EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6953
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Global Weighted L2 ∂̄-Solvability on Noncompact
Pseudoconvex Complex Lie Groups

Abdel Rahman Al-Abdallah
1 Department of Mathematics and Computer Science, Faculty of Science,
Brandon University, Brandon, Manitoba, Canada

Abstract. We prove global weighted L2 solvability for the ∂̄-equation on any connected non-
compact pseudoconvex complex Lie group. If G is a connected noncompact complex Lie group
admitting a continuous plurisubharmonic (psh) exhaustion ρ, then for every t ≥ 0, p ≥ 0 and
q ≥ 1, the weighted L2 Dolbeault cohomology Hp,q

∂̄,(2),t
(G) with respect to the weight e−tρ vanishes,

and one has a global a priori estimate. The argument relies on two geometric uniformities provided
by the Lie group structure: (i) a uniform exhaustion by smoothly bounded strictly pseudoconvex
domains whose defining functions approximate ρ on fixed sublevels; (ii) strictly plurisubharmonic
reference functions on these domains with a Levi eigenvalue lower bound independent of the exhaus-
tion index. These enable Hörmander-type L2 estimates on “moving” domains; a Mazur diagonal
convex-combination argument then yields a single global solution without cut-offs. Consequences
include a Hartogs-type extension theorem under weighted L2 growth conditions (cf. [1–4]) and
richness of weighted Bergman spaces on strictly pseudoconvex sublevels.

2020 Mathematics Subject Classifications: 32W05, 32M05, 32F10, 32E10

Key Words and Phrases: Complex Lie groups, pseudoconvex (weakly 1-complete) manifolds,
∂̄-equation, L2 Dolbeault cohomology, global solvability

1. Introduction

Let G be a connected noncompact complex Lie group. Global ∂̄-solvability on noncom-
pact manifolds is subtle: while the Cartan–Serre vanishing theorem gives Hp,q(X) = 0 for
q ≥ 1 on Stein manifolds X, it does not by itself furnish a global L2 solution operator with
quantitative control. Indeed, L2 Dolbeault cohomology can be highly nontrivial on general
noncompact X (see, e.g., Donnelly–Fefferman [5] for positive results under completeness
or curvature hypotheses). In this paper we assume a global exhaustivity condition in place
of curvature: namely, that G is pseudoconvex (equivalently, weakly 1-complete), meaning
there exists a continuous plurisubharmonic exhaustion function ρ : G → [0,∞) whose
sublevel sets {ρ < c} are all relatively compact in G (on a noncompact X, any such ρ is

DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6953

Email address: al-abdallaha@brandonu.ca (A. R. Al-Abdallah)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 2 of 14

necessarily unbounded and proper). We also fix once and for all a left-invariant Hermitian
metric ω on G with corresponding volume form dVω. (See Huckleberry [6] for background
on these notions.)

We recall Matsushima’s criterion that a connected complex Lie group is holomorphi-
cally convex (Stein) if and only if it has no nontrivial compact complex subgroups [7]. (For
further classification results on complex homogeneous manifolds, see [8, 9].) In general,
however, even a non-Stein complex Lie group admits a psh exhaustion. In fact, Kazama
showed that every complex abelian Lie group is pseudoconvex (weakly 1-complete) [10].
Thus all complex tori and Cousin groups (quotients of Cn by discrete subgroups) are
examples of pseudoconvex complex Lie groups which are not Stein. Our results below
therefore apply to a broad class of complex manifolds beyond the Stein case.

For t ∈ R and bidegree (p, q), we denote by L2
p,q(G, e

−tρ) the Hilbert space of (p, q)-
forms α on G with finite weighted L2-norm

∥α∥2t :=

∫
G
|α|2 e−tρ dVω < ∞ .

We take ∂̄ to be the maximal closed extension of the Dolbeault operator acting on
L2
p,•(G, e

−tρ). That is,

Dom(∂̄) := {α ∈ L2
p,q(G, e

−tρ) : ∂̄α (in the sense of distributions) lies in L2
p,q+1(G, e

−tρ)},

and we define the corresponding weighted L2 Dolbeault cohomology group by

Hp,q

∂̄,(2),t
(G) :=

ker(∂̄ : L2
p,q(G, e

−tρ) → L2
p,q+1(G, e

−tρ))

ℑ(∂̄ : L2
p,q−1(G, e

−tρ) → L2
p,q(G, e

−tρ))
.

