1. Introduction
Let be a domain in Cn. According to the fundamental work of Kohn and Hörmander in the sixties, if is bounded and pseudoconvex then for every 1 ≤ q ≤ n,
the complex Laplacian q on square integrable (0,q) forms on has a bounded
inverse, denoted by Nq. This is the ∂-Neumann operator Nq. The most basic property of Nq is that if v is a ∂ closed (0,q + 1)-form, then u := ∂∗Nq+1v provides
the canonical solution to ∂u = v, namely the one orthogonal to the kernel of ∂ and
so the one with minimal norm (see Corollary 2.10 in [11]). In this paper, we are
interested in the existence and compactness of Nq on (possibly unbounded or nonsmooth) q-convex domains, a generalization of pseudoconvex domains in which
the existence of plurisubharmonic exhaustion function is replaced by existence of
q-subharmonic ones. Our research is motivated from the fact that compactness of
Nq
implies global regularity in the sense of preservation of Sobolev spaces. Up to
now, there is no complete characterization for compactness of Nq even in the case
q = 1 and is a smooth bounded domain in C. However, important progresses in
this direction of research have been made following the ground breaking paper [2]
in which Catlin introduce the notion of domains having the property (Pq). Recently,
Gansberger and Haslinger studied compactness estimates for the ∂-Neumann operator in weighted L2-spaces and the weighted ∂-Neumann problem on unbounded
domains in Cn (see [5] and [6]). We would like to remark that in [5], instead of
using Rellich’s lemma, the author obtained compactness of the weighted Neumann
operator Nq,ϕ under a strong assumption on rapidly increasing of the gradient ∇ϕ
and the Laplacian ϕ at the infinite point and at boundary points of a domain
(Proposition 4.5 in [5]). The main step in Gansberger’s proof is to show that under
the above assumption the embedding H01(,ϕ,∇ϕ) → L2(,ϕ) is compact

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Manuscripta Mathematica
ISSN 0025-2611
manuscripta math.
DOI 10.1007/s00229-014-0661-2
Existence and compactness for the $
${\overline{\partial}}$$ ∂ ¯ -Neumann
operator on q-convex domains
Mau Hai Le, Quang Dieu Nguyen &
Xuan Hong Nguyen
1 23
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manuscripta math. © Springer-Verlag Berlin Heidelberg 2014
Mau Hai Le · Quang Dieu Nguyen · Xuan Hong Nguyen
Existence and compactness for the ∂-Neumann
operator on q-convex domains
Received: 3 April 2012 / Revised: 11 December 2012
Abstract. The aim of this paper is to give a sufficient condition for existence and compact-
ness of the ∂-Neumann operator Nq on L2(0,q)() in the case is an arbitrary q-convex
domain in Cn .
1. Introduction
Let be a domain in Cn . According to the fundamental work of Kohn and Hör-
mander in the sixties, if is bounded and pseudoconvex then for every 1 ≤ q ≤ n,
the complex Laplacian q on square integrable (0, q) forms on has a bounded
inverse, denoted by Nq . This is the ∂-Neumann operator Nq . The most basic prop-
erty of Nq is that if v is a ∂ closed (0, q + 1)-form, then u := ∂∗Nq+1v provides
the canonical solution to ∂u = v, namely the one orthogonal to the kernel of ∂ and
so the one with minimal norm (see Corollary 2.10 in [11]). In this paper, we are
interested in the existence and compactness of Nq on (possibly unbounded or non-
smooth) q-convex domains, a generalization of pseudoconvex domains in which
the existence of plurisubharmonic exhaustion function is replaced by existence of
q-subharmonic ones. Our research is motivated from the fact that compactness of
Nq implies global regularity in the sense of preservation of Sobolev spaces. Up to
now, there is no complete characterization for compactness of Nq even in the case
q = 1 and is a smooth bounded domain in C. However, important progresses in
this direction of research have been made following the ground breaking paper [2]
in which Catlin introduce the notion of domains having the property (Pq). Recently,
Gansberger and Haslinger studied compactness estimates for the ∂-Neumann oper-
ator in weighted L2-spaces and the weighted ∂-Neumann problem on unbounded
domains in Cn (see [5] and [6]). We would like to remark that in [5], instead of
using Rellich’s lemma, the author obtained compactness of the weighted Neumann
operator Nq,ϕ under a strong assumption on rapidly increasing of the gradient ∇ϕ
M. H. Le · Q. D. Nguyen(B)· X. H. Nguyen: Department of Mathematics, Hanoi National
University of Education (Dai hoc Su Pham Ha Noi), 136 Xuan Thuy Street, Caugiay District,
