EXISTENCE AND COMPACTNESS FOR THE ∂ NEUMANN OPERATOR ON qCONVEX DOMAINS

Abstract. The aim of this paper is to give a sufficient condition for existence and compactness of the ∂Neumann operator Nq on L 2 (0,q) (Ω) in the case Ω is an arbitrary qconvex domain in C nAbstract. The aim of this paper is to give a sufficient condition for existence and compactness of the ∂Neumann operator Nq on L 2 (0,q) (Ω) in the case Ω is an arbitrary qconvex domain in C n

∂-NEUMANN OPERATOR ON q-CONVEX DOMAINS

MAU HAI LE, QUANG DIEU NGUYEN AND XUAN HONG NGUYEN

Abstract The aim of this paper is to give a sufficient condition for

existence and compactness of the ∂-Neumann operator N q on L2

(0,q)(Ω)

in the case Ω is an arbitrary q-convex domain inCn.

1 Introduction Let Ω be a domain inCn According to the fundamental work of Kohn and H¨ormander in the sixties, if Ω is bounded and pseudoconvex then for every

1≤ q ≤ n, the complex Laplacian 2 q on square integrable (0, q) forms on Ω has a bounded inverse, denoted by N q This is the ∂-Neumann operator N q

The most basic property of N q is that if v is a ∂ closed (0, q + 1)-form, then

u := ∂ ∗ N q+1 v provides the canonical solution to ∂u = v, namely the one

or-thogonal 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

compact-ness of N q on (possibly unbounded or non-smooth) q-convex domains, a

generalization of pseudoconvex domains in which the existence of

plurisub-harmonic exhaustion function is replaced by existence of q-subplurisub-harmonic ones Our research is motivated from the fact that compactness of N q im-plies global regularity in the sense of preservation of Sobolev spaces Up to

now, there is no complete characterization for compactness of N q even in the

case q = 1 and Ω 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 (P q) Recently, K.Gansberger and F Haslinger studied

com-pactness estimates for the ∂-Neumann operator in weighted L2-spaces and

the weighted ∂-Neumann problem on unbounded domains inCn(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 N q,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

2010 Mathematics Subject Classification Primary 32W05.

Key words and phrases q-subharmonic functions, q-convex domain, ∂-Neumann

op-erator, compactness of ∂-Neumann operator.

1

Ω (Proposition 4.5 in [5]) The main step in Gansberger’s proof is to show

that under the above assumption the embedding H01(Ω, φ, ∇φ) ,→ L2(Ω, φ)

is compact

The paper is organized as follows In Section 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 (P q) introduced earlier in the case of bounded domains by D Catlin in [2] and E J Straube in [11] Roughly speaking, we say that a

(possibly unbounded) domain Ω has the property (P q ′) if Ω 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 ∂Ω We should say that this notion is motivated from Theorem 1.3 in [5] in which the case where q = 1 and ∂Ω is smooth is

studied The main result of the section is Theorem 3.5 which gives sufficient

conditions for domain Ω having the property (P q ′) The existence and

com-pactness of the ∂-Neumann operator N q on q-convex domains are discussed

in Section 4 We start the section with a geometric necessary condition for

compactness of N q The main result of the paper is Theorem 4.3 Here we prove that if Ω⊂ C n is a q-convex domain having property (P q ′) then there

exists a bounded ∂-Neumann operator N q on L2

(0,q) (Ω) and N q is compact

Our proof exploits the property (P q ′) of Ω and some techniques from the book [11]

Acknowledgements This work was partly written during visits of the

second named author first at the Max-Planck Institute in the winter of

2011 and later at the Vietnam Institute for Advanced Mathematics in the winter of 2012 He wishes to thank these institutions for financial support and hospitality We are also grateful to Professor Emile Straube for giving

a key idea for the proof of Proposition 4.2 Finally, we are indebted to the referee for his(her) insightful comments, especially for showing us the proof

of Theorem 4.3 which is simpler than our original one

Notation λ 2n denotes the Lebesgue measure on Cn and B(a, r) is the ball with center a ∈ C n and radius r > 0 For a real valued function u ∈ C2(Ω) we define H q (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

