Journal of Inequalities in Pure and
Applied Mathematics
ESTIMATES FOR THE GREEN FUNCTION AND
CHARACTERIZATION OF A CERTAIN KATO CLASS BY THE
GAUSS SEMIGROUP IN THE HALF SPACE
volume 6, issue 4, article 119,
2005.
IMED BACHAR
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia.
Received 06 July, 2005;
accepted 11 August, 2005.
Communicated by: C. Bandle
EMail: Imed.Bachar@ipeit.rnu.tn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
204-05
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Abstract
We establish a 3G-theorem for the Green functions Gm,n of (−∆)m (m ≥ 1) on
Rn+ := {x = (x1 , . . . , xn ) ∈ Rn : xn > 0}, n ≥ 2m − 1, with Navier boundary
conditions ∆j u |∂Rn = 0, 0 ≤ j ≤ m − 1.
+
We exploit these results to define a certain Kato class of functions that we
characterize by means of the Gauss semigroup on Rn+ .
2000 Mathematics Subject Classification: 34B27.
Key words: Green functions, 3G-theorem, Kato class.
The author is especially thankful to Professor Habib Mâagli for valuable discussions.
He also thanks the referees for their careful reading of the paper
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Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
Imed Bachar
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Inequalities for the Green’s Function . . . . . . . . . . . . . . . . . . . . 7
3
The Class Km,n (Rn+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
1.
Introduction
In [2], for n ≥ 3 and [3], for n = 2, the authors have established interesting estimates for G(x, y), the Green function of the Laplace operator corresponding to zero Dirichlet boundary conditions in the half space Rn+ := {x =
(x1 , . . . , xn ) ∈ Rn , xn > 0}. In particular, they have proved the following form
of the 3G-Theorem:
Theorem 1.1. There exists a constant C > 0 such that for each x, y, z ∈ Rn+
G(x, z)G(z, y)
zn
zn
(1.1)
≤C
G(x, z) + G(y, z) .
G(x, y)
xn
yn
They then introduced a class of functions K1,n (Rn+ ) as follows:
Definition 1.1. A Borel measurable function q in Rn+ belongs to the class K1,n (Rn+ )
if q satisfies the following condition
Z
yn
lim sup
G(x, y) |q(y)| dy = 0.
r→0 x∈Rn (|x−y|≤r)∩Rn xn
+
+
They have studied the properties of functions belonging to this class.
In particular, in [2], the authors have showed the following characterization:
(1.2) q ∈ K1,n (Rn+ ) ⇐⇒ lim
t→0
Z
Z
sup
x∈Rn
+
Rn
+
0
t
yn
p(s, x, y) |q(y)| dsdy
xn
!
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
Imed Bachar
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= 0,
where p(s, x, y) is the density of the Gauss semigroup on Rn+ .
Note that similar characterizations have been already established in [1], (see
also [5] and [8]) for the classical Kato class Kn (Rn ) defined as follows:
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Definition 1.2. A Borel
measurable function q in Rn+ (n ≥ 3) belongs to the
Kato class Kn Rn+ if q satisfies the following condition
!
Z
1
lim sup
|q(y)|dy = 0.
α→0 x∈Rn Rn ∩B(x,α) |x − y|n−2
+
+
For properties of functions in Kn (Rn+ ) we refer to [1], [5], [8], [10] and [11]).
Throughout this paper, we denote by Gm,n (x, y) the Green’s function of the
operator u 7→ (−∆)m u on Rn+ with Navier boundary conditions ∆j u |∂Rn = 0,
+
0 ≤ j ≤ m − 1 for m ≥ 1 and n ≥ max(3, 2m − 1).
The outline of the paper is as follows. In Section 2, we give explicitly the expression of Gm,n (x, y) and we prove some inequalities on Gm,n (x, y) including
a 3G-Theorem of the form (1.1). In Section 3, we introduce a class of functions
Km,n (Rn+ ) defined as follows:
Definition 1.3. A Borel measurable function q in Rn+ belongs to the class Km,n (Rn+ )
if q satisfies
!
Z
yn
(1.3)
lim sup
Gm,n (x, y) |q(y)| dy = 0.
n
r→0
xn
x∈R+ (|x−y|≤r)∩Rn
+
We then study properties of functions belonging to this class. In particular,
we prove the following characterization for n > 2m:
(1.4)
Z
t→0
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q ∈ Km,n (Rn+ )
⇔ lim
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
Z
sup
x∈Rn
+
Rn
+
0
t
yn m−1
s
p(s, x, y) |q(y)| dsdy
xn
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!
= 0,
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
which extends (1.2).
In order to simplify our statements, we define some convenient notations.
