Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 3, Article 70, 2004
¯
ESTIMATES FOR THE ∂−NEUMANN
OPERATOR ON STRONGLY
PSEUDO-CONVEX DOMAIN WITH LIPSCHITZ BOUNDARY
O. ABDELKADER AND S. SABER
M ATHEMATICS D EPARTMENT
FACULTY OF S CIENCE
M INIA U NIVERSITY
E L -M INIA , E GYPT.
M ATHEMATICS D EPARTMENT
FACULTY OF S CIENCE
C AIRO U NIVERSITY
B ENI -S WEF B RANCH , E GYPT.
sayedkay@yahoo.com
Received 29 February, 2004; accepted 19 April, 2004
Communicated by J. Sándor
A BSTRACT. On a bounded strongly pseudo-convex domain X in Cn with a Lipschitz boundary,
¯
we prove that the ∂−Neumann
operator N can be extended as a bounded operator from Sobolev
(−1/2)−spaces to the Sobolev (1/2)−spaces. In particular, N is compact operator on Sobolev
(−1/2)−spaces.
Key words and phrases: Sobolev estimate, Neumann problem, Lipschitz domains.
2000 Mathematics Subject Classification. Primary 35N15; Secondary 32W05.
1. I NTRODUCTION
Let X be a bounded pseudo-convex domain in Cn with the standard Hermitian metric. The
¯
∂−Neumann operator N is the (bounded) inverse of the (unbounded) Laplace-Beltrami opera¯
tor . The ∂−Neumann
problem has been studied extensively when the domain X has smooth
boundaries (see [12], [1], [3], [18], [19], [21], and [22]). Dahlberg [6] and Jerison and Kenig
[17] established the work on the Dirichlet and classical Neumann problem on Lipschitz domains. The compactness of N on Lipschitz pseudo-convex domains is studied in Henkin and
s
Iordan [14]. Let W(p,q)
(X) be the Hilbert spaces of (p, q)−forms with W s (X)−coefficients.
Henkin, Iordan, and Kohn in [15] and Michel and Shaw in [23] showed that N is bounded from
1/2
L2(p,q) (X) to W(p,q) (X) on domains with piecewise smooth strongly pseudo-convex boundary by
1/2
two different methods. Also Michel and Shaw in [24] proved that N is bounded on W(p,q) (X)
when the domain is only bounded pseudo-convex Lipschitz with a plurisubharmonic defining
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
101-04
2
O. A BDELKADER AND S. S ABER
function. Other results in this direction belong to Bonami and Charpentier [4], Straube [26],
Engliš [10], and Ehsani [7], [8], and [9]. In fact, the main aim of this work is to establish the
following:
Theorem 1.1. Let X ⊂⊂ Cn be a bounded strongly pseudo-convex domain with Lipschitz
¯
boundary. For each 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1, the ∂−Neumann
operator
N : L2(p,q) (X) −→ L2(p,q) (X)
satisfies the following estimate: for any ϕ ∈ L2(p,q) (X), there exists a constant c > 0 such that
kN ϕk1/2(X) ≤ ckϕk−1/2(X) ,
(1.1)
where c = c(X) is independent of ϕ; i.e., N can be extended as a bounded operator from
−1/2
1/2
−1/2
W(p,q) (X) into W(p,q) (X). In particular, N is a compact operator on L2(p,q) (X) and W(p,q) (X).
¯
2. N OTATIONS AND THE ∂−N
EUMANN P ROBLEM
We will use the standard notation of Hörmander [16] for differential forms. Let X be a
bounded domain of Cn . We express a (p, q)−form ϕ on X as follows:
X
ϕIJ dz I ∧ dz J ,
ϕ=
I,J
where I and J are strictly increasing multi-indices with lengths p and q, respectively. We denote
by Λ(p,q) (X) the space of differential forms of class C ∞ and of type (p, q) on X. Let
Λ(p,q) (X̄) = {ϕ|X̄ ; ϕ ∈ Λ(p,q) (Cn )},
be the subspace of Λ(p,q) (X) whose elements can be extended smoothly up to the boundary ∂X
of X. For ϕ, ψ ∈ Λ(p,q) (X̄), the inner product and norm are defined as usual by
Z
XZ
2
hϕ, ψi =
ϕIJ ψ IJ dv, and kϕk =
|ϕ|2 dv,
I,J
X
X
where dv is the Lebesgue measure. Let Λ0,(p,q) (X) be the subspace of Λ(p,q) (X̄) whose elements
have compact support disjoint from ∂X.