Ω⊂ C n can be expressed as u = ∑

|J|=q

′

u J dz J , where J are strictly increas-ing multi-indices with lengths q and {u J } are defined functions on Ω Let

C ∞

(0,q)(Ω) be the space of complex-valued differential forms of class C ∞ and

of type (0, q) on Ω By C ∞

0 (Ω) we denote the space of C ∞ functions with

compact support in Ω We use L2(0,q) (Ω) to denote the space of (0, q)-forms

on Ω with square-integrable coefficients If u, v ∈ L2

(0,q)(Ω), the weighted

L2-inner product and norms are defined by

(u, v)Ω =

∫ Ω

∑

|J|=q

′

u J v J dλ 2n and ∥u∥2

Ω = (u, u)Ω.

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 ,

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 be-longs to L2(0,q+1) (Ω) So ∂ is a densely defined closed operator, and therefore has an adjoint operator from L2

(0,q+1) (Ω) into L2

(0,q) (Ω) denoted by ∂ ∗ For

|J|=q+1

′

u J dz J ∈ dom(∂ ∗) one has

∂ ∗ u = − ∑

|K|=q

′∑n

j=1

∂u jK

∂z j dz K .

The complex Laplacian on (0, q)-forms is defined as

2 q := ∂∂ ∗ + ∂ ∗ ∂,

where the symbol 2 q is understood as the maximal closure of the

opera-tor initially defined on (0, q)-forms with coefficients in C ∞

0 (Ω) 2 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 N q is -if

it exists-the bounded inverse of2 q We refer the reader to the monographs

[11] for a complete survey on ∂-Neumann operators and their applications

to other problems in several complex variables Next, we recall the

defini-tion of q-subharmonic funcdefini-tions which is an extension of plurisubharmonic

functions (see [1], [7], [8])

Definition 2.1 Let Ω be a domain in Cn An upper semicontinuous

func-tion u : Ω −→ [−∞, ∞), u ̸≡ −∞ is called q-subharmonic if for every q-dimensional complex plane L in Cn , u | L is a subharmonic function on

L ∩ Ω.

The set of all q-subharmonic functions on Ω is denoted by SH q(Ω)

The function u is called to be strictly q-subharmonic if for every U b Ω

there exists constant C U > 0 such that u(z) − C U |z|2 ∈ SH q (U ).

Remark 2.2 (a) q-subharmonicity and strict q-subharmonicity are local

properties

(b) 1-subharmonicity (resp n −subharmonicity) coincides with

plurisub-harmonicity (resp subplurisub-harmonicity)

We list below basic properties of q-subharmonic functions that will be

used later on (see [7])

Proposition 2.3 Let Ω be an open set in Cn and let q is an integer with

16 q 6 n Then we have.

(a) If u ∈ SH q (Ω) then u ∈ SH r (Ω), for every q 6 r 6 n.

(b) If u, v ∈ SH q (Ω) and α, β > 0 then αu + βv ∈ SH q (Ω).

(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 (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 C n vanishing outside the unit ball and satisfying ∫

Cn ρdV = 1 For u ∈ SH q (Ω) we define

u ε (z) := (u ∗ ρ ε )(z) =

∫

B(0,ε)

u(z − ξ)ρ ε (ξ)dλ 2n (ξ), ∀z ∈ Ω ε , where ρ ε (z) := ε21n ρ(z/ε) and Ω ε = {z ∈ Ω : d(z, ∂Ω) > ε} Then u ε ∈

SH q(Ωε)∩ C ∞(Ω

ε ) and u ε ↓ u as ε ↓ 0.

(g) Let u1, , u p ∈ SH q (Ω) 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)∈ SH q (Ω).

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 Ω be a domain in Cn and let q be an integer with

1 6 q 6 n Let u be a real valued C2-function defined on Ω Then the following are equivalent:

(a) u is a q-subharmonic function Ω.

(b) i∂∂u ∧ (i∂∂|z|2)q −1 > 0 i.e., H q (u)(z) ≥ 0 for every z ∈ Ω.

(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 ∂z k

f jK f kK > 0.