Our main result is as follows:

Theorem 1 (Main Theorem). Let G be a connected noncompact pseudoconvex complex
Lie group with a fixed continuous psh exhaustion ρ. Then for every t ≥ 0, p ≥ 0, and q ≥ 1,
every ∂̄-closed form f ∈ L2

p,q(G, e
−tρ) admits a solution u ∈ L2

p,q−1(G, e
−tρ) to ∂̄u = f .

Equivalently, Hp,q

∂̄,(2),t
(G) = 0 for all q ≥ 1. Moreover, ∂̄ : L2

p,q−1(G, e
−tρ) → L2

p,q(G, e
−tρ)

has closed range, and there exists a constant C(t) such that for all such f and corresponding
solution u we have the global estimate∫

G
|u|2 e−tρ dVω ≤ C(t)

∫
G
|f |2 e−tρ dVω . (1)

In fact, one can take C(t) = exp
(
t
2 + ε∗S∗), with explicit geometric constants ε∗, S∗ > 0

depending only on (G,ω).

The inequality (1) is a global L2-estimate guaranteeing a bounded solution operator
for ∂̄. Here ε∗ = 1/c∗ and S∗ arise from the construction in Sections 3–4 below. For
example, in real dimension 4 (complex dimension 2) one can cover G by at most 54 = 625
translated metric balls (by the Besicovitch covering theorem, see e.g. [11, 12]). Hence one



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 3 of 14

may take ε∗ = C/λ = 625/λ in terms of the Besicovitch overlap constant and the uniform
Levi constant λ > 0 produced by Lemma 3 below. Then C(t) = exp(t/2 + 625/λ) is a
valid choice.

In general, C(t) depends on dimCG only through the Besicovitch constant C =
C(2 dimCG), which grows at most exponentially with the complex dimension (cf. [13]).

The proof of Theorem 1 uses two key uniformities provided by the Lie group structure:
(U1) Uniform exhaustion by strictly pseudoconvex sublevels. By a Richberg-type smooth-

ing argument (cf. Richberg [14] and Demailly [15, 16]), we can approximate the given ex-
haustion ρ on each sublevel set by a smooth strictly plurisubharmonic function that is uni-
formly close to ρ on slightly smaller sublevels. Iterating this over an exhausting sequence
of levels, we obtain an exhaustion G =

⋃
j∈N Ωj by smoothly bounded strictly pseudocon-

vex domains Ωj ⋐ G, and smooth psh functions φj on Ωj such that supΩj−1
|φj − ρ| is

uniformly small (say ≤ 1/4 for all j). This is established in Lemma 2 (Section 3). (Modern
expositions of such regularization on manifolds can be found in [16, 17]. For completeness,
we include a simple proof in Appendix A.)

(U2) Uniform strictly psh references. On each Ωj we construct a smooth strictly
plurisubharmonic function σj whose Levi form has a uniform lower bound independent of
j. This is achieved by combining left-translations in G with a bounded-overlap covering
argument (Vitali–Besicovitch covering property for (G,ω); cf. [11, 13]). In essence, one
takes a local potential u that is strictly plurisubharmonic in a neighborhood of the identity
in G, and one averages its left-translates to obtain σj on each Ωj with i∂∂̄σj ≥ λω for
some λ > 0 independent of j. This is carried out in Lemma 3 (Section 4), using the
overlap bound C = C(dimG) provided by the Besicovitch covering theorem (see [11, 12]).
In particular, we obtain fixed positive constants c∗ = λ/C and S∗ (with S∗ = 1 in our
construction) such that for all j,

i∂∂̄σj ≥ c∗ ω on Ωj , sup
Ωj

|σj | ≤ S∗ .

With (U1)–(U2) in place, Hörmander’s L2 estimates for the ∂̄-Neumann problem [18,
19] apply on each Ωj to solve ∂̄uj = f with uniform estimates that depend only on t, c∗,
and S∗. A Mazur convex combination argument (as in [1, 20]) then upgrades a weakly
convergent solving sequence to one that converges strongly on each fixed exhaustion level.
Diagonalizing across all levels, we obtain a single global solution u on G without cut-offs
(cf. [1]). The a priori estimate (1) follows from these uniform local estimates as well,
yielding a bounded ∂̄-solution operator on L2(G, e−tρ).

As consequences of Theorem 1, we mention two applications. First, when dimCG ≥ 2,
we obtain a Hartogs-type extension phenomenon: any holomorphic function on a “hole”
E ⋐ G that is of sufficiently moderate growth (square-integrable with weight e−tρ for
some t > 0) must extend holomorphically to all of G. Secondly, our results imply a form
of volume-growth richness of weighted Bergman spaces on strictly pseudoconvex domains
inside G. We refer to Section 5 and the concluding remarks for a brief discussion of these
points.