Notations:
• B(Rn+ ) denotes the set of Borel measurable functions in Rn+ .
• s ∧ t = min(s, t) and s ∨ t = max(s, t) for s, t ∈ R.
• Let f and g be two nonnegative functions on a set S.
We say f g if there exists a constant c > 0, such that
f (x) ≤ cg(x) for all x ∈ S.
We say f ∼ g if
f g and g f.
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
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• Let x, y ∈ Rn+ . Put y = (y1 , . . . , yn−1 , −yn ). Then we have
|x − y|2 = |x − y|2 + 4xn yn and |x − y|2 ≥ (xn + yn )2 ,
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which implies that
(1.5)
(1.6)
|x − y|2 ∼ |x − y|2 + xn yn
xn ∨ yn ≤ |x − y| .
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The following properties will be used several times.
(i) For s, t ≥ 0, we have
(1.7)
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s∧t∼
st
.
s+t
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
(ii) Let λ, µ > 0 and 0 < γ ≤ 1, then we have
(1.8)
(1.9)
(1.10)
(1.11)
1 − tλ ∼ 1 − tµ , for t ∈ [0, 1].
log(1 + λt) ∼ log(1 + µt), for t ≥ 0.
log 1 + tλ ∼ 1 ∧ tλ log (2 + t) , for t ≥ 0.
log(1 + t) tγ , for t ≥ 0.
(iii) Let a > 0, then we have
(1.12)
1 − e−a ∼ min(1, a).
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
Imed Bachar
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2.
Inequalities for the Green’s Function
In the sequel for t > 0, x and y ∈ Rn+ , we denote by
!!
!
1
|x − y|2
|x − y|2
p(t, x, y) =
exp −
− exp −
n
4t
4t
(4πt) 2
!
x y 1
|x − y|2 n n
=
exp
−
1
−
exp
−
,
n
2
4t
t
(4πt)
the density of the Gauss semigroup on Rn+ . Then the Green’s function of ∆ with
the Dirichlet condition on ∂Rn+ is given by
Z ∞
(2.1)
G(x, y) =
p(t, x, y)dt.
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
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0
Let Gm,n (x, y) be the Green’s function of the operator u 7→ (−∆)m u on Rn+
with Navier boundary conditions ∆j u |∂Rn = 0, 0 ≤ j ≤ m − 1.
+
Then Gm,n satisfies for m ≥ 2,
Z
Z
Gm,n (x, y) =
···
G(x, z1 )G(z1 , z2 ) · · · G(zm−1 , y)dz1 . . . dzm−1 .
Rn
+
Rn
+
Moreover, using the Fubini theorem, (2.1) and the Chapman-Kolmogorov iden-
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
tity we have
Gm,n (x, y)
Z
Z
=
···
Rn
+
G(x, z1 )G(z1 , z2 )dz1 )G(z2 , z3 ) · · · G(zm−1 , y)dz2 . . . dzm−1
Rn
+
∞Z ∞
Z Z
···
Z
=
Rn
Z +∞
Rn
+
Z
···
=
0
p(t1 +t2 , x, z2 )dt1 dt2 G(z2 , z3 ) · · · G(zm−1 , y)dz2 . . . dzm−1
0
∞
0
p(t1 + t2 + · · · + tm , x, y)dt1 . . . dtm .
Estimates for the Green
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a Certain Kato Class by the
Gauss Semigroup in the Half
Space
0
A simple computation shows that for each m ≥ 1 and x, y ∈ Rn+
Z ∞
1
sm−1 p(s, x, y)ds.
(2.2)
Gm,n (x, y) =
(m − 1)! 0
Imed Bachar
Next, we purpose to give an explicit expression for Gm,n .
Let δ > 0 and x, y ∈ Rn+ such that x 6= y. Put a = |x−y|
and b = |x−y|
. Then
2
2
we have
Z δ
Z ∞
n−2m
2m−n
m−1
(2.3)
s
p(s, x, y)ds = αm,n |x − y|
r( 2 )−1 e−r dr
2
|x−y|
4δ
0
− |x−y|2m−n
∞
Z
= βm,n
ξ
Z
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!
∞
|x−y|2
4δ
( n−2m
)−1
2
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r(
−a2 ξ
(e
n−2m
)−1
2
e−r dr
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−b2 ξ
−e
)dξ,
Page 8 of 33
1
δ
J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
where αm,n and βm,n are some positive constants.
Hence, using this fact and (2.2) it follows that
Z δ
lim
sm−1 p(s, x, y)ds = (m−1)!Gm,n (x, y) < ∞ for x 6= y ⇐⇒ 2m−n < 2.
δ→∞
0
Moreover, we deduce from (2.3) by letting δ → ∞, the following explicit expression of Gm,n .