Hanoi, Vietnam. e-mail: dieu_vn@yahoo.com
M. H. Le: e-mail: mauhai@fpt.vn
X. H. Nguyen: e-mail: xuanhongdhsp@yahoo.com
Mathematics Subject Classification (2010): Primary 32W05
DOI: 10.1007/s00229-014-0661-2
Author's personal copy
M. H. Le et al.
and the Laplacian ϕ at the infinite point and at boundary points of a domain
(Proposition 4.5 in [5]). The main step in Gansberger’s proof is to show that under
the above assumption the embedding H10 (, ϕ,∇ϕ) ↪→ L2(, ϕ) is compact.
The paper is organized as follows. In Sect. 2 we recall basic facts about q-
subharmonic functions and q-convex domains. In particular, we note that Corollary
2.13 in [11] is still valid for bounded q-convex domains in Cn . Section 3 is devoted
to present the property (P ′q), a slight modification of the property (Pq) introduced
earlier in the case of bounded domains by Catlin [2] and Straube [10,11]. Roughly
speaking, we say that a (possibly unbounded) domain has the property (P ′q) if
admits a real valued, bounded, smooth function ϕ having the property that the
sum of the q smallest eigenvalues of the complex Hessian of ϕ goes to infinity
at the infinite point and at ∂. We should say that this notion is motivated from
Theorem 1.3 in [5] in which the case where q = 1 and ∂ is smooth is studied.
The main result of the section is Theorem 3.3 which gives sufficient conditions
for domain having the property (P ′q). The existence and compactness of the
∂-Neumann operator Nq on q-convex domains are discussed in Sect. 4. We start
the section with a geometric necessary condition for compactness of Nq . The main
result of the paper is Theorem 4.3. Here we prove that if ⊂ Cn is a q-convex
domain having property (P ′q) then there exists a bounded ∂-Neumann operator Nq
on L2(0,q)() and Nq is compact. Our proof exploits the property (P
′
q) of and
some techniques from the book [11].
Notation. λ2n denotes the Lebesgue measure on Cn and B(a, r) is the ball with
center a ∈ Cn and radius r > 0. For a real valued function u ∈ C2() we define
Hq(u)(z) to be the sum of the q smallest eigenvalues of the complex Hessian of u
acting at the point z.
2. Preliminaries
A complex-valued differential form u of type (0, q) on an open subset ⊂ Cn can
be expressed as u = ∑|J |=q ′u J dz J , where J are strictly increasing multi-indices
with lengths q and {u J } are defined functions on . Let C∞(0,q)() be the space of
complex-valued differential forms of class C∞ and of type (0, q) on . By C∞0 ()
we denote the space of C∞ functions with compact support in . We use L2(0,q)()
to denote the space of (0, q)-forms on with square-integrable coefficients. If
u, v ∈ L2(0,q)(), the weighted L2-inner product and norms are defined by
(u, v) =
∫
∑
|J |=q
′
u J v J dλ2n and ‖u‖2 = (u, u).
The ∂-operator on (0, q)-forms is given by
∂
( ∑
|J |=q
′
u J dz J
)
=
∑
|J |=q
′
n∑
j=1
∂u J
∂z j
dz j ∧ dz J ,
Author's personal copy
Existence and compactness for the ∂-Neumann operator
where
∑ ′
means that the sum is only taken over strictly increasing multi-indices J .