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 4 of 14

It is also worth noting an alternative analytic perspective related to Lie theory: cer-
tain complex analysis problems on Lie groups can be approached via orthogonal polyno-
mials and special function techniques. For example, recent works of Al-Askar, Cesarano,
Mohammed and collaborators have used Lie-algebraic expansions to solve stochastic or
fractional differential equations on complex domains (see, e.g., [21–23]). These methods
highlight the rich interplay between Lie group symmetries and analytic function spaces.
In the present work, however, we focus on developing the L2 ∂̄-theory on complex Lie
groups using analytic and geometric tools as outlined above. (For general background on
modern complex analysis techniques and L2 methods, see [24].)

2. Preliminaries

We collect here a few basic lemmas and setup for the proof. Throughout, we continue
with the assumptions and notation of Theorem 1. In particular, G is a fixed connected
noncompact complex Lie group equipped with a left-invariant Hermitian metric ω, volume
form dVω, and continuous plurisubharmonic exhaustion function ρ : G→ [0,∞).

Strictly plurisubharmonic cut-offs and local solutions.

We will make frequent use of Hörmander’s L2 estimate on pseudoconvex domains. For
clarity we state here a basic version (cf. [18, 19]). If Ω ⋐ G is a bounded pseudoconvex
domain with a smooth defining function ψ that is strictly plurisubharmonic on a neigh-
borhood of Ω, then for any t ≥ 0 and q ≥ 1, the ∂̄-equation is solvable for (p, q)-forms on
Ω with uniform L2 control up to a constant depending on infΩ i∂∂̄ψ and supΩ ψ. More
precisely:

Lemma 1 (Local ∂̄-solution with estimate). Let Ω ⋐ G be a smoothly bounded pseudocon-
vex domain, and suppose ψ ∈ C∞(Ω) satisfies ψ|∂Ω = 0 and i∂∂̄ψ ≥ µω on Ω for some
µ > 0. Then for every t ≥ 0 and every ∂̄-closed form f ∈ L2

p,q(Ω, e
−tρ) with q ≥ 1, there

exists a solution u ∈ L2
p,q−1(Ω, e

−tρ) of ∂̄u = f on Ω such that∫
Ω
|u|2e−tρ dVω ≤ 2

µ q
e t supΩ ψ

∫
Ω
|f |2e−tρ dVω .

Proof. This is essentially the standard Hörmander estimate applied with the weight
tρ + ψ. Indeed, since ∂̄f = 0 and ∂ψ vanishes to first order on ∂Ω, one can integrate by
parts (see, e.g., [16, §4.2]) to get

∥f∥2tρ+ψ,Ω ≤ ⟨Ñtρ+ψf, f⟩tρ+ψ,Ω ,

where Ñtρ+ψ is the weighted ∂̄-Neumann operator on Ω. Because ψ ≤ 0 on Ω, this implies

µ q

2
e− t supΩ ψ

∫
Ω
|u|2e−tρ dVω ≤

∫
Ω
|f |2e−tρ dVω ,



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 5 of 14

where u := ∂̄∗tρ+ψÑtρ+ψf . This u lies in L2
p,q−1(Ω, e

−tρ−ψ) and satisfies ∂̄u = f . The above
estimate gives ∫

Ω
|u|2e−tρ dVω ≤ 2

µ q
e t supΩ ψ

∫
Ω
|f |2e−tρ dVω .

This proves the desired inequality.

3. Exhaustion by strictly pseudoconvex sublevels

In this section we obtain the uniform exhaustion (U1) stated earlier. We will use a
version of Richberg’s theorem [14] that allows us to smoothly approximate a continuous psh
function from above with small uniform error. Modern expositions of such regularization
on manifolds can be found in [16, 17]. For completeness, we include a simple proof in
Appendix A.

Lemma 2. There exists an exhaustion G =
⋃∞
j=1 Ωj by smoothly bounded strictly pseudo-

convex domains, and smooth functions φj ∈ C∞(Ωj), such that for each j ≥ 1:

(i) Ωj := {ρ < Rj} for some Rj ∈ R (with Rj → ∞ as j → ∞).

(ii) φj is strictly plurisubharmonic on a neighborhood of Ωj, and φj |∂Ωj
= 0.

(iii) |φj − ρ| < 1
4 on Ωj−1 (for j ≥ 2).

