case n ≥ 3 of the following theorem.
Theorem 1.1 ( see [9]). On D2 we have

δ(x)δ(y)
1

min
1
,
if n > 2m,
 |x−y|n−2m
|x−y|2






if n = 2m,
Log 1 + δ(x)δ(y)

|x−y|2

√

p
δ(x)δ(y)
Gm,n (x, y) ∼ Hm,n (x, y) =
δ(x)δ(y) min 1 , |x−y|
if n = 2m − 1,





1

δ(x)δ(y) Log 2 + |x−y|2 +δ(x)δ(y)
if n = 2m − 2,





δ(x)δ(y)
if n < 2m − 2,
where the symbol ∼ is defined in the notations below.
As a second step we will also use (1.1) and (1.3) to give new contributions in the
case n > 2m to the study of the Kato class Km,n (D) defined in [11] for m = 1 and
in [2] for m ≥ 2 as follows.
Definition 1.1. A Borel measurable function q in D belongs to the Kato class Km,n (D)
if q satisfies the following condition
Z
δ(y)
Gm,n (x, y)|q(y)| dy = 0.
(1.4)
lim sup
α→0 x∈D D∩(|x−y|≤α) δ(x)
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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We note that in the case m = 1, the class K1,n (D) properly contains the classical
Kato class Kn (D) introduced in [1] as the natural class of singular functions which
replaces the Lp -Lebesgue spaces in order that the weak solutions of the Shrödinger
equation are continuous and satisfy a Harnack principle. More precisely, it is shown
in [11] that the function ρα (y) = δα1(y) belongs to K1,n (D) if and only if α < 2 but
for 1 ≤ α < 2, ρα ∈
/ Kn (D).
Our second contribution here is to exploit estimates of Theorem 1.1 on the one
hand, to give new examples of functions belonging to the class Km,n (D) and to
characterize this class by means of the density of the Gauss semigroup in D on the
other hand. In particular we will prove the following results for the unit ball.
Proposition 1.2. For λ , µ ∈ R and y ∈ B(0, 1) we put
ρλ, µ (y) =
vol. 8, iss. 1, art. 26, 2007
1
Contents
h
For m ≥ 2 we have
λ < 3 or (λ = 3 and µ > 1).
Theorem 1.3. Let n > 2m and q be a Borel measurable function in D. Then the
following assertions are equivalent:
1) q ∈ Km,n (B(0, 1))
RtR
2) limt→0 supx∈B 0 B
Habib Mâagli and Noureddine Zeddini
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iµ .
2
(1 − |y|)λ Log( 1−|y|
)
ρλ, µ ∈ Km,n (B(0, 1)) if and only if
Iterated Green Functions
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δ(y)
δ(x)
sm−1 p(s, x, y)|q(y)| dyds = 0
We also note that in the case m = 1, similar characterizations have been obtained
by Aizenman and Simon in [1] for the Kato class Kn (Rn ) and by Bachar and Mâagli
in [4] for the half space Rn+ , where they introduce a new Kato class that properly
contains the classical one. This was extended for m ≥ 2 by Mâagli and Zribi [12]
to the class Km,n (Rn ) and by Bachar [3] to the class Km,n (Rn+ ). The density of the
Gauss semigroup in the case of Rn and Rn+ are explicitly known, but this is not the
case for a bounded C 1,1 domain even if D is an open ball .
In order to simplify our statements, we define some convenient notations.
Notations.
i) For x, y ∈ D, we denote by δ(x) the Euclidean distance from x to the boundary
of D, [x, y]2 = |x − y|2 + δ(x)δ(y) and d is the diameter of D.
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
ii) For a, b ∈ R, we denote by a ∧ b = min(a, b) and a ∨ b = max(a, b).
iii) Let f and g be two nonnegative functions on a set S.
We say that f g, if there exists c > 0 such that
f (x) ≤ c g(x)
for all x ∈ S.
We say that f ∼ g, if there exists C > 0 such that
1
g(x) ≤ f (x) ≤ Cg(x)
C
for all x ∈ S.
The following properties will be used several times
iv) For a, b ≥ 0, we have
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(1.5)
ab
ab
≤ min(a, b) ≤ 2
a+b
a+b
(1.6)
(a + b)p ∼ ap + bp for p ∈ R+ .
(1.7)
min(1, a) min(1, b) ≤ min(1, a b) ≤ min(1, a) max(1, b)
(1.8)
a
≤ Log(1 + a)
1+a
Iterated Green Functions
v) Let η, ν > 0 and 0 < γ ≤ 1. Then we have
Habib Mâagli and Noureddine Zeddini
γ
(1.9)
Log(1 + t) t , for t ≥ 0.
(1.10)
Log(1 + η t) ∼ Log(1 + ν t) , for t ≥ 0.
vol. 8, iss. 1, art. 26, 2007
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Finally we note that since for each a ≥ b ≥ 0 and c > 0 we have
(a + 1)(b + 1) −c (b−a)2
a+b
2
e
= 1+
e−c (b−a)
1 + ab
1 + ab
2a + ξ
2
e−c ξ
= 1+
1 + a(a + ξ)
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2
≤ (2 + ξ)e−c ξ ≤ C.
Then, using (1.5) we deduce that for each x , y ∈ D and 0 < t ≤ 1 we have
|δ(x)−δ(y)|2
δ(x)
δ(x)δ(y)
δ(y)
t
min
, 1 ≤ C min √ , 1 min √ , 1 ec
t
t
t
|x−y|2
δ(x)
δ(y)
≤ C min √ , 1 min √ , 1 ec t .
t
t
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So, using this fact, (1.7) and the fact that D is bounded we deduce that estimates
(1.1) can be written as follows:
There exist positive constants c, C and λ such that
(1.11)
1
h 1 (t, x, y) ≤ p (t, x, y) ≤ C hc,λ (t, x, y),
C c,λ
where
(1.12)
Iterated Green Functions