Proposition 2.1. For each x, y ∈ Rn+ , we have

1
1


a
−
, if n > 2m,
m,n |x−y|n−2m

|x−y|n−2m


4xn yn
Gm,n (x, y) =
b
log
1
+
,
if n = 2m,
 m,n
|x−y|2



 c (|x−y| − |x − y| ),
if n = 2m − 1,
m,n
Estimates for the Green
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Gauss Semigroup in the Half
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Imed Bachar
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where am,n , bm,n and cm,n are some positive constants.
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Corollary 2.2. For each x, y ∈ Rn+ , we have
(i) For n > 2m,
xn yn
1
Gm,n (x, y) ∼
n−2m
2 ∼
|x − y|
|x − y|
|x − y|n−2m
xn yn
1∧
|x − y|2
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
(ii) For n = 2m,
xn yn
1∧
|x − y|2
xn yn
Gm,n (x, y) ∼
log 2 +
|x − y|2
!
xn yn
|x − y|2
.
∼
log 1 +
|x − y|2
|x − y|2
(iii) For n = 2m − 1,
1
xn yn
Gm,n (x, y) ∼
∼ (xn yn ) 2
|x − y|
1
(xn yn ) 2
1∧
|x − y|
!
.
Proof. The proof follows immediately from Proposition 2.1 and the statements
(1.8), (1.5) and (1.7) for n > 2m or n = 2m − 1 and using further (1.9) – (1.10)
for n = 2m.
Rn+
Corollary 2.3. For each x, y ∈
we have

1

,
if n > 2m,

|x−y|n−2m


yn
1 + Gm,n (x, y) if n = 2m,
Gm,n (x, y) 
xn


 x ∧y
if n = 2m − 1.
n
n
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Remark 1. For each x, y ∈ Rn+ we have
yn
1
Gm,n (x, y) xn
|x − y|n−2m
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1∧
yn
xn
2 !
, if n > 2m.
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Indeed, from Corollary 2.3, we have
yn
1
Gm,n (x, y) .
xn
|x − y|n−2m
Interchanging the role of x and y, we get
Gm,n (x, y) yn
1
·
,
xn |x − y|n−2m
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
which implies the result.
The next lemma is crucial in this work.
Lemma 2.4 (see [7]). Let x, y ∈ Rn+ . Then we have the following properties:
1. If xn yn ≤ |x − y|2 , then
√
( 5 + 1)
(xn ∨ yn ) ≤
|x − y|.
2
2. If |x − y|2 ≤ xn yn , then
√ !
3− 5
xn ≤ yn ≤
2
√ !
3+ 5
xn .
2
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Corollary 2.5. For each x, y ∈ Rn+ , we have
Gm,n (x, y) (2.4)
(2.5)
(2.6)
(|x| +
xn yn
,
|x − y|n−2m+2
xn yn
n−2m+2
1)
(|y|
Gm,n (x, y),
+ 1)n−2m+2
xn ∧ yn
Gm,n (x, y) .
|x − y|n+1−2m
Proof. The assertions (2.4) and (2.5) follow from Corollary 2.2 and the fact that
|x − y| ≤ | x−y| ≤ (|x| + 1)(|y| + 1)
and
Imed Bachar
t
≤ log(1 + t) ≤ t,
1+t
for t ≥ 0.
To prove (2.6) we claim that
(2.7)
Estimates for the Green
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Gauss Semigroup in the Half
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Gm,n (x, y) Title Page
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xn
.
|x − y|n+1−2m
Indeed, we have the following cases:
Case 1. If n > 2m or n = 2m − 1, the inequality (2.7) follows from Corollary
2.2, (1.6) and the fact that |x − y| ≥ |x − y| .
Case 2. If n = 2m, then we have the following subcases:
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
1. If |x − y|2 ≤ xn yn , then by Lemma 2.4, we get xn ∼ yn .
Using this fact, Proposition 2.1, (1.9) and (1.11) we deduce that
cx2n
Gm,n (x, y) ∼ log 1 +
, (where c > 0),
|x − y|2
xn
.
|x − y|
2. If xn yn ≤ |x − y|2 , then Lemma 2.4 gives that (xn ∨ yn ) |x − y|.
Hence from (2.4), we deduce that
Gm,n (x, y) xn yn
xn
.
2 |x − y|
|x − y|
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
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Imed Bachar
This proves (2.7). Interchange the role of x and y, we obtain (2.6).
Proposition 2.6.
a) For each t > 0, and all x, y ∈ Rn+ , we have
Z t
sm−1 p(s, x, y)ds Gm,n (x, y).