The derivatives are taken in the sense of distributions, and the domain of ∂ consists
of those (0, q)-forms for which the right hand side belongs to L2(0,q+1)(). So ∂
is a densely defined closed operator, and therefore has an adjoint operator from
L2(0,q+1)() into L
2
(0,q)() denoted by ∂
∗
. For u = ∑|J |=q+1 ′u J dz J ∈ dom(∂∗)
one has
∂
∗
u = −
∑
|K |=q
′
n∑
j=1
∂u j K
∂z j
dzK .
The complex Laplacian on (0, q)-forms is defined as
q := ∂∂∗ + ∂∗∂,
where the symbol q is understood as the maximal closure of the operator initially
defined on (0, q)-forms with coefficients in C∞0 (). q is a self adjoint, positive
operator. The associated Dirichlet form is denoted by
Q( f, g) = (∂ f, ∂g) + (∂∗ f, ∂∗g),
for f, g ∈ dom(∂) ∩ dom(∂∗). The weighted ∂-Neumann operator Nq is—if it
exists—the bounded inverse of q . We refer the reader to the monographs [11] for
a complete survey on ∂-Neumann operators and their applications to other prob-
lems in several complex variables. Next, we recall the definition of q-subharmonic
functions which is an extension of plurisubharmonic functions (see [1,7,8]).
Definition 2.1. Let be a domain in Cn . An upper semicontinuous function u :
−→ [−∞,∞), u ≡ −∞ is called q-subharmonic if for every q-dimensional
complex plane L in Cn , u|L is a subharmonic function on L ∩ .
The set of all q-subharmonic functions on is denoted by SHq().
The function u is called to be strictly q-subharmonic if for every U there
exists constant CU > 0 such that u(z) − CU |z|2 ∈ SHq(U ).
Remark 2.2. (a) q-subharmonicity and strict q-subharmonicity are local
properties.
(b) 1-subharmonicity (resp. n-subharmonicity) coincides with plurisubharmonic-
ity (resp. subharmonicity).
We list below basic properties of q-subharmonic functions that will be used
later on (see [7]).
Proposition 2.3. Let be an open set in Cn and let q is an integer with 1 q n.
Then we have.
(a) If u ∈ SHq() then u ∈ SHr (), for every q r n.
(b) If u, v ∈ SHq() and α, β > 0 then αu + βv ∈ SHq().
(c) If {u j }∞j=1 is a family of q-subharmonic functions, u = sup j u j < +∞ and u
is upper semicontinuous then u is a q-subharmonic function.
(d) If {u j }∞j=1 is a family of nonnegative q-subharmonic functions such that u =∑∞
j=1 u j < +∞ and u is upper semicontinuous then u is q-subharmonic.
Author's personal copy
M. H. Le et al.
(e) If {u j }∞j=1 is a decreasing sequence of q-subharmonic functions then so is
u = lim
j→+∞ u j .
(f) Let ρ ≥ 0 be a smooth radial function in Cn vanishing outside the unit ball and
satisfying ∫
Cn
ρdV = 1. For u ∈ SHq() we define
uε(z) := (u ∗ ρε)(z) =
∫
B(0,ε)
u(z − ξ)ρε(ξ)dλ2n(ξ), ∀z ∈ ε,
where ρε(z) := 1ε2n ρ(z/ε) and ε = {z ∈ : d(z, ∂) > ε}. Then uε ∈
SHq(ε) ∩ C∞(ε) and uε ↓ u as ε ↓ 0.
(g) Let u1, . . . , u p ∈ SHq() and χ : Rp → R be a convex function which is
non decreasing in each variable. If χ is extended by continuity to a function
[−∞,+∞)p → [−∞,∞), then χ(u1, . . . , u p) ∈ SHq().
The property (g) in the cases q = 1 and q = n are given in Theorem 5.6 and
Theorem 4.16 in [4]. These proofs can be easily extended to the general case. We
will use (f) and (g) in the proof of Theorem 3.5 to produce a version of Richberg’s
regularization lemma for continuous strictly q-subharmonic functions. We should
remark that for 2 ≤ q ≤ n, the class of q-subharmonic functions is not invariant
under biholomorphic changes of coordinates.