2
 min δ(x)δ(y) , 1 e−c |x−y|
t
,
n
t
t2
hc,λ (t, x, y) :=

δ(x)δ(y)e−λt ,
if 0 < t ≤ 1
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
if t > 1.
Throughout the paper, the letter C will denote a generic positive constant which may
vary from line to line.
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2.
Proof of Theorem 1.1
First we need the following lemma.
Lemma 2.1. For each x, y ∈ D we have
a) For n ≥ 2m
δ(x)δ(y)
δ(x)δ(y) ≤ min 1 ,
|x − y|2
dn−2m+2
.
|x − y|n−2m
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
b)
δ(x)δ(y)
δ(x)δ(y) ≤ d min 1 ,
|x − y|2
2
δ(x)δ(y)
≤ 2d Log 1 +
|x − y|2
2
.
Now we will give the proof of Theorem 1.1. More precisely, using (1.3) and
(1.11) we will prove that for each c > 0, we have
Z ∞
tm−1 hc,λ (t, x, y)dt ∼ Hm,n (x, y) .
0
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Without loss of generality we will assume that λ = 1 , c = 1 and denote by
h1,1 (t, x, y) = h(t, x, y). Hence, using a change of variable, we obtain
Z ∞
tm−1 h(t, x, y)dt
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0
− |x−y|2
1
δ(x)δ(y)
e t
= C δ(x)δ(y) +
tm−1 min
,1
dt
n
t
t2
0
Z ∞
n
δ(x)δ(y)
2m−n
−m−1
r, 1 e− r dr.
= C δ(x)δ(y) + |x − y|
r2
min
2
|x
−
y|
2
|x−y|
Z
n
Since we will sometimes omit e−r and we need to integrate the functions r → r 2 −m−1
n
and r → r 2 −m near zero or near ∞, we will discuss the following cases
Case 1. n > 2m. Using (1.7) we obtain
δ(x)δ(y)
δ(x)δ(y)
min
, 1 min(r, 1) ≤ min
r, 1
|x − y|2
|x − y|2
δ(x)δ(y)
, 1 max(r , 1)).
≤ min
|x − y|2
Hence the lower bound follows from the fact that
Z ∞
Z ∞
n
n
−m−1
−
r
e dr ≥
min(1, r)r 2 −m−1 e− r dr = C
min(1, r)r 2
d2
|x−y|2
and the upper bound follows from Lemma 2.1.
Case 2. n = 2m. In this case
Z ∞
Z ∞
m−1
t
h(t, x, y)dt = C δ(x)δ(y) +
|x−y|2
0
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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min
δ(x)δ(y) 1
,
|x − y|2 r
e− r dr.
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So using (1.5) and the fact that
|x − y|2 + (2d2 + 1)δ(x)δ(y)
≥ |x − y|2 + δ(x)δ(y) ,
1 + δ(x)δ(y)
we obtain
Z
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∞
m−1
t
0
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Z
2d2 +1
δ(x)δ(y)
dr
2
|x−y|2 |x − y| + rδ(x)δ(y)
|x − y|2 + (2d2 + 1)δ(x)δ(y)
= C Log
|x − y|2 (1 + δ(x)δ(y))
h(t, x, y)dt ≥
δ(x)δ(y)
≥ C Log 1 +
|x − y|2
.
To prove the upper inequality we use (1.5), (1.11) and (1.10) to obtain
Z ∞
δ(x)δ(y) 1 − r
min
,
e dr
|x − y|2 r
|x−y|2
Z ∞
δ(x)δ(y)
≤C
e− r dr
2
|x−y|2 δ(x)δ(y)r + |x − y|
Z d2 +1
Z
δ(x)δ(y)
δ(x)δ(y) ∞ − r
≤C
dr + C
e dr
2
[x, y]2 1+d2
|x−y|2 δ(x)δ(y)r + |x − y|
|x − y|2 + (d2 + 1)δ(x)δ(y)
δ(x)δ(y)
= C Log
+
C
|x − y|2 (1 + δ(x)δ(y))
[x, y]2
2
(d + 1)δ(x)δ(y)
δ(x)δ(y)
≤ C Log 1 +
+C
2
|x − y| (1 + δ(x)δ(y))
|x, y|2 + δ(x)δ(y)
δ(x)δ(y)
≤ C Log 1 +
.
|x − y|2
Hence the result follows from Lemma 2.1.
Case 3. n = 2m − 1. In this case
Z ∞
tm−1 h(t, x, y) dt
0
Z ∞
δ(x)δ(y) 1 − r
− 21
= Cδ(x)δ(y) + |x − y|
r min
,
e dr
|x − y|2 r
|x−y|2
Z ∞
δ(x)δ(y)
δ(x)δ(y)
− 12 − r
d+
.
≤C
r e dr = C
|x − y|
|x − y|
0