The operator ∂¯ : Λ(p,q−1) (X) −→ Λ(p,q) (X) is defined by
X X ∂ϕIJ
¯ =
∂ϕ
dz̄ k ∧ dz I ∧ dz̄ J .
k
∂
z̄
IJ
k
The formal adjoint operator δ of ∂¯ is defined by :
¯
hδϕ, ψi = ϕ, ∂ψ
for any ϕ ∈ Λ(p,q) (X) and ψ ∈ Λ0,(p,q−1) (X). It is easily seen that ∂¯ is a closed, linear, densely
defined operator, and ∂¯ forms a complex, i.e., ∂¯2 = 0. We denote by L2(p,q) (X) the Hilbert space
of all (p, q) forms with square integrable coefficients. We denote again by ∂¯ : L2(p,q−1) (X) −→
¯ Then ∂¯ is a closed, linear, densely defined
L2(p,q) (X) the maximal extension of the original ∂.
operator, and forms a complex, i.e., ∂¯2 = 0. Therefore, the adjoint operator ∂¯? : L2(p,q) (X) −→
L2(p,q−1) (X) of ∂¯ is also a closed, linear, defined operator. We denote the domain and the range
¯ and Range(p,q) (∂)
¯ respectively.
of ∂¯ in L2(p,q) (X) by Dom(p,q) (∂)
We define the Laplace-Beltrami operator
= ∂¯∂¯∗ + ∂¯∗ ∂¯ : L2 (X) −→ L2 (X)
(p,q)
J. Inequal. Pure and Appl. Math., 5(3) Art. 70, 2004
(p,q)
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¯
E STIMATES FOR THE ∂−N
EUMANN O PERATOR
3
on
¯ ∩ Dom(p,q) (∂¯? ); ∂ϕ
¯ ∈ Dom(p,q+1) (∂¯? )
Dom(p,q) () = {ϕ ∈ Dom(p,q) (∂)
¯
and ∂¯? ϕ ∈ Dom(p,q−1) (∂)}.
Let
¯ ∩ Dom(p,q) (∂¯? ); ∂ϕ
¯ = 0 and ∂¯? ϕ = 0}.
Ker(p,q) () = {ϕ ∈ Dom(p,q) (∂)
Definition 2.1. A domain X ⊂⊂ Cn is said to be strongly pseudo-convex with C ∞ −boundary
if there exist an open neighborhood U of the boundary ∂X of X and a C ∞ function λ : U −→ <
having the following properties:
(i) X ∩ U = {z ∈ U ; λ(z) < 0}.
P
∂ 2 λ(z) α β
≥ L(z)|η|2 ; z ∈ U , η = (η 1 , . . . , η n ) ∈ Cn and L(z) > 0.
(ii) nα,β=1 ∂z
α ∂ z̄ β η η̄
∂λ(z)
∂λ(z) ∂λ(z)
(iii) The gradient ∇λ(z) = ∂λ(z)
,
,
.
.
.
,
,
6= 0
∂x1
∂y 1
∂xn
∂y n
for z = (z 1 , . . . , z n ) ∈ U ; z α = xα + iy α .
Let f : <2n−1 −→ < be a function that satisfies the Lipschitz condition
(2.1)
|f (x) − f (x0 )| ≤ T |x − x0 | for all
x, x0 ∈ <2n−1 .
The smallest T in which (2.1) holds is called the bound of the Lipschitz constant. By choosing
finitely many balls {Vj } covering ∂X, the Lipschitz constant for a Lipschitz domain is the
smallest T such that the Lipschitz constant is bounded in every ball {Vj }.
Definition 2.2. A bounded domain X in Cn is called a strongly pseudo-convex domain with
Lipschitz boundary ∂X if there exists a Lipschitz defining function % in a neighborhood of X̄
such that the following condition holds:
(i) Locally near every point of the boundary ∂X, after a smooth change of coordinates, ∂X
is the graph of a Lipschitz function.