We give some equivalent conditions for q-subharmonicity which is similar to
plurisubharmonicity (see [1,8]).
Proposition 2.4. Let be a domain in Cn and let q be an integer with 1 q n.
Let u be a real valued C2-function defined on . Then the following are equivalent:
(a) u is a q-subharmonic function .
(b) i∂∂u ∧ (i∂∂|z|2)q−1 0 i.e., Hq(u)(z) ≥ 0 for every z ∈ .
(c) For every (0, q)-form f = ∑|J |=q ′ f J dz J we have
∑
|K |=q−1
′
n∑
j,k=1
∂2u
∂z j∂zk
f j K f kK 0.
We also have the following simple result about smoothing q-subharmonic functions.
Proposition 2.5. Let be an open set in Cn and let u ∈ SHq() such that u −
δ|idCn |2 ∈ SHq() for some δ > 0. Then for every ε > 0 we have uε − δ|idCn |2 ∈
SHq(ε), where ε := {z ∈ : d(z, ∂) > ε}.
Proof. By Proposition 2.3 (f) we have (u − δ|idCn |2)ε ∈ SHq(ε). Since
(u − δ|idCn |2)ε(z) = uε(z) − δ
∫
B(0,ε)
|z − w|2ρε(w)dV (w)
= uε(z) − δ|z|2 − δ
∫
B(0,ε)
(2(z,−w) + |w|2)ρε(w)dV (w)
= uε(z) − δ|z|2 − v(ε)(z),
Author's personal copy
Existence and compactness for the ∂-Neumann operator
where v(ε)(z) := δ
∫
B(0,ε)(2(z,−w) + |w|2)ρε(w)dV (w) is a pluriharmonic
function in Cn . Hence, uε − δ|idCn |2 = (u − δ|idCn |2)ε + v(ε) ∈ SHq(ε). This
completes the proof. unionsq
The following definition is an extension of pseudoconvexity.
Definition 2.6. A domain ⊂ Cn is said to be q-convex if there exists a q-
subharmonic exhaustion function on . Moreover, a C2 smooth bounded domain
is called strictly q-convex if it admits a C2 smooth defining function which is
strictly q-subharmonic on a neighbourhood of .
It is not clear if we can find a smooth strictly q-subharmonic exhaustion function
on a q-convex domain. However by Proposition 2.7 in [7], every q-convex domain
can be written as an increasing union of bounded q-convex domains j such that
each j has a smooth strictly q-convex exhaustion function. Using Sard theorem
this result can be refined as follows.
Proposition 2.7. Let be a q-convex domain in Cn. Then can be written as
=
∞⋃
j=1
j such that j j+1 and each j is a strictly q-convex domain.
According to a classical result of Green-Wu, every domain in Cn is n-convex (see
Theorem 9.3.5 in [4] for an elegant proof). In the case where 1 < q < n, there
exists no geometric characterization for q-convexity. However, in analogy with
the classical Kontinuitassatz principle for pseudoconvexity we have the following
partial result.
Proposition 2.8. Let be a domain in Cn and p ∈ ∂. Assume that there exist
r > 0, a sequence {p j } ⊂ , p j → p, and a sequence of q-dimensional complex
subspaces L j satisfying the following conditions:
(a) p j ∈ L j for every j ≥ 1,
(b) ∂B(p, r) ∩ L j is contained in a fixed compact subset K of .
Then is not q-convex.
Proof. Suppose that there exists a q-subharmonic exhaustion function u on .
Since u|L j is subharmonic, by the maximum principle and the assupmtions (a) and
(b) we get for every j ≥ 1 the inequality u(p j ) ≤ sup
K
u. By letting j → ∞ we
arrive at a contradiction. unionsq
Remark 2.9. (i) Let := {z ∈ C3 : 1 < |z| < 3} and p = (1, 0, 0) ∈ ∂.