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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On the other hand, an integration by parts shows that
Z ∞
δ(x)δ(y) 1 − r
− 12
|x − y|
r min
,
e dr
|x − y|2 r
|x−y|2
Z 2 |x−y|2
Z ∞
δ(x)δ(y) d δ(x)δ(y) − 1
− 32 − r
≤C
r 2 dr + |x − y|
r
e dr
|x−y|2
|x − y| 0
d2 δ(x)δ(y)
i∞
h
p
− 12 − r
≤ C δ(x)δ(y) + |x − y| −2r e
|x−y|2
d2 δ(x)δ(y)
p
≤ C δ(x)δ(y).
Hence
Z
∞
δ(x)δ(y)
t
h(t, x, y)dt ≤ C min
δ(x)δ(y) ,
|x − y|
0
For the lower inequality we discuss two subcases
m−1
p
0
4d2 |x−y|2
(δ(x)δ(y))
Z
vol. 8, iss. 1, art. 26, 2007
.
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|x−y|2
δ(x)δ(y)
≥C
|x − y|
Habib Mâagli and Noureddine Zeddini
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• If δ(x)δ(y) ≤ |x − y|2 . Then from (1.7) we have
Z ∞
Z ∞
δ(x)δ(y)
m−1
− 23 − r
t
h(t, x, y)dt ≥ |x − y| min 1,
r
e dr
|x − y|2
0
1+d2
δ(x)δ(y)
=C
.
|x − y|
• If |x − y|2 ≤ δ(x)δ(y). Then
Z ∞
Z
m−1
t
h(t, x, y)dt ≥ |x − y|
Iterated Green Functions
δ(x)δ(y) 1
∧
|x − y|2
r
4d2 |x−y|2
(δ(x)δ(y))
|x−y|2
1
r− 2 e− r dr
1
r− 2 e− r dr
δ(x)δ(y)
≥C
|x − y|
Z
4d2 |x−y|2
(δ(x)δ(y))
1
r− 2 dr
|x−y|2
"
≥ C δ(x)δ(y) p
#
2d
−1
δ(x)δ(y)
h
i
p
p
≥ C δ(x)δ(y) 2d − δ(x)δ(y)
p
≥ C δ(x)δ(y).
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
Case 4. n = 2m − 2. In this case, we use (1.5) to deduce that
Z ∞
tm−1 h(t, x, y)dt
0
Z ∞ δ(x)δ(y)
1
2
= Cδ(x)δ(y) + |x − y|
∧ 2 e− r dr.
2
r|x − y|
r
|x−y|2
Z ∞ 1
δ(x)δ(y)
∼ δ(x)δ(y) + δ(x)δ(y)
−
e− r dr.
2
r
δ(x)δ(y)r
+
|x
−
y|
2
|x−y|
To prove the upper estimates we remark first that
δ(x)δ(y) ≤ C δ(x)δ(y) Log 2 +
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1
[x, y]2
and we discuss the following subcases
1
1
2
• If 12 ≤ δ(x)δ(y) 1 + [x,y]
2 . Then 1 + δ(x)δ(y) ≤ 4 + [x,y]2 . So
Z ∞ 1
δ(x)δ(y)
−
e− r dr
2
r
δ(x)δ(y)r
+
|x
−
y|
|x−y|2
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Z
∞
1
δ(x)δ(y)
≤
−
r δ(x)δ(y)r + |x − y|2
|x−y|2
1
= Log 1 +
δ(x)δ(y)
1
≤ Log 2 + Log 2 +
[x, y]2
1
≤ C Log 2 +
.
[x, y]2
• If δ(x)δ(y) 1 +
1
[x,y]2
dr
≤ 12 . Then δ(x)δ(y) ([x, y]2 + 1) ≤
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
1
2
[x, y]2 , which
implies that δ(x)δ(y) ≤ |x − y|2 and consequently [x, y]2 ≤ 2|x − y|2 . Hence
Z ∞ 1
δ(x)δ(y)
−
e− r dr
r δ(x)δ(y)r + |x − y|2
|x−y|2
Z ∞
r
− |x−y|2
e− r dr
≤ C Log (1 + δ(x)δ(y)) e
+C
Log
2
δ(x)δ(y)r
+
|x
−
y|
2
|x−y|
Z ∞
r
2
−|x−y|2
≤ C Log(1 + d ) e
+C
e− r dr
Log
δ(x)δ(y)r + |x − y|2
|x−y|2
Z ∞
(1 + d2 )r
≤C
Log
e− r dr
2
δ(x)δ(y)r
+
|x
−
y|
|x−y|2
Z ∞
(1 + d2 )r
≤C
Log
e− r dr
2 (1 + δ(x)δ(y))
|x
−
y|
2
|x−y|
Z ∞
1 + d2
1
+C
Log
r e− r dr
≤ C Log
2
|x − y|
1 + δ(x)δ(y)
|x−y|2
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≤ C Log
1 + d2
|x − y|2
2
Z
∞
+C
Log(r)e− r dr
|x−y|2
∞
Z
1+d
+C
Log(r)e− r dr
|x − y|2
1
1 + d2
1
≤ C + C Log
≤ C Log 2 +
.
|x − y|2
[x, y]2
≤ C Log
Iterated Green Functions
Hence
Habib Mâagli and Noureddine Zeddini
Z
∞
m−1
t
0
1
h(t, x, y)dt ≤ C δ(x)δ(y) Log 2 +
[x, y]2
vol. 8, iss. 1, art. 26, 2007
.
Next we prove the lower estimates.
Z ∞
tm−1 h(t, x, y)dt
0
Z ∞ 1
δ(x)δ(y)
∼ δ(x)δ(y) + δ(x)δ(y)
−
e− r dr
2
r
δ(x)δ(y)r
+
|x
−
y|
2
|x−y|
Z 2d2 1
δ(x)δ(y)