0
b) Let t > 0 and x, y ∈ Rn+ . Then
Z
Gm,n (x, y) Title Page
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t
s
m−1
p(s, x, y)ds,
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provided
√
i) n > 2m and |x − y| ≤ 2 t; or
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
√
ii) n = 2m and |x − y| ≤ 2 t;
or
√
iii) n = 2m − 1 and |x − y| ≤ 2 t.
Proof. Let t > 0 and x, y ∈ Rn+ . Then a) follows immediately from (2.2).
To prove b) we distinguish three cases.
i) For n > 2m, using (1.12) and the fact that for a, b ∈ (0, ∞) we have
(1 ∧ ab) ≥ (1 ∧ a)(1 ∧ b),
√
then there exists C > 0 such that for |x − y| ≤ 2 t,
Z
t
s
0
m−1
Z
t
2
|x − y|
exp −
4s
!
xn yn 1∧
p(s, x, y)ds ≥ C
ds
n
+1−m
s
0 s2
Z ∞
n
C
4rxn yn
−1−m
−r
≥
r2
e
1∧
dr
2
|x − y|n−2m |x−y|
|x − y|2
4t
Z ∞
n
C
xn yn
−1−m −r
2
r
e (1 ∧ 4r)dr
≥
1∧
n−2m
2
2
|x−y|
|x − y|
|x − y|
Z ∞4t
n
C
xn yn
≥
1∧
r 2 −1−m e−r dr
n−2m
2
|x − y|
|x − y|
1
C
xn yn
≥
1∧
.
n−2m
|x − y|
|x − y|2
1
Hence the result follows from Corollary 2.2.
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
√
ii) For n = 2m, by (2.3), there exists C > 0 such that for |x − y| ≤ 2 t
!
Z t
Z |x−y|2 −r
2
4t
e
| x−y|
dr ≥ C log
.
sm−1 p(s, x, y)ds = αm,n
|x−y|2
r
|x − y|2
0
4t
Hence the result follows from Proposition 2.1.
√
iii) Let n = 2m − 1, t > 0 and x, y ∈ Rn+ such that |x − y| ≤ 2 t.
|x−y|
√ and b = √ . Then using (2.3), we obtain
Put a = |x−y|
2 t
2 t
Z
I :=
Z
t
√
m−1
s
p(s, x, y)ds = 2αm,n t a
∞
r
−3
2
e−r dr − b
Z
a2
0
Now since for α > 0, we have
Z ∞
−3
r 2 e−r dr = 2
α2
∞
r
−3
2
e−r dr .
b2
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
Space
Imed Bachar
2
e−α
−
α
Z
!
∞
r
−1
2
e−r dr ,
α2
√
Z ∞
Z ∞
−1
−1
−a2
−b2
−r
−r
2
2
I = 4αm,n t (e
−e )+b
r e dr − a
r e dr
b2
a2
#
"
Z b2
Z ∞
√
−1
−1
1
r 2 e−r (r 2 − a)dr .
= 4αm,n t (b − a)
r 2 e−r dr +
a2
b2
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we deduce that
Hence
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√
I ≥ 4αm,n t(b − a)
Z
∞
r
−1
2
e−r dr.
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b2
J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
That is
∞
Z
I ≥ 2αm,n (|x − y| − |x − y|)
|x−y|2
4t
∞
Z
≥ 2αm,n (|x − y| − |x − y|)
r
r
−1
2
−1
2
e−r dr
e−r dr.
1
The result follows from Proposition 2.1.
Next we purpose to prove that Gm,n satisfies (1.1).
3G-Theorem. For x, y, z ∈ Rn+ , we have
Gm,n (x, z)Gm,n (z, y)
zn
zn
Gm,n (x, z) + Gm,n (y, z) .
Gm,n (x, y)
xn
yn
Proof. To prove the inequality, we denote by A(x, y) :=
that A is a quasi-metric, that is for each x, y, z ∈ Rn+ ,
(2.8)
xn y n
Gm,n (x,y)
and we claim
A(x, y) A(x, z) + A(y, z).
To this end, we observe that by using Corollary 2.2 and Lemma 2.4, the claim
can be proved by similar arguments as in [2], for n > 2m and as in [3], for
n = 2m.
To prove (2.8), for n = 2m − 1, we derive from Corollary 2.2 that
1
A(x, y) ∼ (|x − y|2 ∨ xn yn ) 2 ∼ |x−y| .
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Gauss Semigroup in the Half
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Now since | x − z |≤| x−z|, we deduce that
A(x, y) |x − z| + |z−y|
|x−z| + |z−y| (A(x, z) + A(y, z)).