Denote by L the hyperplane tangent to ∂ at the point p. Consider the
sequences of points p j := (1 + 1/j, 0, 0), j ≥ 1 tending to p and hyper-
planes L j := L + (1/j, 0, 0) passing through the points p j . Using the above
result, we can see that is not 2-convex.
(ii) In [8], the following generalization of Levi convexity was introduced. We say
that a bounded domain in Cn with a C2 smooth defining function ρ is weakly
q-convex if for every p ∈ ∂, for every (0, q)-form f = ∑|J |=q ′ f J dz J
Author's personal copy
M. H. Le et al.
satisfying
n∑
i=1
∂ρ
∂zi
(p) fi K = 0 ∀|K | = q − 1,
we have
∑
|K |=q−1
′
n∑
j,k=1
∂2ρ
∂z j∂zk
f j K f kK 0.
It follows from Theorem 2.4 in [8] that every C∞ smooth bounded weakly
q-convex domain is q-convex in our sense. Unfortunately, we do not know if
the reverse implication is true.
The following proposition similar as Corollary 2.13 in [11] is still valid for bounded
q-convex domains in Cn .
Proposition 2.10. Let be a bounded q-convex domain in Cn, u = ∑|J |=q ′u J dz¯ J
∈ dom(∂¯) ∩ dom(∂¯∗) ⊂ L2(0,q)(). Then for all b ∈ C2(), b 0 the following
holds
∑
|K |=q−1
′
n∑
j,k=1
∫
eb
∂2b
∂z j∂ z¯k
u j K ukK dλ2n ‖∂u‖2 + ‖∂¯∗u‖2.
3. The property (P′q)
First we recall an important concept introduced and investigated by Catlin in [2]
and Straube in [11] (see also [9]). We say that a compact set K in Cn has the
property (Pq) if for every M > 0, there exist a neighborhood UM of K , a C2
smooth function λM on UM such that 0 λM (z) 1, z ∈ UM and for any
z ∈ UM , the sum of the smallest q eigenvalues of the complex Hessian of λ is at
least M (or, equivalently, λM − Mq |z|2 ∈ SHq(UM )). Moreover, given a closed set
E (not necessarily bounded), we say that E locally has the property (Pq) if for
every z0 ∈ E we can find a compact neighborhood K of z0 in E such that K has
the (Pq) property.
Using Kohn–Morrey–Hörmander formula in [11], it is not hard to prove (see
[2] for the case q = 1 and [11] for general q) that if is a smoothly bounded
pseudoconvex domain in Cn with the boundary ∂ having the property (Pq) then
the ∂-Neumann operator Nq is compact.
The following notion is the key to our research on compactness of Nq in the
case where is unbounded.
Definition 3.1. Let be a domain in Cn . We say that has the property (P ′q) if
there exists a real valued, bounded C2 smooth function ϕ on such that for every
positive number M , we have ϕ(z) − M |z|2 ∈ SHq(\KM ) for some compact
subset KM of .
Author's personal copy
Existence and compactness for the ∂-Neumann operator
Remark 3.2. (i) The function ϕ is not assumed to be q-subharmonic on the whole
. We will prove, however, that ϕ can be chosen to have this additional prop-
erty.
(ii) If has the property (P ′q) then for every complex space L of dimension
q, ∩ L is quasibounded (in L) i.e., ∩ L contains only a finite number
of disjoint balls with fixed radii (see Definition 1.4 in [5]). Indeed, it suffices
to prove the statement for the case q = n. Assume for the sake of seeking a
contradiction that we can find a sequence of disjoint balls B(z j , r) contained
in with |z j | → ∞. By passing to a subsequence, we can find a sequence m j
of real numbers such that m j → +∞ and ϕ(z)− m j |z|2 ∈ SHn(B(z j , r)) for
every j ≥ 1. Now we let θ ≥ 0 be a smooth function with compact support
in B(0, r) such that θ = 1 on B(0, r/2). Set θ j (z) := θ(z − z j ). By Stoke’s
theorem we have
∫
B(z j ,r/2)
i∂∂ϕ ∧ (i∂∂|z|2)n−1 ≤
∫
B(z j ,r)
θ j i∂∂ϕ ∧ (i∂∂|z|2)n−1
=
∫
B(z j ,r)
iϕ∂∂θ j ∧ (i∂∂|z|2)n−1
≤ C‖ϕ‖λ2n(B(z j , r)).