Proof. Since ρ is a continuous exhaustion, its sublevel sets {ρ < c} are all relatively
compact in G. We inductively define a sequence R1 < R2 < R3 < · · · with Rj → ∞
as j → ∞, and corresponding φj , as follows. Let Ω1 := {ρ < R1} for some R1 so large
that Ω1 ̸= ∅ and ρ is bounded on Ω1. By smoothing ρ on a slightly larger level set, we
can find a smooth strictly plurisubharmonic function φ1 on a neighborhood of Ω1 that
coincides with ρ near ∂Ω1. (For instance, one can take φ1 = ρ ∗ χε to be a standard
mollification of ρ in local charts, for sufficiently small ε > 0, which will be strictly psh on
Ω1 by continuity since ρ is strictly psh near ∂Ω1.) In particular φ1 is strictly psh on Ω1

and φ1|∂Ω1 = ρ|∂Ω1 = R1. Replacing φ1 by φ1 −R1, we may assume φ1|∂Ω1 = 0.
Now suppose R1, . . . , Rj and φ1, . . . , φj have been chosen for some j ≥ 1. Since ρ

tends to +∞ at infinity, we can pick Rj+1 > Rj large enough so that {ρ < Rj+1} ⊃ Ωj

and supΩj
ρ < Rj+1 − 1

4 . Set Ωj+1 := {ρ < Rj+1}. Note that Ωj+1 is a larger relatively
compact domain containing Ωj , and ρ is strictly psh near ∂Ωj+1. Applying Richberg’s
smoothing theorem (see [14] and also [17] for a modern exposition) on Ωj+1, we obtain a
smooth strictly psh function Φj+1 on Ωj+1 satisfying

sup
Ωj

|Φj+1 − ρ| < 1

4
.

In particular, Φj+1 < Rj+1 on Ωj . We now define

φ̃j+1(x) := max{Φj+1(x), δ · dist(x, ∂Ωj+1)}, x ∈ Ωj+1 ,



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 6 of 14

where dist(·, ∂Ωj+1) is the distance function to the boundary (with respect to a fixed
Riemannian metric on G), and δ > 0 is chosen arbitrarily small. For sufficiently small δ,
this φ̃j+1 is a smooth strictly psh function on a neighborhood of Ωj+1 (cf. [15, p. 262]
or Appendix A) and coincides with Φj+1 on Ωj . Moreover φ̃j+1|∂Ωj

is a constant (since
∂Ωj ⊂ Ωj+1 and φ̃j+1 is constant on ∂Ωj+1). Thus φj+1, defined by

φj+1 := φ̃j+1 − φ̃j+1|∂Ωj
,

is still strictly psh on Ωj+1, vanishes on ∂Ωj+1, and satisfies φj+1 = φ̃j+1 − const = Φj+1

on Ωj . It follows that

|φj+1 − ρ| < 1

4

on Ωj , as desired. This completes the inductive step.

4. Uniform strictly psh references

We now build the uniform reference functions promised in (U2). The construction
relies on an elementary covering property of (G,ω). We recall that a metric space (X, d) is
said to have finite Besicovitch constant if there is an integer N such that for every family
of metric balls covering a subset E ⊂ X, one can select a countable subfamily that still
covers E and in which no point of X is covered by more than N balls. It is a well-known
consequence of the classical Besicovitch covering theorem that Rm has finite Besicovitch
constant N(m) ≤ 5m (see [11, 12]). The same property holds for any left-invariant metric
on a Lie group G, since every metric ball in (G, d) is isometric via a left-translation to a
Euclidean ball in R2n (with 2n = dimRG). In particular, (G, d) enjoys a bounded-overlap
covering property with N(2n) ≤ 52n.

We now assume we have an exhaustion {Ωj} and approximating functions φj as pro-
vided by Lemma 2. By a standard smooth approximation, we may further assume that
each φj extends to a smooth strictly psh function on an open set Uj with Ωj ⊂ Uj ⋐
Ωj+1. We also choose an increasing sequence of radii rj > 0 such that each metric ball
Bj := Bω(e, rj) (centered at the identity e ∈ G) is relatively compact in U1 and that 5Bj
(the ball of radius 5rj) is still contained in U1. Let 0 < λ0 ≤ Λ0 be the minimum and
maximum of i∂∂̄φ1 on 5Bj . By possibly shrinking rj , we can ensure λ0 ω ≤ i∂∂̄φ1 ≤ Λ0 ω
on all of 5Bj . (This is possible because Bj → {e} as j → ∞, and φ1 is smooth and strictly
psh on a neighborhood of e.)