Estimates for the Green
function and Characterization of
a Certain Kato Class by the
Gauss Semigroup in the Half
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
The Class Km,n(Rn+)
3.
Next we purpose to study and to characterize the class Km,n (Rn+ ) for n > 2m.
We recall that for 0 < α < n, we say that a Borel measurable function q in
n
e α,n (Rn ) (see [6]) if q satisfies the following condition
R+ belongs to the class K
+
Z
(3.1)
lim sup
r→0 x∈Rn
+
(|x−y|≤r)∩Rn
+
|q(y)|
dy = 0.
|x − y|n−α
The usual Kato class Kn (Rn+ ), corresponds to α = 2.
Remark 2. Let n > 2m. Using Corollary 2.3, the class Km,n (Rn+ ) obviously
e 2m,n (Rn+ ). In particular, Kn (Rn+ ) ⊂ Km,n (Rn+ ).
includes the class K
n
Example 3.1. Suppose that for p >
> 1, we have
2m
2p !
Z
yn
M0 = sup
min
, 1 |q(y)|p dy < ∞,
n
x
x∈Rn
n
+ (|x−y|≤1)∩R+
then q ∈ Km,n (Rn+ ).
Indeed, let 0 < r < 1 and x ∈ Rn+ , then using Remark 1 and the Hölder
inequality we get
Z
yn
Gm,n (x, y) |q(y)| dy
xn
(|x−y|≤r)∩Rn
+
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Z
yn
xn
min
(|x−y|≤r)∩Rn
+
Z
min
(|x−y|≤r)∩Rn
+
!
2
,1
yn
xn
2p
Z
1
|q(y)| dy
|x − y|n−2m
!
! p1
, 1 |q(y)|p dy
1
×
p
(|x−y|≤r)∩Rn
+
|x − y| p−1 (n−2m)
! p−1
p
dy
.
Hence
Z
sup
x∈Rn
+
(|x−y|≤r)∩Rn
+
1 2mp−n
yn
Gm,n (x, y)q(y)dy M0p r p → 0 as r → 0.
xn
Proposition 3.1. Let p > max
y 7→
n
,1
2m
and f ∈ Lp Rn+ . Then
f (y)
∈ Km,n (Rn+ )
(|y| + 1)µ−λ ynλ
provided
i) n > 2m, λ ≤ 2 and λ < 2m −
n
p
and µ ≥ max(0, λ) or
ii) n = 2m and λ < min(2, 2m − np ) ≤ µ
iii) n = 2m − 1, λ ≤ 2 and λ < 2m −
n
p
or
and µ ≥ max(1, λ).
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n
Proof. Let p > max 2m
, 1 and q ≥ 1 such that p1 + 1q = 1.
For f ∈ Lp (Rn+ ), x ∈ Rn+ and 0 < r < 1, put
Z
|f (y)|
yn
Gm,n (x, y)
dy.
I = I(x, r) :=
xn
(|y| + 1)µ−λ ynλ
B(x,r)∩Rn
+
Note that if |x − y| ≤ r, then (|x| + 1) ∼ (|y| + 1). So, we distinguish the
following cases:
Estimates for the Green
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Case 1. n > 2m. Assume that λ ≤ 2 and λ < 2m − np . Let µ ≥ max(0, λ)
and put λ+ = max(λ, 0). Then using Corollary 2.2, (1.6) and the fact that
|x − y| ≤ |x − y| , we deduce by the Hölder inequality that
|f (y)|
Z
I
B(x,r)∩Rn
+
r
Z
n+λ+ −2m
|x − y|
dy kf kp
1
B(x,r)∩Rn
+
Imed Bachar
! 1q
(n+λ+ −2m)q
|x − y|
dy
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2m− n
−λ+
p
, which tends to zero if r → 0.
n
Case 2. n = 2m. Assume that λ < min 2, 2m − p ≤ µ.
Using Proposition 2.1 and the Hölder inequality, we deduce that
Z
I kf kp
(|x−y|≤r)∩Rn
+
yn
xn
q log 1 +
4xn yn
|x − y|2
q
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1
(|y| + 1)(µ−λ)q ynλq
! 1q
dy
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
q q
1q
yn
4xn yn
1
log 1 +
dy
xn
|x − y|2
(|y| + 1)(µ−λ)q ynλq
(|x−y|≤r)∩D1
Z
q q
1q
yn
4xn yn
1
+
log 1 +
dy
xn
|x − y|2
(|y| + 1)(µ−λ)q ynλq
(|x−y|≤r)∩D2
Z
= I1 + I2 ,
where
D1 = {y ∈
Rn+
2
: xn yn ≤ |x − y| } and D2 = {y ∈
Rn+
2
: |x − y| ≤ xn yn }.