Here C > 0 is a constant depends only on the second derivatives of θ . It
follows that there exists C ′ > 0 depends only on n such that m j ≤ C ′‖ϕ‖
for every j ≥ 1. By letting j → ∞ we get a contradiction.
It is easy to see that finite intersection of domains possessing the (P ′q) property
still has this property. The main result of the section provides a substantial class of
domains satisfying the property (P ′q). More precisely, we have
Theorem 3.3. Let be an open set in Cn with ∂ locally has the property (Pq).
Assume that there exist negative q-subharmonic functions ρ, ρ˜ on satisfying the
following conditions.
(a) ρ ∈ C2(), ρ(z) − |z|2 ∈ SHq().
(b) lim|z|→∞ Hq (ρ)(z)1+ρ(z)2 = ∞.
(c) ρ˜ is strictly q-subharmonic on and satisfies lim
z→ξ ρ˜(z) = 0 for every ξ ∈ ∂.
Then has the property (P ′q).
We first need the following result which generalizes in part Theorem 2.1 in [9]
where the case q = 1 was treated.
Lemma 3.4. Let be a bounded domain in Cn with ∂ has the property (Pq).
Assume that there exists a negative strictly q-subharmonic exhaustion function ϕ
of . Then for every real valued continuous function f on ∂ the function
P B f,(z) := sup{u(z) : u ∈ SHq(), lim sup
x→ξ
u(x) ≤ f (ξ) ∀ξ ∈ ∂}
belongs to SHq() ∩ C() and P B f,|∂ = f.
Author's personal copy
M. H. Le et al.
Proof. Since ∂ has the property (Pq), by Proposition 4.10 in [11], there exists
a sequence f j of continuous q−subharmonic functions on neighborhoods of ∂
such that f j converges uniformly to f on ∂. By Proposition 2.3 (f), after taking
convolution with a smoothing kernel we may achieve that f j is q−subharmonic
and C2 smooth near ∂ for every j . Now we fix j ≥ 1 and choose a real valued
C2 smooth function θ j on Cn with compact support such that θ j = 1 on a small
neighborhood of ∂. Since ϕ is strictly q-subharmonic on , by taking a constant
M j > 0 large enough the function
Fj (z) := M jϕ(z) + θ j (z) f j (z)
will belong to SHq() ∩ C() and satisfies Fj |∂ = f j . So the function
P B f j ,(z) := sup{u(z) : u ∈ SHq(), lim sup
x→ξ
u(x) ≤ f j (ξ) ∀ξ ∈ ∂}
satisfies
lim inf
z→ξ P B f j ,(z) ≥ limz→ξ Fj (z) = f j (ξ) ∀ξ ∈ ∂.
On the other hand, since ϕ is a negative subharmonic exhaustion function of , by
a well known result in potential theory we know that is regular with respect to the
Dirichlet problem for Laplacian. So we can find a real valued continuous function
Hj on which is harmonic on and satisfies Hj = f j on ∂. By the maximum
principle for subharmonic functions we obtain P B f j , ≤ Hj on . Therefore
lim sup
z→ξ
P B f j ,(z) ≤ lim
z→ξ Hj (z) = f j (ξ) ∀ξ ∈ ∂.
Summing up, we have proved that
lim
z→ξ P B f j , = f j (ξ) ∀ξ ∈ ∂.
Now we apply Lemma 1 in [12] to conclude that P B f j , is in fact continuous on
. Notice that Walsh’s lemma is proved only in the case q = 1, however since
q-subharmonicity is invariant both under taking finite maximum and translates of
variables we can check that his proof works also for general q. Finally, from the
definition of