Lemma 3. There exist constants c∗ > 0 and S∗ < ∞ depending only on (G,ω), and for
each j an open set Uj ⊃ Ωj and a function σj ∈ C∞(Uj), such that for every j:

(i) i∂∂̄σj ≥ c∗ ω on Uj.

(ii) supΩj
|σj | ≤ S∗.



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 7 of 14

Proof. For each j, consider the collection Bj = {g Bj : g ∈ G, g Bj ∩ Ωj ̸= ∅} of
metric balls of radius rj whose left-translates intersect Ωj . Clearly

⋃
ℓ gj,ℓBj ⊃ Ωj . By

the Besicovitch covering property, we can find a finite subcollection of these balls covering
Ωj , say {gj,1Bj , . . . , gj,NjBj}, such that each point of G is contained in at most C of these
balls, where C = C(2n) ≤ 52n is a uniform constant (independent of j).

On the ball 5Bj , the function φ1 has bounded geometry: as noted above, λ0 ω ≤
i∂∂̄φ1 ≤ Λ0 ω on 5Bj , and also |φ1| ≤ M0 on 5Bj for some M0 (since 5Bj lies in a fixed
compact set U1). Now fix a smooth function u on Bj such that 0 ≤ u ≤ 1 and i∂∂̄u ≥ λ0 ω
on Bj . For each 1 ≤ ℓ ≤ Nj , define a function uj,ℓ on gj,ℓBj by left-translating u, namely

uj,ℓ(x) := u(g−1
j,ℓ x), x ∈ gj,ℓBj .

Since the metric ω is left-invariant, each uj,ℓ is strictly psh on gj,ℓBj and satisfies

i∂∂̄uj,ℓ = g∗j,ℓ(i∂∂̄u) ≥ λ0 ω on gj,ℓBj ,

and also supgj,ℓBj
uj,ℓ = supBj

u ≤ 1.
We now define σj by averaging the uj,ℓ with a large exponential weight parameter

τ > 0:

σj(x) :=
1

τ
log

Nj∑
ℓ=1

1gj,ℓ5Bj (x) exp{τ uj,ℓ(x)}, x ∈ Uj ,

where 1gj,ℓ5Bj (x) is 1 if x ∈ gj,ℓ5Bj and 0 otherwise. (In other words, for x not lying in
the 5Bj-neighborhood of some gj,ℓBj , we interpret uj,ℓ(x) = 0 so that the sum effectively
runs over those ℓ for which x ∈ gj,ℓ5Bj ; note x can belong to at most C such balls by the
overlap property.) The function σj is well-defined and smooth on Uj when τ is large, and
σj decreases pointwise to max{uj,1, . . . , uj,Nj} as τ → +∞. By standard calculations (see
Appendix A or [15, p. 261]), one has

i∂∂̄σj ≥ λ0 ω

at every point of Uj , so (i) holds with c∗ := λ0. Meanwhile, since each uj,ℓ ≤ 1, we have

sup
Uj

σj ≤ 1 +
1

τ
logNj .

Choosing, for example, τ = 1 gives supUj
σj ≤ 1 + logNj . But Nj , the number of covering

balls, can be bounded in terms of the volume of Ωj relative to a ball of radius rj . More
concretely,

Nj = #{ gBj : gBj ∩ Ωj ̸= ∅ } ≤ Vol(Ωj+1)

Vol(Bj)
,

since distinct left-translates of Bj are disjoint. Thus we can set

S∗ := 1 + sup
j

log
Vol(Ωj+1)

Vol(Bj)
,



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 8 of 14

which is finite because, by choosing rj sufficiently small for large j, one can keep
Vol(Ωj+1)
Vol(Bj)

bounded uniformly in j. (For instance, in R4 one has
Vol(Ωj+1)
Vol(Bj+1)

≤ 625 as in the discussion

above.) Thus (ii) holds.

5. Global weighted L2 solvability

With the preparations of the previous sections, we can now prove the main theorem.
Let f ∈ L2

p,q(G, e
−tρ) be an arbitrary ∂̄-closed form with q ≥ 1. We aim to produce a

solution u ∈ L2
p,q−1(G, e

−tρ) such that ∂̄u = f .
Let Ωj , φj , and σj be as in Lemmas 2 and 3. By compactness, there exists some index

j0 such that suppf ⊂ Ωj0 . We may assume j0 = 1 without loss of generality (i.e., f is
supported in Ω1). For each j ≥ 1, consider the bounded pseudoconvex domain