(ii) There exists a constant c1 > 0 such that,
n
X
∂2% α β
(2.2)
η η̄ ≥ c1 |η|2 ,
η = (η 1 , . . . , η n ) ∈ Cn ,
α
β
∂z ∂ z̄
α,β
where (2.2) is defined in the distribution sense.
Let W s (X), s ≥ 0, be defined as the space of all u|X such that u ∈ W s (Cn ).We define the
norm of W s (X) by
kuks(X) = inf{kvks(Cn ) , v ∈ W s (Cn ), v|X = u}.
s
We use W(p,q)
(X) to denote Hilbert spaces of (p, q)−forms with W s (X) coefficients and their
norms are denoted by k ks(X) . Let W0s (X) be the completion of C0∞ (X)−functions under
the W s (X)−norm. Restricting to a small neighborhood U near a boundary point, we shall
choose special boundary coordinates t1 , . . . , t2n−1 , λ such that t1 , . . . , t2n−1 restricted to ∂X
are coordinates for ∂X. Let Dtj = ∂/∂tj , j = 1, . . . , 2n − 1, and Dλ = ∂/∂λ. Thus Dtj ’s
are the tangential derivatives on ∂X, and Dλ is the normal derivative. For a multi-index β =
(β1 , . . . , β2n−1 ), where each βj is a nonnegative integer, Dtβ denotes the product of Dtj ’s with
β2n−1
order |β| = β1 + · · · + β2n−1 , i.e., Dtβ = Dtβ11 · · · Dt2n−1
. For any φ ∈ C0∞ (X̄) with compact
support in U , we define the tangential Fourier transform for φ in a special boundary chart by
Z
e
φ(ν, λ) =
e−iht,νi φ(t, λ)dt,
R2n−1
J. Inequal. Pure and Appl. Math., 5(3) Art. 70, 2004
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4
O. A BDELKADER AND S. S ABER
where ν = (ν1 , . . . , ν2n−1 ) and ht, νi = t1 ν1 + · · · + t2n−1 ν2n−1 . We define the tangential
Sobolev norms k|·|ks by
Z
Z 0
e λ)|dλdν.
k|φk|s =
(1 + |ν|2 )s |φ(ν,
R2n−1
−∞
¯
We recall the L existence theorem for the ∂−Neumann
operator on any bounded pseudon
2
convex domain X ⊂ C . Following Hörmander L − estimates for ∂¯ on any bounded pseudoconvex domains, one can prove that has closed range and Ker(p,q) () = {0}. The
¯
∂−Neumann
operator N is the inverse of . In fact, one can prove
2
Proposition 2.1 (Hörmander [16]). Let X be a bounded pseudo-convex domain in Cn , n ≥ 2.
For each 0 ≤ p ≤ n and 1 ≤ q ≤ n, there exists a bounded linear operator
N : L2(p,q) (X) −→ L2(p,q) (X)
such that we have the following:
(i) Range(p,q) (N ) ⊂Dom(p,q) () and N = N = I on Dom(p,q) ().
¯ ϕ.
(ii) For any ϕ ∈ L2(p,q) (X), ϕ = ∂¯∂¯? N ϕ + ∂¯? ∂N
(iii) If δ is the diameter of X,we have the following estimates:
eδ 2
kN ϕk ≤
kϕk
q
s
2
¯ ϕk ≤ eδ kϕk
k∂N
q
s
eδ 2
k∂¯? N ϕk ≤
kϕk
q
for any ϕ ∈ L2(p,q) (X).
For a detailed proof of this proposition see Shaw [25], Proposition 2.3, and Chen and Shaw
[5], Theorem 4.4.1.
Theorem 2.2 (Rellich Lemma). Let X be a bounded domain in Cn with Lipschitz boundary. If
s > t ≥ 0, the inclusion W s (X) ,→ W t (X) is compact.
3. P ROOF OF THE M AIN T HEOREM
To prove the main theorem we first obtain the following estimates on each smooth subdomain.
As Lemma 2.1 in Michel and Shaw [23], we prove the following lemma:
Lemma 3.1. Let X ⊂⊂ Cn be a bounded strongly pseudo-convex domain with Lipschitz boundary. Then, there exists an exhaustion {Xµ } of X with the following conditions:
(i) {Xµ } is an increasing sequence of relatively compact subsets of X and ∪µ Xµ = X.