≥ Cδ(x)δ(y) + Cδ(x)δ(y)
−
dr
r δ(x)δ(y)r + |x − y|2
|x−y|2
2d2 (1 + δ(x)δ(y))
.
= Cδ(x)δ(y) + Cδ(x)δ(y) Log
|x − y|2 + 2d2 δ(x)δ(y)
2
Let α > 1 such that α 2d2d2 +1 > 2[x, y]2 + 1; ∀x, y ∈ D. Then we have
2α d2
1
2α d2 (1 + δ(x)δ(y))
≥
≥2+
.
2
2
2
2
|x − y| + 2d δ(x)δ(y)
(1 + 2d )[x, y]
[x, y]2
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Hence
Z
∞
tm−1 h(t, x, y)dt
0
2d2 (1 + δ(x)δ(y))
≥ Cδ(x)δ(y) Log α + Log
|x − y|2 + 2d2 δ(x)δ(y)
2α d2 (1 + δ(x)δ(y))
≥ Cδ(x)δ(y) Log
|x − y|2 + 2d2 δ(x)δ(y)
1
≥ Cδ(x)δ(y) Log 2 +
.
[x, y]2
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
Case 5. n < 2m − 2. In this case we need only to prove the upper inequality.
Z ∞
n
δ(x)δ(y) 1 − r
2m−n
−m
2
|x − y|
r
min
,
e dr
|x − y|2 r
|x−y|2
Z ∞
Z d2 |x−y|2
δ(x)δ(y)
n
n
−m
2m−n
2m−n−2
2
≤ δ(x)δ(y)|x − y|
r
dr + |x − y|
r 2 −m−1 dr
2
d2 δ(x)δ(y)
#
n
m−1− 2
δ(x)δ(y)
2
δ(x)δ(y)
+
2
d
2m − n
d2
m−1− n2
2
2
δ(x)δ(y)
≤
δ(x)δ(y) + 2
δ(x)δ(y)
2m − n − 2
d (2m − n)
d2
≤ C δ(x)δ(y).
≤
2
δ(x)δ(y) 1 −
2m − n − 2
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|x−y|
|x−y|2
"
Title Page
Page 16 of 33
m− n2
This completes the proof of the theorem.
Now using estimates of Theorem 1.1 and similar arguments as in the proof of
Corollary 2.5 in [5], we obtain the following.
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Corollary 2.2. Let r0 > 0. For each x, y ∈ D such that |x − y| ≥ r0 , we have
(2.1)
Gm,n (x, y) δ(x)δ(y)
.
r0n+2−2m
Moreover, on D2 the following estimates hold
( δ(x)∧δ(y)
δ(x)δ(y) Gm,n (x, y) |x−y|n+1−2m
,
δ(x) ∧ δ(y) ,
for n ≥ 2m
for n ≤ 2m − 1.
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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3.
The Kato Class Km,n (D)
To give new examples of functions belonging to this class we need the following
lemma
Lemma 3.1. For λ, µ ∈ R and x ∈ D, let ρλ, µ (x) =
ρλ, µ ∈ L1 (D) if and only if
1
µ
2d
δ λ (x)[Log( δ(x)
)]
. Then
λ < 1 or (λ = 1 and µ > 1).
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
Proof. Since for λ < 0 the function ρλ, µ is continuous and bounded in D we need
only to prove the result for λ ≥ 0.
Since D is a bounded C 1,1 domain and the function t 7→ tλ Log(1 2d ) µ is decreasing
[
t ]
near 0 for λ > 0, then the proof of the lemma on page 726 in [10] can be adapted.
vol. 8, iss. 1, art. 26, 2007
Title Page
Contents
Proposition 3.2. Let m ≥ 2 and p ∈ [1, ∞]. Then ρλ , µ (·)Lp (D) ⊂ Km,n (D),
provided that:
i) For n ≥ 2m − 1, we have λ < 2 +
ii) For n = 2m − 2, we have λ < 2 +
iii) For n < 2m − 2, we have λ < 3 −
p
2(m−1)
n
−
n−1
n
1
p
1
p
−
1
p
and
and
n
2(m−1)
n
n−1
< p.
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Page 18 of 33
< p.
.
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1
+ 1q
p
Proof. Let h ∈ L (D) and q ∈ [1, ∞] such that
= 1. For x ∈ D and α ∈ (0, 1),
we put
Z
δ(y)
I = I(x, α) :=
Gm,n (x, y)ρλ , µ (y)h(y) dy.
B(x,α)∩D δ(x)
Taking account of Theorem 1.1, we will discuss the following cases:
Close
Case 1. n ≥ 2m − 1. In this case we have
Z
h(y)
dy
h
I
n−2(m−1)
B(x,α)∩D |x − y|
δ(y)λ−2 Log
2d
δ(y)
iµ .
It follows from the Hölder inequality that
 1q