We also have the following simple result about smoothing q-subharmonic

functions

Proposition 2.5 Let Ω be an open set in Cn and let u ∈ SH q (Ω) such

that u − δ|idCn |2 ∈ SH q (Ω) for some δ > 0 Then for every ε > 0 we have

u ε − δ|idCn |2 ∈ SH q(Ωε ), where Ω ε :={z ∈ Ω : d(z, ∂Ω) > ε}.

Proof By Proposition 2.3 (f) we have (u − δ|idCn |2)ε ∈ SH q(Ωε) Since

(u − δ|idCn |2

)ε (z) = u ε (z) − δ

∫

B(0,ε)

|z − w|2

ρ ε (w)dV (w)

= u ε (z) − δ|z|2− δ

∫

B(0,ε)

(2ℜ(z, −w) + |w|2)ρ ε (w)dV (w)

= u ε (z) − δ|z|2− v (ε) (z),

where v (ε) (z) := δ ∫

B(0,ε)

(2ℜ(z, −w) + |w|2)ρ ε (w)dV (w) is a pluriharmonic

function inCn Hence, u ε − δ|idCn |2 = (u − δ|idCn |2)ε + v (ε) ∈ SH q(Ωε) This

The following definition is an extension of pseudoconvexity

Definition 2.6 A domain Ω⊂ C n is said to be q-convex if there exists a

q-subharmonic exhaustion function on Ω Moreover, a C2 smooth bounded

domain Ω is called strictly q-convex if it admits a C2 smooth defining

func-tion which is strictly q-subharmonic on a neighbourhood of Ω.

It is not clear if we can find a smooth strictly q-subharmonic exhaustion function on a convex domain However by Proposition 2.7 in [7], every q-convex domain Ω can be written as an increasing union of bounded q-q-convex

domains Ωj such that each Ωj has a smooth strictly q-convex exhaustion

function Using Sard theorem this result can be refined as follows

Proposition 2.7 Let Ω be a q-convex domain inCn Then Ω can be written

as Ω = ∪∞

j=1

Ωj such that Ω j b Ωj+1 and each Ω j is a strictly q-convex domain.

According to a classical result of Green-Wu, every domain inCn 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 Ω be a domain in Cn and p ∈ ∂Ω Assume that there exist r > 0, a sequence {p j } ⊂ Ω, 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 Ω.

Then Ω is not q-convex.

Proof Suppose that there exists a q-subharmonic exhaustion function u on

Ω Since u | Lj is subharmonic, by the maximum principle and the

assupm-tions (a) and (b) we get for every j ≥ 1 the inequality u(p j) ≤ sup

K

u By

Remark 2.9 (i) Let Ω := {z ∈ C3 : 1 < |z| < 3} and p = (1, 0, 0) ∈ ∂Ω.

Denote by L the hyperplane tangent to ∂Ω at the point p Consider the sequences of points p j := (1 + 1/j, 0, 0), j ≥ 1 tending to p and hyperplanes

L j := L + (1/j, 0, 0) passing through the points p j Using the above result,

we can see that Ω is not 2-convex

(ii) In [8], the following generalization of Levi convexity was introduced We say that a bounded domain Ω in Cn with a C2 smooth defining function ρ

is weakly q-convex if for every p ∈ ∂Ω, for every (0, q)-form f = ∑

|J|=q

′

f J dz J

satisfying

n

∑

i=1

∂ρ

∂z i (p)f iK = 0 ∀|K| = q − 1,

we have

∑

|K|=q−1

′ ∑n

j,k=1

∂2ρ

∂z j ∂z k f jK 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 inCn

Proposition 2.10 Let Ω be a bounded q-convex domain inCn , u = ∑

|J|=q

′ u

J d¯ z J ∈ dom( ¯ ∂) ∩ dom(¯∂ ∗)⊂ L2

(0,q) (Ω) Then for all b ∈ C2(Ω), b 6 0 the following

holds

∑

|K|=q−1

′ n

∑

j,k=1

∫ Ω

e b ∂

2b

∂z j ∂ ¯ z k u jK u kK dλ 2n 6 ∥∂u∥2 +∥¯∂ ∗ u ∥2.