(ii) Each {Xµ } has a C ∞ plurisubharmonic defining Lipschitz function λµ such that
n
X
∂ 2 λµ (z) α β
η η̄ ≥ c1 |η|2
α ∂ z̄ β
∂z
α,β=1
for z ∈ ∂Xµ and η ∈ Cn , where c1 > 0 is a constant independent of µ.
(iii) There exist positive constants c2 , c3 such that c2 ≤ |∇λµ | ≤ c3 on ∂Xµ , where c2 , c3
are independent of µ.
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¯
E STIMATES FOR THE ∂−N
EUMANN O PERATOR
5
Proof. Let ℵ = {z ∈ X| − δ0 < %(z) < 0}, where δ0 > 0 is sufficiently small. Thus, there
exists a constant c1 > 0 such that the function σ0 (z) = %(z)−c1 |z|2 is a plurisubharmonic on ℵ.
Let δµ be a decreasing sequence such that δµ & 0, and we define Xδµ = {z ∈ X|%(z) < −δµ }.
Then {Xδµ } is a sequence of relatively compact subsets of X with union equal to X. Let
Ψ ∈ C0∞ (Cn ) be a function depending only on |z1 |, . . . , |zn | and such that
(i) Ψ ≥ 0.
(ii) Ψ
R = 0 when |z| > 1.
(iii) Ψdλ = 1, where dλ is the Lebesgue measure.
1
We define Ψε (z) = ε2n
Ψ( zε ) for ε > 0.
For each z ∈ Xδµ ,, 0 < ε < δµ , we define
Z
%ε (z) = % ? Ψε (z) = %(z − εζ)Ψ(ζ)dλ(ζ).
Then %ε ∈ C ∞ (Xδµ ) and %ε & % on Xδµ when ε & 0. Since
∂ 2 %ε (z)
∂2
=
{(σ0 (z) + c1 |z|2 ) ? Ψε (z)}
∂z α ∂ z̄ β
∂z α ∂ z̄ β
∂2
= α β {σ0 (z) ? Ψε (z) + c1 |z|2 ? Ψε (z)}
∂z ∂ z̄
∂ 2 |z|2
∂ 2 σ0 (z)
= α β ? Ψε (z) + c1 α β ? Ψε (z)
∂z ∂ z̄
∂z ∂ z̄
n
for z ∈ Xδµ ∩ ℵ, and η ∈ C it follows that
n
n
n
X
X
X
∂ 2 %ε (z) α β
∂ 2 σ0 (z) α β
∂ 2 |z|2 α β
η
η̄
=
η
η̄
?
Ψ
(z)
+
c
η η̄ ? Ψε (z)
ε
1
α ∂ z̄ β
α ∂ z̄ β
α ∂ z̄ β
∂z
∂z
∂z
α,β=1
α,β=1
α,β=1
≥ c1 |η|2 .
Each %εµ is well defined if 0 < εµ < δµ+1 for z ∈ Xδµ+1 . Let c3 = supX̄ |∇%|, then for εµ
sufficiently small, we have %(z) < %εµ (z) < %(z) + c3 εµ on Xδµ+1 . For each µ, we choose
εµ = 2c13 (δµ−1 − δµ ) and ζµ ∈ (δµ+1 , δµ ). We define Xµ = {z ∈ Cn | %εµ < −ζµ }. Since
%(z) < %εµ (z) < −ζµ < −δµ+1 , we have that Xµ ⊂ Xδµ+1 . Also, if z ∈ Xδµ−1 , then %εµ (z) <
%(z) + c3 εµ < −δµ < −ζµ . Thus we have
Xδµ+1 ⊃ Xµ ⊃ Xδµ−1
and (i) is satisfied. Then the function λµ = %εµ + ζµ satisfies (ii). Now, we prove (iii). First,
since a Lipschitz function is almost everywhere differentiable (see Evans and Gariepy [11] for
a proof of this fact), the gradient of a Lipschitz function exists almost everywhere and we have
|∇%| ≤ c3 a.e. in X̄ and |∇λµ | ≤ c3 on ∂Xµ . Secondly, we show that |∇λµ | is uniformly
bounded from below. To do that, since ∂X is Lipschitz from our assumption, then there exists
a finite covering {Vj }1≤j≤m of ∂X such that V̄j ⊂ Ūj for 1 ≤ j ≤ m, a finite set of unit
vectors {χj }1≤j≤m and c2 > 0 such that the inner product (∇%, χj ) ≥ c2 > 0 a.e. for z ∈ Vj ,
1 ≤ j ≤ m. Since this is preserved by convolution, (iii) is proved. Moreover, we have ∇λµ 6= 0
in a small neighborhood of ∂Xµ . Thus ∂Xµ is smooth. Then, the proof is complete.