Z
I khkp 
1
B(x,α)∩D
h
iqµ  .
|x − y|(n−2(m−1))q δ(y)(λ−2)q Log 2d
δ(y)
< p, then λ − 2 < 1q − n−2(m−1)
and
n
n
n
1
q < n−2(m−1)
so that q q 0 < n−2(m−1)
. Hence we can choose q 0 > max 1, 1−(λ−2)q
Since λ < 2 +
2(m−1)
n
dy
−
1
p
and
n
2(m−1)
1
.
r
1
q0
and (λ − 2)q < 1 − :=
We apply the Hölder inequality again and Lemma 3.1 to deduce that

Z

I khkp
D
 qr1
dy
h
δ(y)(λ−2)qr Log
0
2d
δ(y)
iqrµ 
αn−(n−2m+2)qq .
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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Hence sup I(x, α) → 0 as α → 0.
x∈D
Case 2. n = 2m − 2. Assume that λ < 2 + n−1
− p1 and
n
and q < n.
Iterated Green Functions
n
n−1
< p, then λ − 2 < 1q − n1
Close
Using (1.9) , (1.6) and the Hölder inequality we obtain
Z
h(y)
1
h
iµ dy
I
1+
[x, y] δ(y)λ−2 Log 2d
B(x,α)∩D
δ(y)
Z
h(y)
1
h
iµ dy
2d
B(x,α)∩D |x − y| δ(y)λ−2 Log
δ(y)

 1q
Z
1
1
h
iqµ dy  .
khkp 
q
2d
B(x,α)∩D |x − y| δ(y)(λ−2)q Log
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
δ(y)
q0
Let us choose q 0 > 1 and r = q0 −1 such that qq 0 < n and (λ − 2)qr < 1. Then, using
the Hölder inequality again and Lemma 3.1 we obtain

 qr1
Z
dy
0
h
iqrµ  αn−qq .
I khkp 
(λ−2)qr
2d
D δ(y)
Log δ(y)
Hence sup I(x, α) → 0 as α → 0.
x∈D
Case 3. n < 2m − 2. Using Theorem 1.1 and the Hölder inequality we obtain
Z
h(y)
h
iµ dy
I
2d
B(x,α)∩D δ(y)λ−2 Log
δ(y)