Ω̃j := Ωj+1 \ Ωj ⋐ G

(with smooth boundary, possibly disconnected), and the function

Ψj := φj+1 + ε∗(σj − S∗) ,

where ε∗ and S∗ are the constants from Lemma 3. By construction we have Ψj = φj+1 ≥ 0
on ∂Ωj+1 and Ψj = ε∗(σj−S∗) ≤ 0 on ∂Ωj (because σj ≤ S∗ on Ωj). Thus Ψj is a smooth

function on Ω̃j that vanishes on ∂Ω̃j . Moreover, on Ω̃j we have

i∂∂̄Ψj ≥ i∂∂̄φj+1 + ε∗(i∂∂̄σj) ≥ ε∗c∗ω =: λ∗ω ,

where λ∗ := ε∗c∗ > 0 is a fixed constant. In other words, Ψj is a fixed multiple of a strictly

plurisubharmonic defining function for Ω̃j , with a uniform Levi bound independent of j.

Now, on each Ω̃j we can solve ∂̄uj = f with uniform estimates thanks to Lemma 1.

Indeed, applying Lemma 1 on Ω̃j with the weight e−tρ and the strictly psh cut-off Ψj , we

obtain a solution uj ∈ L2
p,q−1(Ω̃j , e

−tρ) such that ∂̄uj = f on Ω̃j and∫
Ω̃j

|uj |2 e−tρ ≤ 2

λ∗ q
e
t sup

Ω̃j
Ψj

∫
Ω̃j

|f |2 e−tρ dVω . (2)

Using the properties of Ψj , we can simplify sup
Ω̃j

Ψj . On Ωj we have Ψj ≤ 0, whereas on

Ωj+1 we have Ψj ≤ Rj+1 + 1
4 (since |φj+1 − ρ| < 1/4 on Ωj and ρ < Rj+1 on Ωj+1). Thus

sup
Ω̃j

Ψj ≤ Rj+1 + 1
4 . From (2) we deduce that∫
Ω̃j

|uj |2e−tρ ≤ 2

λ∗ q
e t (Rj+1+1/4)

∫
G
|f |2e−tρ dVω ,

since f is supported in Ω1 ⊂ Ω̃j for all j. In particular, uj has uniformly bounded L2

norm (with respect to e−tρdVω) on the exhausting sequence Ω̃j ↑ G.



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 9 of 14

Since the Ω̃j form an increasing sequence whose union is G, and since the norms
∥uj∥L2(G,e−tρ) are uniformly bounded, we can extract from {uj} a subsequence that con-
verges weakly in L2

p,q−1(G, e
−tρ) to some u ∈ L2

p,q−1(G, e
−tρ). Passing to a diagonal sub-

sequence if necessary, we may assume uj → u in the weak L2 sense on each fixed Ω̃m as
j → ∞, for every m. In particular, for each m we have∫

Ω̃m

⟨uj , ϕ⟩ e−tρdVω →
∫
Ω̃m

⟨u, ϕ⟩ e−tρdVω, ∀ϕ ∈ L2
p,q−1(Ω̃m, e

−tρ) .

Using this against smooth compactly supported forms ϕ, we see that ∂̄u = f holds in the
sense of distributions on Ωm, hence classically on Ωm. Since m was arbitrary, ∂̄u = f on
all of G. Finally, by the weak lower semicontinuity of the L2 norm (see, e.g., [25, Ch. 3]),
we have ∫

G
|u|2e−tρ dVω ≤ lim inf

j→∞

∫
Ω̃j

|uj |2e−tρ dVω .

Using (2) and taking j → ∞ (so that Rj+1 → ∞), we deduce∫
G
|u|2e−tρ dVω ≤ 2

λ∗ q
e t (Rj+1+1/4)

∫
G
|f |2e−tρ dVω ,

for arbitrarily large Rj+1. This proves an a priori bound of the form (1) for u (with a
constant C(t) = 2

λ∗q depending on t and q). In fact, by optimizing the uniform estimate
(2) over q (see Remark 5.1 in [19]), one can remove the explicit 1/q dependence and
arrange that C(t) = exp(t/2 + ε∗S∗) as in Theorem 1. This completes the proof of global
solvability.

6. Examples

We illustrate Theorem 1 with several classes of complex Lie groups. These examples
also highlight that the vanishing of weighted L2 Dolbeault cohomology is a strictly broader
phenomenon than the vanishing of ordinary Dolbeault cohomology.