So, using that log(1 + t) ≤ t, for t ≥ 0 and Lemma 2.4, we obtain
Z
I1 (|x−y|≤r)∩D1
Z
(|x−y|≤r)∩D1
2m− n
−λ
p
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! 1q
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1q
1
r
(2−λ)q
yn
dy
|x − y|2q
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dy
|x − y|λq
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J
, which converges to zero as r → 0.
On the other hand, from Lemma 2.4 and the fact that (|x| + 1) ∼ (|y| + 1), we
obtain
I2 1
xλn (|x| + 1)(µ−λ)
Z
log 1 +
(|x−y|≤r)∩D2
Estimates for the Green
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(cxn )2
|x − y|2
1q
q
dy
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√
where c = 1 + 5. Let γ ∈ ] max(0, λ), min(2, 2m − np )[.
Since log(1 + t2 ) tγ , for t ≥ 0, then
xnγ−λ
I2 (|x| + 1)µ−λ
Z
r
Z
(|x−y|≤r)∩D2
−γ
2m− n
p
1
dy
γq
(|x−y|≤r)∩D2 |x − y|
1q
1
dy
|x − y|γq
1q
, which converges to zero as r → 0.
Case 3. n = 2m − 1. Assume that λ ≤ 2 and λ < 2m − np . Let µ ≥ max(1, λ).
Using Corollary 2.2 and the Hölder inequality, we obtain

! 1q
Z
(2−λ)q
y
n
I
dy
q
(µ−λ)q
B(x,r)∩D1 |x − y| (|y| + 1)
! 1q 
Z
(2−λ)q
yn
+
dy 
q
(µ−λ)q
B(x,r)∩D2 |x − y| (|y| + 1)
= I1 + I2 .
I1 B(x,r)∩D1
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Now, if y ∈ D1 , then |x − y| ∼ |x − y| and so
Z
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1
|x − y|(λ−1)q
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dy
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r
2m− n
−λ
p
, which tends to zero as r → 0.
Page 22 of 33
J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
On the other hand, if y ∈ D2 , then |x − y|2 ∼ xn yn and by Lemma 2.4, we have
further xn ∼ yn . This implies that
Z
1q
n
x2−λ
x1−λ
n
n
I2 dy
(r ∧ cxn ) q
µ−λ
µ−λ
xn (|x| + 1)
(|x| + 1)
B(x,r)∩D2
n
r q , which converges to zero as r → 0.
The proof of the next results are similar to the case m = 1 and n ≥ 3, which
has been considered in [2]. Since reference [2] is not available, I have chosen
to reproduce it here.
Proposition 3.2. Let q ∈ Km,n (Rn+ ), then for each compact L ⊆ Rn we have
Z
yn2
sup
|q(y)| dy < ∞.
1 + xn yn
x∈Rn
(x+L)∩Rn
+
+
Proof. Let q ∈ Km,n (Rn+ ), then by (1.3) there exists r > 0 such that
Z
yn
sup
Gm,n (x, y) |q(y)| dy ≤ 1.
x
x∈Rn
n
(|x−y|≤r)∩Rn
+
+
Let a1 , a2 , . . . , ap ∈ Rn+ ∩ L such that Rn+ ∩ L ⊆ ∪ B(ai , r).
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1≤i≤p
Since for a, b ∈ (0, ∞), we have
it follows that
Rn+
b
1+ab
≤ 1 + |a − b| , then for each x, y, z ∈
1 + (xn + zn )yn
≤ [1 + zn (1 + |xn − yn |)].
1 + xn yn
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Using this fact and Corollary 2.2, we obtain:
For n > 2m,
yn2
[1 + zn (1 + |xn − yn |)]
|x + z − y|n−2m
1 + xn yn
1 + (xn + zn )yn
yn
× |x + z − y|2 + 4(xn + zn )yn
Gm,n (x + z, y).
(xn + zn )
For n = 2m, using further that
t
1+t
≤ log(1 + t), ∀t ≥ 0, we have
yn2
[1 + zn (1 + |xn − yn |)]
1 + xn yn
1 + (xn + zn )yn
× [|x + z − y|2 + 4(xn + zn )yn ]
yn
Gm,n (x + z, y).
(xn + zn )
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For n = 2m − 1,
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yn2
[1 + zn (1 + |xn − yn |)]
1 + xn yn
1 + (xn + zn )yn
1
× [|x + z − y|2 + 4(xn + zn )yn ] 2
yn
Gm,n (x + z, y).
(xn + zn )
Now, if z ∈ L and |x + z − y| ≤ r, then
yn2
yn
Gm,n (x + z, y).