 1q
Z
1
h
iqµ dy  .
khkp 
2d
B(x,α)∩D δ(y)(λ−2)q Log
δ(y)
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As in the preceding cases we choose q 0 > 1 so that (λ − 2)qq 0 < 1 to deduce from
the Hölder inequality and Lemma 3.1 that sup I(x, α) → 0 as α → 0.
x∈D
This completes the proof of the proposition.
Next, we will prove Proposition 1.2. So we need the following results
Lemma 3.3 (see [5]). Let x, y ∈ D. Then the following properties are satisfied:
Iterated Green Functions
2
1) If δ(x)δ(y) ≤ |x − y| then
Habib Mâagli and Noureddine Zeddini
√
1+ 5
|x − y|.
max(δ(x) , δ(y)) ≤
2
vol. 8, iss. 1, art. 26, 2007
Title Page
2
2) If |x − y| ≤ δ(x)δ(y) then
√
√
(3 + 5)
(3 − 5)
δ(x) ≤ δ(y) ≤
δ(x).
2
2
Lemma 3.4. Let q ∈ Km,n (D). Then the function : x → δ 2 (x)q(x) is in L1 (D).
Proof. Let q ∈ Km,n (D). Then by (1.4), there exists α > 0 such that for all x ∈ D
we have
Z
δ(y)
Gm,n (x, y)|q(y)| dy ≤ 1.
(|x−y|≤α)∩D δ(x)
S
Let x1 , x2 , . . . , xp ∈ D such that D ⊂ pi=1 B(xi , α). Then by Corollary 2.2, there
exists C > 0 such that ∀ y ∈ B(xi , α) ∩ D we have
δ 2 (y) ≤ C
δ(y)
Gm,n (xi , y).
δ(x)
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Hence
Z
2
δ (y)|q(y)|dy ≤ C
D
p Z
X
i=1
B(xi ,α)∩D
δ(y)
Gm,n (xi , y)|q(y)| dy
δ(xi )
≤ C p < ∞.
Iterated Green Functions
Proof of Proposition 1.2. It follows from Lemmas 3.1 and 3.4 that a necessary condition for ρλ , µ to belong to Km,n (B) is that λ < 3 or (λ = 3 and µ > 1). Let us
prove that this condition is sufficient.
For λ ≤ 2 the results follow from Proposition 3.2 by taking p = ∞. Hence we
need only to prove the results for
2 < λ < 3 or (λ = 3 and µ > 1).
µ
For x ∈ D and α ∈ 0, 4e− λ , we put
Z
δ(y)
I = I(x, α) :=
Gm,n (x, y)ρλ , µ (y) dy
B(x,α)∩D δ(x)
Z
Gm,n (x, y)
h
iµ dy .
=
4
B(x,α)∩D δ(x)δ(y)λ−1 Log
δ(y)
Taking account of Theorem 1.1 we distinguish the following cases.
Case 1. n ≥ 2m − 1. Then we have
Z
1
1
h
iµ dy
I
n−2(m−1)
4
B(x,α)∩D1 |x − y|
(δ(y))λ−2 Log δ(y)
Z
1
1
h
iµ dy
+
n−2(m−1)
4
λ−2
B(x,α)∩D2 |x − y|
(δ(y))
Log δ(y)
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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= I1 + I2 ,
where
D1 = {x ∈ D : |x − y|2 ≤ δ(x)δ(y)}
and D2 = {x ∈ D : δ(x)δ(y) ≤ |x − y|2 }.
• If y ∈ D1 , then from Lemma 3.3, we have δ(x) ∼ δ(y) and so |x − y| δ(y).
Hence
Z
1
iµ dy
h
I1 C
B(x,α) |x − y|n−2m+λ Log
|x−y|
Z α 2m−(λ+1)
r
µ dr,
Log Cr
0
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
Title Page
which tends to zero as α → 0.
Contents
√
• If y ∈ D2 , then using Lemma 3.3, we have max(δ(x), δ(y)) ≤ 1+2 5 |x − y|.
Hence,
Z
Z 1
tn−1
dσ(ω)
µ
I2 dt ,
√
4
n−2(m−1)
λ−2 Log
1−α( 1+2 5 ) (1 − t)
S n−1 |x − tω|
1−t
where σ is the normalized measure on the unit sphere S n−1 of Rn .
Now by elementary calculus, we have
Z
dσ(ω)
1
t2(m−1)−n .
n−2(m−1)
(|x| ∨ t)n−2(m−1)
S n−1 |x − tω|
So
Z
1
I2 1−α
√ 1+ 5
2
t2m−3
(1 − t)λ−2 Log
which tends to zero as α tends to zero.
4
1−t
µ dt,
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Case 2. n = 2m − 2. In this case we have
Z
1
1
h
iµ dy
I
Log 2 +
2
λ−2
4
[x, y] δ(y)
B(x,α)∩D1
Log δ(y)
Z
1
1
h
iµ dy
+
Log 2 +
2
[x, y] δ(y)λ−2 Log 4
B(x,α)∩D2
δ(y)
= I1 + I2 .
Habib Mâagli and Noureddine Zeddini
√
• If y ∈ D1 , it follows from the fact that Log(2 + t) ≤ t for t ≥ 2 that
Z
1
h
iµ dy
I1 4
B(x,α)∩D1 |x − y|(δ(y))λ−2 Log
δ(y)
Z
1
h
iµ dy
4
B(x,α)∩D1 |x − y|λ−1 Log
|x−y|
Z α
rn−λ
µ dr,
Log 4r
0
which tends to zero as α tends to zero.
• If y ∈ D2 , then
Z
I2 Iterated Green Functions
vol. 8, iss. 1, art. 26, 2007
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1
1
h
iµ dy
Log 2 +
2
|x − y| δ(y)λ−2 Log 4
B(x,α)∩D2
δ(y)
Z
1
h
iµ dy
4
B(x,α)∩D2 |x − y|2 δ(y)λ−2 Log
δ(y)
Close
Z
1
√
5
1−α( 1+2
Z
1
√
1−α( 1+2 5 )
Z
)
1
√
1−α( 1+2 5 )
tn−1
(1 − t)λ−2 Log
4
1−t
µ
tn−1
(1 − t)λ−2 Log
4
1−t
µ
tn−3
(1 − t)λ−2 Log
4
1−t
Z
S n−1
dσ(ω)
|x − tω|2
dt
1
dt
(|x| ∨ t)2
µ dt,
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
which tends to zero as α tends to zero.
Case 3. n < 2m − 2. In this case
Z
I
B(x,α)∩D
vol. 8, iss. 1, art. 26, 2007
Title Page
1
h
(δ(y))λ−2 Log
4
δ(y)
iµ dy.
Hence the result follows from Lemma 3.3, using similar arguments as in the above
cases.