3 The property (P q ′) First we recall an important concept introduced and investigated by D Catlin in [2] and E J Straube in [11] (see also [9]) We say that a compact set

K inCn has the property (P q ) if for every M > 0, there exist a neighborhood

U M of K, a C2 smooth function λ M on U M such that 06 λ M (z) 6 1, z ∈ U M

and for any z ∈ U M , the sum of the smallest q eigenvalues of the complex Hessian of λ is at least M (or, equivalently, λ M − M

q |z|2 ∈ SH q (U M))

Moreover, given a closed set E (not necessarily bounded), we say that E locally has the property (P q ) if for every z0 ∈ E we can find a compact

neighborhood K of z0 in E such that K has the (P q) property

Using Kohn-Morrey-H¨ormander formula in [11], it is not hard to prove

(see [2] for the case q = 1 and [11] for general q) that if Ω is a smoothly

bounded pseudoconvex domain in Cn with the boundary ∂Ω having the property (P q ) then the ∂-Neumann operator N q is compact

The following notion is the key to our research on compactness of N q in the

case where Ω is unbounded.

Definition 3.1 Let Ω be a domain inCn We say that Ω has the property

(P q ′) if there exists a real valued, boundedC2 smooth function φ on Ω such that for every positive number M , we have φ(z) − M|z|2 ∈ SH q(Ω\ K M)

for some compact subset K M of Ω

Remark 3.2 (i) The function φ is not assumed to be q-subharmonic on

the whole Ω We will prove, however, that φ can be chosen to have this

additional property

(ii) If Ω has the property (P q ′ ) then for every complex space L of dimension

q, Ω ∩ L is quasibounded (in L) i.e., Ω ∩ 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 ballsB(z j , r) contained

in Ω 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 ∈ SH n(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∥Ωλ 2n(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∥Ω

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 Ω be an open set in Cn with ∂Ω locally has the property

(P q ) Assume that there exist negative q-subharmonic functions ρ, ˜ ρ on Ω satisfying the following conditions.

(a) ρ ∈ C2(Ω), ρ(z) − |z|2 ∈ SH q (Ω).

(b) lim |z|→∞ Hq 1+ρ(z) (ρ)(z)2 =∞.

(c) ˜ ρ is strictly q-subharmonic on Ω and satisfies lim

z →ξ ρ(z) = 0 for every˜

ξ ∈ ∂Ω.

Then Ω 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 Ω be a bounded domain in Cn with ∂Ω has the property

(P q ) Assume that there exists a negative strictly q-subharmonic exhaustion

function φ of Ω Then for every real valued continuous function f on ∂Ω the function

P B f,Ω (z) := sup {u(z) : u ∈ SH q (Ω), lim sup

x →ξ u(x) ≤ f(ξ) ∀ξ ∈ ∂Ω} belongs to SH q(Ω)∩ C(Ω) and P B f,Ω | ∂Ω = f.

Proof Since ∂Ω has the property (P q), by Proposition 4.10 in [11], there

ex-ists a sequence f j of continuous q −subharmonic functions on neighborhoods

of ∂Ω such that f j converges uniformly to f on ∂Ω 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 ∂Ω for every j Now we fix j ≥ 1

and choose a real valued C2 smooth function θ j on Cn with compact

sup-port such that θ j = 1 on a small neighborhood of ∂Ω Since φ is strictly

q-subharmonic on Ω, by taking a constant M j > 0 large enough the function

F j (z) := M j φ(z) + θ j (z)f j (z) will belong to SH q(Ω)∩ C(Ω) and satisfies F j | ∂Ω = f j So the function

P B fj,Ω (z) := sup {u(z) : u ∈ SH q (Ω), lim sup

x →ξ u(x) ≤ f j (ξ) ∀ξ ∈ ∂Ω}

satisfies

lim inf

z →ξ P B fj,Ω (z) ≥ lim

z →ξ F j (z) = f j (ξ) ∀ξ ∈ ∂Ω.

On the other hand, since φ is a negative subharmonic exhaustion function

of Ω, by a well known result in potential theory we know that Ω is regular with respect to the Dirichlet problem for Laplacian So we can find a real

valued continuous function H j on Ω which is harmonic on Ω and satisfies

H j = f j on ∂Ω By the maximum principle for subharmonic functions we obtain P B fj,Ω ≤ H j on Ω Therefore

lim sup

z →ξ P B fj,Ω (z) ≤ lim

z →ξ H j (z) = f j (ξ) ∀ξ ∈ ∂Ω.

