Journal of Inequalities in Pure and
Applied Mathematics
ESTIMATES FOR THE ∂−NEUMANN OPERATOR ON STRONGLY
PSEUDO-CONVEX DOMAIN WITH LIPSCHITZ BOUNDARY
volume 5, issue 3, article 70,
2004.
O. ABDELKADER AND S. SABER
Mathematics Department
Faculty of Science
Minia University
El-Minia, Egypt.
Mathematics Department
Faculty of Science
Cairo University
Beni-Swef Branch, Egypt.
EMail: sayedkay@yahoo.com
Received 29 February, 2004;
accepted 19 April, 2004.
Communicated by: J. Sándor
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
101-04
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Abstract
On a bounded strongly pseudo-convex domain X in Cn with a Lipschitz bound¯
ary, 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.
2000 Mathematics Subject Classification: Primary 35N15; Secondary 32W05.
Key words: Sobolev estimate, Neumann problem, Lipschitz domains.
Contents
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
O. Abdelkader and S. Saber
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
¯
2
Notations and the ∂−Neumann
Problem . . . . . . . . . . . . . . . . . 5
3
Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
Let X be a bounded pseudo-convex domain in Cn with the standard Hermi¯
tian metric. The ∂−Neumann
operator N is the (bounded) inverse of the (un¯
bounded) Laplace-Beltrami operator . 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
s
(X) be the Hilbert spaces of
is studied in Henkin and Iordan [14]. Let W(p,q)
s
(p, q)−forms with W (X)−coefficients. Henkin, Iordan, and Kohn in [15] and
1/2
Michel and Shaw in [23] showed that N is bounded from L2(p,q) (X) to W(p,q) (X)
on domains with piecewise smooth strongly pseudo-convex boundary by two
different methods. Also Michel and Shaw in [24] proved that N is bounded
1/2
on W(p,q) (X) when the domain is only bounded pseudo-convex Lipschitz with
a plurisubharmonic defining 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:
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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)
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satisfies the following estimate: for any ϕ ∈ L2(p,q) (X), there exists a constant
c > 0 such that
(1.1)
kN ϕk1/2(X) ≤ ckϕk−1/2(X) ,
O. Abdelkader and S. Saber
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where c = c(X) is independent of ϕ; i.e., N can be extended as a bounded op−1/2
1/2
erator from W(p,q) (X) into W(p,q) (X). In particular, N is a compact operator
−1/2
on L2(p,q) (X) and W(p,q) (X).
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
O. Abdelkader and S. Saber
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2.
¯
Notations and the ∂−Neumann
Problem
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
ϕ=
ϕIJ dz I ∧ dz J ,
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 )},
O. Abdelkader and S. Saber
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
|ϕ|2 dv,
hϕ, ψi =
ϕIJ ψ IJ dv, and kϕk =
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
¯ =
∂ϕ
X X ∂ϕIJ
k
IJ
∂ z̄ k
dz̄ k ∧ dz I ∧ dz̄ J .
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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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) −→ L2(p,q) (X)
¯ Then ∂¯ is a closed, linear, densely
the maximal extension of the original ∂.
defined 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 of ∂¯ in L2(p,q) (X) by Dom(p,q) (∂)
¯ respectively.
and Range(p,q) (∂)
We define the Laplace-Beltrami operator
O. Abdelkader and S. Saber
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= ∂¯∂¯∗ + ∂¯∗ ∂¯ : L2(p,q) (X) −→ L2(p,q) (X)
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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) (∂¯? );
Ker(p,q) () = {ϕ ∈ Dom(p,q) (∂)∩
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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¯ = 0 and
∂ϕ
∂¯? ϕ = 0}.
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:
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(i) X ∩ U = {z ∈ U ; λ(z) < 0}.
(ii)
∂ 2 λ(z) α β
α,β=1 ∂z α ∂ z̄ β η η̄
Pn
≥ L(z)|η|2 ; z ∈ U , η = (η 1 , . . . , η n ) ∈ Cn and
L(z) > 0.
(iii) The gradient ∇λ(z) =
∂λ(z) ∂λ(z)
, ∂y1 , . . . , ∂λ(z)
, ∂λ(z)
∂x1
∂xn
∂y n
6= 0
for z = (z 1 , . . . , z n ) ∈ U ; z α = xα + iy α .
Let f : <2n−1 −→ < be a function that satisfies the Lipschitz condition
|f (x) − f (x0 )| ≤ T |x − x0 | for all
(2.1)
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,
(2.2)
n
X
α,β
∂2% α β
η η̄ ≥ c1 |η|2 ,
∂z α ∂ z̄ β
Estimates for the ∂−Neumann
Operator On Strongly
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Lipschitz Boundary
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η = (η 1 , . . . , η n ) ∈ Cn ,
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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
β2n−1
product of Dtj ’s with 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
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
k|φk|s =
R2n−1
−∞
e λ)|dλdν.
(1 + |ν|2 )s |φ(ν,
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
O. Abdelkader and S. Saber
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¯
We recall the L2 existence theorem for the ∂−Neumann
operator on any bounded
n
pseudo-convex domain X ⊂ C . Following Hörmander L2 − 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
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).
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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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.
Estimates for the ∂−Neumann
Operator On Strongly
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Lipschitz Boundary
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3.
Proof of the Main Theorem
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.
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
Estimates for the ∂−Neumann
Operator On Strongly
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Lipschitz Boundary
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(i) Ψ ≥ 0.
(ii) Ψ = 0 when |z| > 1.