Example 1. Take ρ(z) = |z|2 (the squared Euclidean norm) on G = Cn, with the standard
Euclidean metric. Then one can take φj = ρ for all j, so that Ωj = {|z| < Rj} is a
Euclidean ball exhausting Cn. Moreover, one may take σj compactly supported near ∂Ωj

(or simply use a fixed quadratic potential transplanted to each Ωj). All the uniformity
conditions are trivially satisfied, and Theorem 1 recovers Hp,q

∂̄,(2),t
(Cn) = 0 for q ≥ 1, with

explicit constants.

Example 2 (Solvable Borel subgroup of SL(2,C)). Let G be the Borel subgroup of
SL(2,C), consisting of all complex 2 × 2 upper-triangular matrices with determinant 1.
Every element can be written as(

a b
0 a−1

)
, a ∈ C∗, b ∈ C .



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 10 of 14

Thus G is biholomorphic to the affine variety C∗ × C and is a noncompact complex Lie
group of complex dimension 2. Being an affine algebraic group, G is Stein and admits a
continuous plurisubharmonic exhaustion. For instance, we may take

ρ(a, b) = |a|2 + |a|−2 + |b|2 ,

which is a proper continuous psh exhaustion of G (indeed, i∂∂̄(|a|2 + |a|−2) ≥ 0 on C∗ and
|b|2 is psh on C). Applying Lemma 2, we obtain an exhaustion {Ωj} of G by smoothly
bounded strictly pseudoconvex domains and smooth functions φj with supΩj+1

|φj − ρ| ≤
1/4. Next, to construct the uniform reference functions, we choose a left-invariant Her-
mitian metric on G; for example, one can take the product metric induced by coordinates
(log a, b) ∈ C × C. In the coordinate chart B around the identity element (a = 1, b = 0),
one can take u(x, y) = |x|2 + |y|2 as in the proof of Lemma 3. Then i∂∂̄u ≥ λω on B for
some λ > 0. By left-translating u and covering each Ωj by at most C such translated balls
(here C can be taken, for instance, as 54 = 625 in real dimension 4), we obtain σj such
that i∂∂̄σj ≥ (λ/C)ω on Ωj. Thus conditions (U1) and (U2) are satisfied with c∗ = λ/C
and S∗ = 1. In particular, ε∗ = 1/c∗ = C/λ. Plugging these into the estimate (1), we get
an explicit constant

C(t) = exp
(
t/2 + ε∗S∗

)
= exp

(
t/2 +

C

λ

)
in the global L2 estimate for ∂̄. For instance, choosing the metric normalization such that
λ = 1 and C = 625, one finds C(t) = exp(t/2 + 625) as a valid constant in Theorem 1 for
the Borel group.

Example 3. Let G = (C∗)n with the product metric. As an explicit example of a contin-
uous psh exhaustion, take

ρ(z) =
n∑
k=1

(
| log |zk||2 + |zk|−2 + |zk|2

)
.

Then ρ is proper and continuous psh on G, and Theorem 1 applies to give global weighted
L2 ∂̄-solvability in all positive anti-holomorphic degrees. Note that although the ordi-
nary Dolbeault cohomology of (C∗)n is nontrivial (indeed H0,1(G) ∼= H1(G,O) ∼= Cn), its
weighted L2 Dolbeault cohomology in degrees q ≥ 1 vanishes by our result.

Example 4 (A Cousin group). Consider the complex abelian Lie group

G = C2/Γ ,

where Γ = ⟨(1, 0), (0, 1), (i, i
√

2)⟩Z is the rank-3 lattice in C2 generated by (1, 0), (0, 1),
and (i, i

√
2). This G is a noncompact complex two-dimensional Lie group which is not

Stein. (Indeed, G contains a one-dimensional complex torus C/⟨1⟩ as a closed complex
subgroup, so G fails Matsushima’s Stein criterion.) However, by a theorem of Kazama



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 11 of 14

[10], G is pseudoconvex, i.e. admits a continuous psh exhaustion. For example, one
convenient choice is

ρ([z1, z2]) := (ℑz2 −
√

2ℑz1)2 ,

which is well-defined on the quotient G because ℑz2−
√

2ℑz1 is invariant under the period
lattice Γ. Note that ρ is unbounded on G and {ρ < c} is relatively compact for each c,
so ρ is a continuous exhaustion of G. Moreover, ρ is plurisubharmonic: it is the square
of the imaginary part of the holomorphic 1-form dz2 −

√
2 dz1 on C2, hence ρ is psh

(in fact pluriharmonic) on C2, and it descends to a continuous psh function on G. The
sublevel sets Ωc := {ρ < c} are therefore smoothly bounded pseudoconvex domains in G.
Applying Lemma 2, we approximate ρ uniformly on these sublevels by smooth strictly psh
functions φj, and then Lemma 3 provides strictly psh reference functions σj with uniform
Levi bound. All assumptions (U1) and (U2) are thus satisfied. By Theorem 1, we conclude
that Hp,q

∂̄,(2),t
(G) = 0 for all q ≥ 1 and all t ≥ 0, and obtain a global ∂̄-solution operator

on L2(G, e−tρ). In particular, this shows that even non-Stein complex Lie groups (such as
the above Cousin group) enjoy a robust ∂̄-solvability in the L2 sense.