Summing up, we have proved that

lim

z →ξ P B fj,Ω = f j (ξ) ∀ξ ∈ ∂Ω.

Now we apply Lemma 1 in [12] to conclude that P B fj,Ωis in fact continuous

on Ω 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 the envelopes we deduce easily that

∥P B fj,Ω − P B fk,Ω ∥Ω =∥f j − f k ∥ ∂Ω

It follows that P B fj,Ω converges uniformly to P B f,Ω on Ω In particular

P B f,Ω ∈ SH q(Ω)∩ C(Ω) and satisfies P B f,Ω | ∂Ω = f We are done. 

Remark The above lemma is false if the assumption on the existence of φ

is omitted Indeed, consider the punctured disk Ω :={z ∈ C : 0 < |z| < 1}.

Then ∂Ω has the property (P1) Now we let f (z) = 0 for |z| = 1 and

f (0) = 1 Suppose that there exists u ∈ SH1(Ω)∩ C(Ω) such that u = f on

∂Ω Since u is bounded near 0, it extends to a subharmonic function on the

whole disk{z : |z| < 1} Notice that

u(0) = 1 > 0 = sup

|z|=1

u(z).

This is a violation of the maximum principle for subharmonic function

Proof We split the proof into three steps.

Step 1 We show that there exists a real valued, bounded function ψ ∈

SH q(Ω)∩ C(Ω) having the following property: For every M > 0 and every

point ξ ∈ ∂Ω there exists r ξ,M > 0 such that ψ(z) − M|z|2 is q-subharmonic

on Ω∩ B(ξ, r ξ,M ) For this, we fix M, ξ as above and define for j ≥ 1 the

do-main Ωj := Ω∩B(0, j) Since ∂Ω locally has the (P q) property and since this property is preserved after taking finite union of compact sets, we infer that

the compact set ∂Ω j has the (P q) property as well In order to apply Lemma 3.4 we observe that the function max{˜ρ(z), |z|2 − j2} is negative strictly

q-subharmonic exhaustion for Ωj Now we define the following function on a

part of ∂Ω j

φ j (z) = −|z|2

for z ∈ B(0, j−1)∩∂Ω; φ j (z) = ρ(z) −|z|2

for z ∈ ∂B(0, j)∩Ω.

Extend φ j to a real valued continuous function (still denoted by φ j ) on ∂Ω j

such that

ρ(z) − |z|2 ≤ φ j (z) ≤ −|z|2

, z ∈ ∂Ω j

It follows from Lemma 3.4 that P B φj,Ωj ∈ SH q(Ωj)∩ C(Ω j ) and P B φj,Ωj =

φ j on ∂Ω j Furthermore, using the assumption (a) we also get

P B φj,Ωj (z) ≥ ρ(z) − |z|2 on Ωj

We define ρ j = P B φj,Ωj on Ωj and ρ j (z) = ρ(z) − |z|2 on Ω\ Ω j Thus for

j ≥ 2 we have ρ j ∈ SH q(Ω)∩ C(Ω) Moreover

ρ j (z) + |z|2 < 0 ∀z ∈ Ω, ρ j (z) + |z|2 = 0 ∀z ∈ B(0, j − 1) ∩ ∂Ω.

Set ˜ρ j (z) := ρ j (z) + |z|2 on Ω Following a construction given in the example

after Proposition 5.3 in [5] we define

ψ(z) :=∑

j ≥2

e2j ρj˜(z)

2j

It is easy to see that ψ is real valued, bounded, continuous and strictly q-subharmonic on Ω For j ≥ 2, we have the following simple estimate on Ω j

which is taken in the sense of distribution

i∂∂

(e2j ρj˜(z)

2j

)

∧(i∂∂|z|2)q−1 ≥ i∂∂ ˜ρ j ∧(i∂∂|z|2)q−1 e2j ρj˜(z) ≥ (i∂∂|z|2)q e2j ρj˜(z)