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In the sequel we aim at proving Theorem 1.3. Below we present some preliminary
results which we will need later.
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Proposition 3.5.
Close
a) For each t > 0 and all x, y ∈ D, we have
Z t
sm−1 p(s, x, y) ds Gm,n (x, y).
0
b) Let 0 < t ≤ 1 and x, y ∈ D. Then
Z
Gm,n (x, y) t
sm−1 p(s, x, y) ds ,
0
provided that
√
i) n > 2m and |x − y| ≤ t; or
ii) n = 2m and [x, y]2 ≤ t ; or
iii) n = 2m − 1 and |x − y|2 + 2δ(x)δ(y) ≤ t.
Proof.
a) Follows from (1.3).
b) We deduce from (1.11) and (1.12) that
Z t
Z ∞
m−1
2m−n
s
p(s, x, y) ds ∼ |x − y|
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
Title Page
Contents
|x−y|2
t
0
Iterated Green Functions
min
n
δ(x)δ(y)
r, 1 r 2 −m−1 e− r dr.
2
|x − y|
Next, we distinguish the following cases
0
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Page 26 of 33
i) n > 2m. In this case the result follows from (1.7) and Theorem 1.1.
ii) n = 2m. Using (1.5) we have
Z t
Z
m−1
s
p(s, x, y) ds ≥ C
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2
|x−y|2
t
≥ C Log
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δ(x)δ(y)
dr
δ(x)δ(y)r + |x − y|2
[x, y]2 + δ(x)δ(y)
t
·
δ(x)δ(y) + t
|x − y|2
Now since [x, y]2 ≤ t and the function t 7→
the result follows from Theorem 1.1.
t
δ(x)δ(y)+t
Close
.
is nondecreasing, then
iii) n = 2m − 1. As in the proof of Theorem 1.1 we distinguish two cases
• If δ(x)δ(y) ≤ |x − y|2 . In this case the result follows from (1.7).
• If δ(x)δ(y) > |x − y|2 . Then
t
Z
s
m−1
0
δ(x)δ(y)
p(s, x, y) ds ≥ C
|x − y|
Z
|x−y|2
δ(x)δ(y)
|x−y|2
t
1
r− 2 dr.
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
Since |x − y|2 + 2δ(x)δ(y) ≤ t, then
1
1− √
2
|x − y|
|x − y|
p
+ √
t
δ(x)δ(y)
vol. 8, iss. 1, art. 26, 2007
!2
≤
|x − y|2
.
δ(x)δ(y)
Title Page
Contents
Hence
Z
t
s
0
m−1
δ(x)δ(y) |x − y|
p
p(s, x, y) ds ≥ C
|x − y|
δ(x)δ(y)
p
= C δ(x)δ(y)
≥ C Gm,n (x, y).
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Proposition 3.6. Let q ∈ Km,n (D). Then for each fixed α > 0, we have
Z
δ(y)
p (t, x, y)|q(y)| dy := M (α) < ∞.
(3.1)
sup sup
t≤1
x∈D (|x−y|>α)∩D δ(x)
Close
Proof. Let 0 < t < 1, q ∈ Km,n (D) and 0 < α < 1. Then using (1.11) and (1.12)
we have
Z
Z
|x−y|2
δ(y)
1
p (t, x, y)|q(y)| dy n +1
δ 2 (y)e− t |q(y)| dy
t2
(|x−y|>α)∩D δ(x)
(|x−y|>α)∩D
α2 Z
e− t
n +1
δ 2 (y)|q(y)| dy.
t2
D
Hence the result follows from Lemma 3.4.
Proof of Theorem 1.3. 2) ⇒ 1) Assume that
Z Z t
δ(y) m−1
lim sup
s
p(s, x, y)|q(y)| ds dy = 0.
t→0 x∈D D 0 δ(x)
Then by Proposition 3.5, there exists C > 0 such that for α > 0 we have
Z
Z Z α2
δ(y)
δ(y) m−1
Gm,n (x, y)|q(y)| dy ≤ C
s
p(s, x, y) dsdy,
δ(x)
(|x−y|≤α)∩D δ(x)
D 0
which shows that q satisfies (1.4).
1) ⇒ 2) Suppose that q ∈ Km,n (D) and let ε > 0. Then there exists 0 < α < 1
such that
Z
δ(y)
sup
Gm,n (x, y)|q(y)| dy ≤ ε.
x∈D (|x−y|≤α)∩D δ(x)
On the other hand, using Proposition 3.5 and (3.1), we have for 0 < t < 1
Z Z t
δ(y) m−1
s
p(s, x, y)|q(y)| dsdy
D 0 δ(x)
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
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Z
Z
t
δ(y) m−1
s
p(s, x, y)|q(y)| dsdy
(|x−y|≤α)∩D 0 δ(x)
Z
Z t
δ(y) m−1
s
p(s, x, y)|q(y)| dsdy
+
(|x−y|>α)∩D 0 δ(x)
Z
δ(y)
Gm,n (x, y)|q(y)| dy
(|x−y|≤α)∩D δ(x)
Z tZ
δ(y)
+
p(s, x, y)|q(y)| dsdy
0
(|x−y|>α)∩D δ(x)
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
vol. 8, iss. 1, art. 26, 2007
ε + t M (α).
Title Page
This achieves the proof.
Next we assume that m = 1 and we will give another characterization of the class
K1,n (D).
Corollary 3.7. Let n ≥ 3 and q be a measurable function. For α > 0, put
Z Z ∞
δ(y)
Gα q(x) =
e−α s
p(s, x, y)|q(y)| dsdy, for x ∈ D
δ(x)
D 0
and
Z
1
α
Z
a(α) = sup
x∈D
0
D
δ(y)
p(s, x, y)|q(y)| dyds.
δ(x)
Then there exists C > 0 such that
1
a(α) ≤ kGα qk∞ ≤ C a(α) ,
e
where kGα qk∞ = sup |Gα q(x)|.
x∈D
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In particular, we have
q ∈ K1,n (D) ⇐⇒ lim kGα qk∞ = 0.
α→∞
Proof. Let α > 0. Then using the Fubini theorem, we obtain for x ∈ D
Z t Z
Z ∞
δ(y)
−α t
Gα q(x) =
αe
p(s, x, y)|q(y)| dyds dt
0
0
D δ(x)
#
"Z t Z
Z ∞
α
δ(y)
p(s, x, y)|q(y)| dyds dt.
=
e−t
0
0
D δ(x)
Hence, 1e a(α) ≤ kGα qk∞ .
On the other hand if we denote by [t] the integer part of t, then we have