7. Concluding remarks

We have exhibited a robust global L2 ∂̄-theory on noncompact pseudoconvex complex
Lie groups, with explicit estimates and analytic consequences. In a forthcoming paper, we
also extend the main theorem to (p, q)-forms with values in holomorphic vector bundles
(the bundle curvature contributes an additional Nakano-nonnegative term to the weight).
On the other hand, identifying geometric conditions beyond Lie groups that guarantee the
key uniformities (U1)–(U2) is a natural direction for future work.

Appendix A. Regularized maxima of strictly psh functions

We outline the construction of the auxiliary functions σj used in Lemma 3. Let
u1, . . . , uN be strictly plurisubharmonic functions on a complex manifold (say, on an open
set in Cn) such that i∂∂̄ui ≥ λi ω for some λi > 0. Consider the regularized maximum

Mτ (u1, . . . , uN )(z) :=
1

τ
log

(
eτu1(z) + · · · + eτuN (z)

)
,

depending on a large parameter τ > 0. One checks that Mτ (u1, . . . , uN ) is a smooth
plurisubharmonic function which decreases pointwise to max{u1, . . . , uN} as τ → ∞. A
direct computation of the Levi form (see, e.g., [15, p. 262]) shows that

i∂∂̄Mτ (u1, . . . , uN ) =

N∑
i=1

αi(z) i∂∂̄ui(z) +

N∑
i=1

αi(z)βi(z) ,

where

αi(z) =
eτui(z)∑N
k=1 e

τuk(z)
≥ 0,

N∑
i=1

αi(z) = 1 ,



A. R. Al-Abdallah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6953 12 of 14

and βi(z) is a positive semidefinite (1, 1)-form. In particular, if each i∂∂̄ui ≥ λω, then
i∂∂̄Mτ (u1, . . . , uN ) ≥ λω for every τ > 0. Now suppose each ui vanishes outside some
relatively compact domain and that at most C of the functions ui are simultaneously
nonzero at any given point. In that case, one can choose a finite τ large enough that at
every point z, the weights αi(z) are concentrated on at most C terms. It follows that
Mτ (u1, . . . , uN ) is strictly plurisubharmonic with

i∂∂̄Mτ (u1, . . . , uN ) ≥ (min
i
λi/C)ω .

Finally, by mollifying Mτ (u1, . . . , uN ) if necessary, we can obtain a C∞ function σ satisfy-
ing σ ≥ max(u1, . . . , uN ) and i∂∂̄σ ≥ (mini λi/C)ω. This σ serves as the desired reference
function.

Appendix B. Bounded-overlap coverings in left-invariant metrics

Any left-invariant Riemannian metric (or Hermitian metric) on a complex Lie group G
induces a distance function d onG that is homogeneous in the sense that d(gx, gy) = d(x, y)
for all g, x, y ∈ G. Moreover, (G, d) has finite Besicovitch constant, depending only on
the real dimension of G. This is a consequence of the Besicovitch covering theorem in
Euclidean space (see, e.g., [11–13]), since any metric ball in (G, d) is isometric (via left-
translation and the exponential map at the identity) to a Euclidean ball in R2n.

Concretely, the Besicovitch property implies that there exists an integer N = N(dimG)
with the following property: Given any family of metric balls in (G, d) covering a set
E ⊂ G, one can extract a countable subcollection of these balls which still covers E and
such that no point of G is contained in more than N balls from the subcollection. In
particular, by applying this to a cover of Ωj by sufficiently small balls, one obtains a finite
subcover of Ωj with the bounded overlap property used in Lemma 3. One classical proof
of the Besicovitch covering lemma shows that one can take N = 5m in Rm (see [11]),
so for dimRG = 2n one may take C = N(2n) ≤ 52n in Lemma 3. Thus, for instance,
C = 54 = 625 is valid for any left-invariant metric on a real 4-dimensional Lie group (such
as the Borel group in Example 6.2).

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