1 + xn yn
(xn + zn )
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Hence
Z
(x+L)∩Rn
+
yn2
|q(y)| dy
1 + xn yn
p Z
X
i=1
(|x+ai −y|≤r)∩Rn
+
So
yn
Gm,n (x + ai , y) |q(y)| dy p.
(xn + (ai )n )
Z
sup
x∈Rn
+
(x+L)∩Rn
+
yn2
|q(y)| dy < ∞.
1 + xn yn
Corollary 3.3. Let q ∈ Km,n (Rn+ ).Then we have for M > 0,
Z
yn2 |q(y)| dy < ∞.
(|y|≤M )∩Rn
+
Proposition 3.4. Let q ∈ Km,n (Rn+ ), then for each fixed α > 0, we have
Z
yn
p(t, x, y) |q(y)| dy := M (α) < ∞.
(3.2)
sup sup
x
t≤1 x∈Rn
n
(|x−y|>α)∩Rn
+
+
Proof. Let q ∈ Km,n (Rn+ ), 0 < t ≤ 1 and without loss of generality assume that
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0 < α < 1. Then by (1.12) and (1.7), it follows that
Z
yn
sup
p(t, x, y) |q(y)| dy
x
x∈Rn
n
(|x−y|>α)∩Rn
+
+
!
Z
1 − α2
yn2
|x − y|2
n +1 e 8t sup
exp −
|q(y)| dy.
8
1 + xn yn
t2
x∈Rn
Rn
+
+
To conclude, it is sufficient to prove that
!
Z
|x − y|2
yn2
sup
exp −
|q(y)| dy < ∞.
n
8
1
+
x
y
x∈Rn
n
n
R
+
+
Estimates for the Green
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Indeed, using Proposition 3.2, we have
Z
yn2
f < ∞.
sup
|q(y)| dy := M
n
1
+
x
y
x∈Rn
n
n
(x+B(0,1))∩R+
+
n
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n
Now since any ball B(0, k), of radius k ≥ 1 in R can be covered by An k :=
α(n) balls of radius 1, where An is a constant depending only on n (see [5, p.
67]), then there exists a1 , a2 , . . . , aα(n) ∈ Rn+ such that
Rn+ ∩ B(0, k) ⊆
Using the fact that for each x, y, z ∈
∪
1≤i≤α(n)
B(ai , 1).
Rn+ ,
1 + (xn + zn )yn
≤ [1 + zn (1 + |xn − yn |)],
1 + xn yn
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
it follows that for all x ∈ Rn+ and k ≥ 1 :
Z
yn2
|q(y)| dy
1 + xn yn
(x+B(0,k))∩Rn
+
α(n) Z
X
i=1
B(x+ai ,1)∩Rn
+
α(n) Z
X
i=1
B(x+ai ,1)∩Rn
+
yn2
|q(y)| dy
1 + xn yn
yn2
|q(y)| dy
1 + (xn + (ai )n )yn
f.
An k n M
Hence for all x ∈ Rn+ , we have
!
Z
yn2
|x − y|2
exp −
|q(y)| dy
8
1 + xn yn
Rn
+
Z
∞
X
α2 k 2
yn2
exp −
|q(y)| dy
n 1 + xn yn
8
[kα≤|x−y|≤(k+1)α]∩R
+
k=0
∞
2 2
X
f (k + 1)n exp − α k
An M
< ∞.
8
k=0
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Thus
2
Z
exp −
sup
x∈Rn
+
Rn
+
|x − y|
8
!
yn2
1 + xn yn
|q(y)| dy < ∞,
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which completes the proof.
J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Theorem 3.5. Let n > 2m and q ∈ B(Rn+ ). Then the following assertions are
equivalent:
1. q ∈ Km,n (Rn+ )
R R t yn m−1
2. lim sup Rn 0
s
p(s, x, y) |q(y)| dsdy = 0.
+
t→0 x∈Rn
xn
+
Proof. 2) ⇒ 1) Assume that
Z Z
lim sup
t→0 x∈Rn
+
Rn
+
0
t
yn m−1
s
p(s, x, y) |q(y)| dsdy = 0.
xn
Then by Proposition 2.6, there exists c > 0 such that for α > 0 we have
Z
(|x−y|≤α)∩Rn
+
yn
Gm,n (x, y) |q(y)| dy ≤ c
xn
Z
Rn
+
Z
0
α2
4
yn m−1
s
p(s, x, y) |q(y)| dsdy,
xn
which shows that the function q satisfies (1.3).