Z ∞
[t] Z k+1 Z
X
α
δ(y)
p(s, x, y)|q(y)| dyds dt
Gα q(x) ≤
e−t 
k
δ(x)
D
0
k=0 α


Z ∞
[t] Z 1 Z
X
α
δ(y)
k
≤
e−t 
p s + , x, y |q(y)| dyds dt.
α
0
D δ(x)
k=0 0
Now, using the Chapmann-Kolmogorov identity and the Fubini theorem we obtain
Z 1Z
α
δ(y)
k
p s + , x, y |q(y)|dyds
α
0
D δ(x)
!
Z 1Z
Z
α
δ(y)
δ(z)
k
=
p(s, z, y)|q(y)|dyds
p
, x, z dz
δ(x)
α
D
0
D δ(z)
Z
δ(z)
k
≤ a(α)
p
, x, z dz.
α
D δ(x)
Iterated Green Functions
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vol. 8, iss. 1, art. 26, 2007
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Since the first eigenfunction ϕ1 associated to −∆ satisfies ϕ1 (x) ∼ δ(x) and
Z
p(t, x, z)ϕ1 (z)dz = e−λ1 t ϕ1 (x) ≤ ϕ1 (x),
D
then
kGα qk∞ ≤ Ca(α).
So, the last assertion follows from Theorem 1.3.
Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
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References
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Iterated Green Functions
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vol. 8, iss. 1, art. 26, 2007
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Iterated Green Functions
Habib Mâagli and Noureddine Zeddini
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