We use a subscript µ to indicate operators on Xµ .
Proposition 3.2. Let {Xµ } be the same as in Lemma 3.1. There exists a constant c4 > 0, such
that for any ϕ ∈ Λ(p,q) (X̄µ )∩Dom(p,q) (∂¯µ? ), 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1,
? 2 2
2
¯
(3.1)
kϕk1/2(Xµ ) ≤ c4 ∂ϕ Xµ + ∂¯µ ϕX ,
µ
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6
O. A BDELKADER AND S. S ABER
where c4 is independent of ϕ and µ. If ϕ ∈ Λ(p,q) (X̄µ )∩ Dom(p,q) (µ ), then
kϕk21/2(Xµ ) ≤ c4 kµ ϕk2Xµ ,
(3.2)
where c4 is independent of ϕ and µ.
Proof. Since |∇λµ | =
6 0 on a neighborhood W of ∂Xµ , then the function ηµ = λµ /|∇λµ | is
defined on W . We extend ηµ to be negative smoothly inside Xµ . Then ηµ is a defining function
in a neighborhood of X̄µ such that ηµ < 0 on Xµ , ηµ = 0 on ∂Xµ and |∇ηµ | = 1 on W .
Then, by simple calculation as in Lemma 2.2 in Michel and Shaw [23] and by using the identity
of Morrey-Kohn-Hörmander which was proved in Chen and Shaw [5], Proposition 4.3.1, and
from (ii)and (iii) in Lemma 3.1, it follows that there exists a constant c5 > 0 such that for any
ϕ ∈ Λ(p,q) (X̄µ )∩ Dom(p,q) (∂¯µ? ),
Z
X X ∂ϕIJ 2
2
2
?
2
¯
¯
+
(3.3)
|ϕ|
ds
≤
c
k
∂ϕk
+
k
∂
ϕk
µ
5
Xµ
µ
Xµ .
∂ z̄ k I,J
k
∂Xµ
Let z ∈ ∂Xµ and U be a special boundary chart containing z. From Kohn
3.10
Pn [20], Proposition
j
and Chen and Shaw [5], Lemma 5.2.2, the tangential Sobolev norm j=1 |||D ϕ|||−1 , and the
¯ Dom(∂¯? ) where the support of
ordinary Sobolev norm ||ϕ|| are equivalent for ϕ ∈ Dom(∂)∩
ϕ lies in U ∩ X̄µ , Dj ϕ = ∂ϕ/∂xj , (j = 1, 2, . . . , 2n), and > 0. Then, from Folland and Kohn
[12], Theorems 2.4.4 and 2.4.5, it follows that there exists a neighborhood V ⊂ U of z and a
positive constant c6 such that
!
Z
X X ∂ϕIJ 2
2
(3.4)
kϕk21/2(Xµ ) ≤ c6
|ϕ|2 dsµ ,
∂ z̄ k + kϕkXµ +
∂Xµ
I,J k
for ϕ ∈ Λ0,(p,q) (V ∩ X̄µ ). Since Xµ is a Lipschitz domain, then c6 depends only on the Lipschitz
constant. Also from Lemma 3.1, if {Xµ }∞
µ=1 is uniformly Lipschitz, then the constant c6 can be
chosen to depend only on the Lipschitz constant of ∂Xµ , which is independent of µ. Now cover
∂Xµ by finitely many charts {Vi }m
i=1 such that this conclusion holds on each chart, and choose
m
V0 so that Xµ − ∪1 Vi ⊂ V0 ⊂ V̄0 ⊂ Xµ . Then, the estimate (3.4) holds for all ϕ ∈ Λ0,(p,q) (V0 ).
Using a partition of unity subordinate to {Vi }m
0 , the estimate (3.4) now reads
!