R
(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
= α β {(σ0 (z) + c1 |z|2 ) ? Ψε (z)}
α
β
∂z ∂ z̄
∂z ∂ z̄
∂2
= α β {σ0 (z) ? Ψε (z) + c1 |z|2 ? Ψε (z)}
∂z ∂ z̄
∂ 2 σ0 (z)
∂ 2 |z|2
= 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
X
∂ 2 %ε (z) α β
η η̄
α ∂ z̄ β
∂z
α,β=1
=
n
X
2
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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n
X
2
2
∂ σ0 (z) α β
∂ |z| α β
η
η̄
?
Ψ
(z)
+
c
η η̄ ? Ψε (z)
ε
1
α ∂ z̄ β
α ∂ z̄ β
∂z
∂z
α,β=1
α,β=1
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≥ c1 |η|2 .
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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 .
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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 + ∂¯? ϕ2 ,
(3.1)
kϕk21/2(Xµ ) ≤ c4 ∂ϕ
µ
Xµ
X
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µ
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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 MorreyKohn-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| dsµ ≤ c5 k∂ϕkXµ + k∂µ ϕkXµ .
∂ z̄ k +
I,J
k
∂Xµ
Let z ∈ ∂Xµ and U be a special boundary chart containing z. From Kohn [20],
Proposition
3.10 and Chen and Shaw [5], Lemma 5.2.2, the tangential Sobolev
P
norm nj=1 |||Dj ϕ|||−1 , and the ordinary Sobolev norm ||ϕ|| are equivalent for
¯ Dom(∂¯? ) where the support of ϕ lies in U ∩ X̄µ , Dj ϕ = ∂ϕ/∂xj ,
ϕ ∈ Dom(∂)∩
(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
Estimates for the ∂−Neumann
Operator On Strongly
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Lipschitz Boundary
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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 V0
so that Xµ − ∪m
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
(3.5)
kϕk21/2(Xµ ) ≤ c6
|ϕ|2 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
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
kϕk2Xµ ≤
¯ 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
operator on L2(p,q) (Xµ ). Since X is a strongly
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
O. Abdelkader and S. Saber
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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). 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 ≤
and
eδ 2
eδ 2
kϕkXµ ≤
kϕkX ,
q
q
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
¯ ∈ L2
to some element ψ ∈ L2(p,q) (X) and ∂ψ
(p,q+1) (X). This implies that ψ ∈
¯
Dom(p,q) (∂). Now, we show that ψ ∈ Dom(p,q) (∂¯µ? ) as follows: for any u ∈
¯ ∩ L2
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−→∞
Estimates for the ∂−Neumann
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Lipschitz Boundary
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≤ kϕkX kukX .
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¯ ∈ Dom(p,q+1) (∂¯? ) and ∂¯? ψ ∈
Thus ψ ∈ Dom(p,q) (∂¯µ? ). Also, we show that ∂ψ
µ
µ
¯ as follows: by using (ii) in Proposition 2.1, we have
Dom(p,q−1) (∂)
(3.6)
?
¯ µ ϕ2 = kϕk2 ≤ kϕk2 .
∂¯∂¯µ Nµ ϕ2 + ∂¯µ? ∂N
Xµ
X
X
X
µ
µ
Thus ∂¯∂¯µ? ψ is the L2 weak limit of some subsequence of ∂¯∂¯µ? Nµ ϕ and ∂¯µ? ψ ∈
¯ By using (3.6), we have, for any v ∈ Dom(p,q) (∂)
¯ ∩ L2 (X),
Dom(p,q−1) (∂).
(p,q)
¯ ∂v
¯
¯ µ ϕ, ∂v
¯
∂ψ,
= lim ∂N
X
Xµ
µ−→∞
¯ µ ϕ, v
= lim ∂¯µ? ∂N
X
µ−→∞
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
µ
≤ kϕkX kvkX .
¯ ∈ Dom(p,q+1) (∂¯? ) and ∂¯? ∂ψ
¯ is the weak limit of a subsequence of
Thus ∂ψ
µ
µ
?¯
¯
∂µ ∂Nµ ϕ. This 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 ) ,
O. Abdelkader and S. Saber
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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
Xµ
µ
Xµ
= c4 hϕ, µ ϕiXµ
(3.7)
≤ c4 kϕk1/2(Xµ ) kµ ϕk−1/2(Xµ ) ,
where c4 is independent of ϕ and µ. Substituting Nµ ϕ into (3.7), we have
(3.8)
kNµ ϕk1/2(Xµ ) ≤ c4 kµ Nµ ϕk−1/2(Xµ ) = c4 kϕk−1/2(Xµ ) ,
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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 ,
O. Abdelkader and S. Saber
Rµ : W 1/2 (Xµ ) −→ W 1/2 (Cn )
Contents
such that for each ϕ ∈ W 1/2 (Xµ ), Rµ ϕ = ϕ on Xµ and
(3.9)
kRµ ϕk1/2(Cn ) ≤ c5 kϕk1/2(Xµ ) ,
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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
Page 18 of 22
kRµ Nµ ϕk1/2(X) ≤ kRµ Nµ ϕk1/2(Cn ) ≤ c5 kNµ ϕk1/2(Xµ ) ≤ ckϕk−1/2(Xµ ) ,
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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
1/2
which converges weakly in W(p,q) (X). Since Rµ Nµ ϕ converges weakly to N ϕ
1/2
in L2(p,q) (X), we conclude that N ϕ ∈ 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).
Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
O. Abdelkader and S. Saber
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Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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Estimates for the ∂−Neumann
Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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Operator On Strongly
Pseudo-Convex Domain With
Lipschitz Boundary
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