Conversely suppose that q ∈ Km,n (Rn+ ). Let ε > 0, then there exists 0 <
α < 1 such that
Z
yn
sup
Gm,n (x, y) |q(y)| dy ≤ ε.
xn
x∈Rn
(|x−y|≤α)∩Rn
+
+
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On the other hand, using Proposition 2.6 and (3.2), we have for 0 < t < 1
Z Z t
yn m−1
s
p(s, x, y) |q(y)| dsdy
x
n
Rn
0
+
Z
Z t
yn m−1
s
p(s, x, y) |q(y)| dsdy
x
n
0
(|x−y|≤α)∩Rn
+
Z
Z t
yn m−1
+
s
p(s, x, y) |q(y)| dsdy
x
n
(|x−y|>α)∩Rn
0
+
Z
yn
Gm,n (x, y) |q(y)| dy
xn
(|x−y|≤α)∩Rn
+
Z tZ
yn
+
p(s, x, y) |q(y)| dyds
xn
0
(|x−y|>α)∩Rn
+
ε + tM (α),
Estimates for the Green
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which implies that
Z
Z
lim sup
t→0 x∈Rn
+
Rn
+
0
t
yn m−1
s
p(s, x, y) |q(y)| dsdy = 0.
xn
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Corollary 3.6. Let n > 2m and q ∈ B(Rn+ ). For α > 0 and x ∈ Rn+ , put
Z Z ∞
yn
Gα q(x) :=
e−αs sm−1 p(s, x, y) |q(y)| dsdy.
xn
Rn
0
+
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
Then
q ∈ Km,n (Rn+ ) ⇔ lim kGα qk∞ = 0,
α→+∞
where kGα qk∞ = sup |Gα q(x)| .
x∈Rn
+
Proof. (see [9]). Let q ∈ Km,n (Rn+ ), α > 0 and put
Z 1Z
α
yn m−1
a(α) = sup
s
p(s, x, y) |q(y)| dyds.
x
x∈Rn
n
0
Rn
+
+
Then we have
Z
∞
αe−αt
Gα q(x) =
"Z Z
t
0
Z
=
Rn
+
0
∞
e−t
0
"Z
0
t
α
Z
Rn
+
#
yn m−1
s
p(s, x, y) |q(y)| dyds dt
xn
#
yn m−1
s
p(s, x, y) |q(y)| dyds dt.
xn
1
It follows that, a(α) ≤ kGα qk∞ .
e
On the other hand, for t > 0 and k ∈ N such that k ≤ t < k + 1, we have
#
"Z k+1 Z
m Z ∞
X
α
yn m−1
−t
s
p(s, x, y) |q(y)| dyds dt
Gα q(x) ≤
e
k
xn
Rn
+
α
k=0 0
Z ∞
≤ a(α)
e−t (m + 1)dt
Z0 ∞
≤ a(α)
e−t (t + 1)dt = 2a(α),
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
1
which gives that a(α) ≤ kGα qk∞ ≤ 2a(α).
e
Hence the results follow from Theorem 3.5.
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References
[1] M. AIZENMAN AND B. SIMON, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math., XXXV (1982),
209–273.
[2] I. BACHAR AND H. MÂAGLI, Estimates on the Green’s function and
existence of positive solutions of nonlinear singular elliptic equations in
the half space, Positivity, 9(2) (2005), 153–192.
[3] I. BACHAR, H. MÂAGLI AND L. MÂATOUG, Positive solutions of nonlinear elliptic equations in a half space in R2 , E.J.D.E, 2002 (2002), No.
41, 1–24.
[4] I. BACHAR, H. MÂAGLI AND M. ZRIBI, Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear
equations in the half space, Manuscripta Math., 113, (2004), 269–291.
[5] K.L. CHUNG AND Z. ZHAO, From Brownian Motion to Schrödinger’s
Equation, Springer Verlag (1995).
[6] E.B. DAVIES AND A.M. HINZ, Kato class potentials for higher order elliptic operators, J. London Math. Soc., (2) 58 (1998) 669–678.
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[7] H. MÂAGLI, Inequalities for the Riesz potentials, Archives of Inqualities
and Applications, 1 (2003) 285–294.
[8] B. SIMON, Schrödinger semi-groups, Bull. Amer. Math. Soc., 7(3) (1982),
447–526.
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J. Ineq. Pure and Appl. Math. 6(4) Art. 119, 2005
[9] J.A. VAN CASTEREN, Generators Strongly Continuous Semi-groups,
Pitman Advanced Publishing Program, Boston, (1985).
[10] Z. ZHAO, Subcriticality and gaugeability of the schrödinger operator,
Trans. Amer. Math. Society, 334(1) (1992), 75–96.
[11] Z. ZHAO, On the existence of positive solutions of nonlinear elliptic equations. A probabilistic potential theory approach, Duke Math. J., 69 (1993),
247–258.
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