Z
X X ∂ϕIJ 2
2
2
2
(3.5)
kϕk1/2(Xµ ) ≤ c6
|ϕ| dsµ ,
∂ z̄ k + kϕkXµ +
∂Xµ
I,J k
for any ϕ ∈ Λ(p,q) (X̄µ )∩ Dom(p,q) (∂¯µ? ). It follows from Proposition 2.1 that
eδ 2 ¯ 2
kϕk2Xµ ≤
k∂ϕkXµ + k∂¯? ϕk2Xµ .
q
2
Therefore, by taking c4 = c6 eδq + c5 , and by using (3.3) and (3.5) inequality (3.1) is proved.
Also, since
¯ 2 + k∂¯? ϕk2 ≤ kµ ϕkXµ kϕkXµ ,
k∂ϕk
Xµ
µ
Xµ
when ϕ ∈ Λ(p,q) (X̄µ )∩Dom(p,q) (µ ). Then, (3.2) is proved also.
Proof of Theorem 1.1. We shall apply the Michel and Shaw technique in [23] with the suitable
¯
modifications required. Let {Xµ } be the same as in Lemma 3.1 and Nµ denote the ∂−Neumann
2
operator on L(p,q) (Xµ ). Since X is a strongly pseudo-convex domain with Lipschitz boundary,
then by using Lemma 3.1, it can be approximated by domains with smooth boundary which are
uniformly Lipschitz. Then, Xµ is a Lipschitz domain, and so C ∞ (X̄µ ) is dense in W s (Xµ ) in the
W s (Xµ )−norm. Then, to prove this theorem, it suffices to prove (1.1) for any ϕ ∈ Λ(p,q) (X̄).
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E STIMATES FOR THE ∂−N
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7
By using the boundary regularity for Nµ which was established by Kohn [19], we have Nµ ϕ ∈
¯
Λ(p,q) (X̄µ )∩ Dom(p,q) (µ ). The ∂−Neumann
operator N is the inverse of the operator . By
using (iii) in Proposition 2.1, we have
kNµ ϕkXµ ≤
eδ 2
eδ 2
kϕkXµ ≤
kϕkX ,
q
q
and
s
¯ µ ϕkXµ + k∂¯? Nµ ϕkXµ ≤ 2
k∂N
µ
s
eδ 2
eδ 2
kϕkX .
kϕkXµ ≤ 2
q
q
Then, there is no loss of generality if we assume that
Nµ ϕ = 0 in
X\Xµ .
Then there is a subsequence of Nµ ϕ, still denoted by Nµ ϕ, converging weakly to some element
¯ ∈ L2
¯
ψ ∈ L2(p,q) (X) and ∂ψ
(p,q+1) (X). This implies that ψ ∈ Dom(p,q) (∂). Now, we show that
¯ ∩ L2
ψ ∈ Dom(p,q) (∂¯µ? ) as follows: for any u ∈ Dom(p,q−1) (∂)
(p,q−1) (X),
¯
¯
ψ, ∂u
= lim | Nµ ϕ, ∂u
|
X
Xµ
µ−→∞
¯?
= lim ∂µ Nµ ϕ, u X µ
µ−→∞
≤ kϕkX kukX .
¯ ∈ Dom(p,q+1) (∂¯? ) and ∂¯? ψ ∈ Dom(p,q−1) (∂)
¯ as
Thus ψ ∈ Dom(p,q) (∂¯µ? ). Also, we show that ∂ψ
µ
µ
follows: by using (ii) in Proposition 2.1, we have
?
¯ µ ϕ2 = kϕk2 ≤ kϕk2 .
∂¯∂¯µ Nµ ϕ2 + ∂¯µ? ∂N
(3.6)
Xµ
X
X
X
µ
µ
¯
Thus ∂¯∂¯µ? ψ is the L2 weak limit of some subsequence of ∂¯∂¯µ? Nµ ϕ and ∂¯µ? ψ ∈ Dom(p,q−1) (∂).
¯ ∩ L2 (X),
By using (3.6), we have, for any v ∈ Dom(p,q) (∂)
(p,q)
¯ µ ϕ, ∂v
¯
¯ ∂v
¯
= lim ∂N
∂ψ,
Xµ
X
µ−→∞
¯? ¯
= lim ∂µ ∂Nµ ϕ, v X µ
µ−→∞
≤ kϕkX kvkX .
¯ is the weak limit of a subsequence of ∂¯? ∂N
¯ µ ϕ. This
¯ ∈ Dom(p,q+1) (∂¯? ) and ∂¯? ∂ψ
Thus ∂ψ
µ
µ
µ
implies that ψ ∈ Dom(p,q) (µ ) and µ ψ = ϕ. Since N is one to one on L2(p,q) (X), then we
conclude that ψ = N ϕ. Since Xµ is a Lipschitz domain. Hence Λ(X̄) are dense in W s (X) in
W s (X)−norm. If s ≤ 1/2, we can show that Λ0 (X̄) are dense in W s (X) as in Theorem 1.4.2.4
in Grisvard [13]. Thus
1/2
W 1/2 (X) = W0 (X).
It follows from the Generalized Schwartz inequality (see Proposition (A.1.1) in Folland and
Kohn [12]) that
hh, f iXµ ≤ khk1/2(Xµ ) kf k−1/2(Xµ ) ,
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8
O. A BDELKADER AND S. S ABER
−1/2
1/2
for any h ∈ W(p,q) (Xµ ) and f ∈ W(p,q) (Xµ ). By using (3.1), there exists a constant c4 > 0
such that for any ϕ ∈ L2(p,q) (X̄µ )∩ Dom(p,q) (µ ), 0 ≤ p ≤ n and 1 ≤ q ≤ n,
¯ 2 + k∂¯? ϕk2 )
kϕk21/2(Xµ ) ≤ c4 (k∂ϕk
µ
Xµ
Xµ
= c4 hϕ, µ ϕiXµ
≤ c4 kϕk1/2(Xµ ) kµ ϕk−1/2(Xµ ) ,
(3.7)
where c4 is independent of ϕ and µ. Substituting Nµ ϕ into (3.7), we have
kNµ ϕk1/2(Xµ ) ≤ c4 kµ Nµ ϕk−1/2(Xµ ) = c4 kϕk−1/2(Xµ ) ,
(3.8)
where c4 is independent of ϕ and µ. By using the extension operator on Euclidean space (see
Theorem 1.4.3.1 in Grisvard [13]), it follows that for any Lipschitz domain Xµ ⊂ Cn ,
Rµ : W 1/2 (Xµ ) −→ W 1/2 (Cn )
such that for each ϕ ∈ W 1/2 (Xµ ), Rµ ϕ = ϕ on Xµ and
kRµ ϕk1/2(Cn ) ≤ c5 kϕk1/2(Xµ ) ,
(3.9)
for some positive constant c5 . The constant c5 in (3.9) can be chosen independent of µ since
extension exists for any Lipschitz domain (see Theorem 1.4.3.1 in Grisvard [13]). By applying
Rµ to Nµ ϕ component-wise, we have, by using (3.8) and (3.9), that
kRµ Nµ ϕk1/2(X) ≤ kRµ Nµ ϕk1/2(Cn ) ≤ c5 kNµ ϕk1/2(Xµ ) ≤ ckϕk−1/2(Xµ ) ,
1/2
where c > 0 is independent of µ. Since W(p,q) (X) is a Hilbert space, then from the weak
compactness for Hilbert spaces, there exists a subsequence of Rµ Nµ ϕ which converges weakly
1/2
in W(p,q) (X). Since Rµ Nµ ϕ converges weakly to N ϕ in L2(p,q) (X), we conclude that N ϕ ∈
1/2
W(p,q) (X) and
kN ϕk1/2(X) ≤ lim kRµ Nµ ϕk1/2(Xµ ) ≤ ckϕk−1/2(X) .
µ−→∞
−1/2
1/2
Thus, N can be extended as a bounded operator from W(p,q) (X) to W(p,q) (X).
To prove that N is compact, we note that for any bounded domain X with Lipschitz boundary
there exists a continuous linear operator
R : W 1/2 (X) −→ W 1/2 (Cn )
such that Rφ|X = φ. Also, we note that the inclusion map
W 1/2 (X) −→ L2 (X) = W 0 (X)
is compact. Thus, by using the Rellich Lemma for Cn , we conclude that
W 1/2 (X) ,→ W −1/2 (X)
−1/2
is compact and this proves that N is compact on W(p,q) (X) and L2(p,q) (X).
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